Physics Book - User contributions [en]

Standing waves

2015-12-06T01:15:13Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The vibration associated with resonance in objects like strings and air columns have characteristic patterns called '''Standing waves'''. These standing waves arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of this constructive interference is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. Standing waves occur in transverse waves, but also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for air columns.

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

== See also ==

[[Mechanical waves]]

[[Transverse and longitudinal waves]]

[[interference]]

[[Gravitational waves]]

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

==References==

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

"Standing Waves." Standing Waves. N.p., n.d. Web. 05 Dec. 2015.

"Fundamental and Harmonics." Fundamental and Harmonic Resonances. N.p., n.d. Web. 05 Dec. 2015.

[[Category: Waves]]

Standing waves

2015-12-06T01:13:52Z

Anvikshabusa:

Standing waves

2015-12-06T01:03:24Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The vibration associated with resonance in objects like strings and air columns have characteristic patterns called '''Standing waves'''. These standing waves arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of this constructive interference is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. Standing waves occur in transverse waves, but also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for air columns.

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

==References==

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

"Standing Waves." Standing Waves. N.p., n.d. Web. 05 Dec. 2015.

"Fundamental and Harmonics." Fundamental and Harmonic Resonances. N.p., n.d. Web. 05 Dec. 2015.

[[Category: Waves]]

Standing waves

2015-12-06T01:00:43Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

==References==

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

"Standing Waves." Standing Waves. N.p., n.d. Web. 05 Dec. 2015.

"Fundamental and Harmonics." Fundamental and Harmonic Resonances. N.p., n.d. Web. 05 Dec. 2015.

[[Category: Waves]]

Standing waves

2015-12-06T00:59:25Z

Anvikshabusa: /* External links */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

==References==

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

"Standing Waves." Standing Waves. N.p., n.d. Web. 05 Dec. 2015.

"Fundamental and Harmonics." Fundamental and Harmonic Resonances. N.p., n.d. Web. 05 Dec. 2015.

[[Category: Waves]]

Standing waves

2015-12-06T00:59:14Z

Anvikshabusa: /* References */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

==References==

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

"Standing Waves." Standing Waves. N.p., n.d. Web. 05 Dec. 2015.

"Fundamental and Harmonics." Fundamental and Harmonic Resonances. N.p., n.d. Web. 05 Dec. 2015.

[[Category: Waves]]

Standing waves

2015-12-06T00:58:55Z

Anvikshabusa: /* External links */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

==References==

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:58:36Z

Anvikshabusa: /* References */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

==References==

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:56:29Z

Anvikshabusa: /* References */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf

https://en.wikipedia.org/wiki/Standing_wave

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:56:05Z

Anvikshabusa: /* External links */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Characteristics of Stationary Waves." Characteristics of Stationary Waves. N.p., n.d. Web. 05 Dec. 2015.

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:54:10Z

Anvikshabusa: /* Further reading */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

6, 1. Of, Stanford Physics Dept., Standing Waves On A String, and P28 Lab 6. Standing Waves on a String (n.d.): n. pag. Web.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:51:21Z

Anvikshabusa: /* See also */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:50:21Z

Anvikshabusa: /* External links */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

===External links===

"What Are Some Examples of Stationary Waves in Real Life?" - Quora. N.p., n.d. Web. 05 Dec. 2015.

"Harmonics and Patterns." Harmonics and Patterns. N.p., n.d. Web. 05 Dec. 2015.

"Standing Wave Patterns." Standing Wave Patterns. N.p., n.d. Web. 05 Dec. 2015.

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:20:35Z

Anvikshabusa: /* Further reading */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Tsokos, K. A. Physics for the IB Diploma. Cambridge: Cambridge UP, 2010. Print.

Georgi, Howard. The Physics of Waves. Englewood Cliffs, NJ: Prentice Hall, 1993. Print.

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:13:57Z

Anvikshabusa: /* Standing Wave Patterns */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:13:23Z

Anvikshabusa: /* A Computational Model */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

Click on the following link to understand standing waves better:

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:12:04Z

Anvikshabusa: /* A Computational Model */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:11:20Z

Anvikshabusa: /* A Computational Model */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

[https://trinket.io/embed/glowscript/759832b92d Demo program on Standing Waves]

<iframe src="https://trinket.io/embed/glowscript/759832b92d" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:10:46Z

Anvikshabusa: /* A Computational Model */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

[http://www.example.com link title]

<iframe src="https://trinket.io/embed/glowscript/759832b92d" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:07:23Z

Anvikshabusa: /* A Computational Model */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

[<iframe src="https://trinket.io/embed/glowscript/759832b92d?outputOnly=true" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>]

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:05:48Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

<iframe src="https://trinket.io/embed/glowscript/759832b92d" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-06T00:02:48Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

lamb = 2
scene.width = 1000
scene.height = 600
scene.background = color.white
scene.range = 3*lamb
scene.center = vector(5*lamb,0,0)
scene.fov = 0.01
scene.userspin = False
scene.userzoom = False

omega = pi
k = 2*pi/lamb
v = omega/k
dx = 0.02
r = 0.05
A = 2
curve(pos=[vector(0,0,0), vector(10*lamb,0,0)], color=color.gray(0.9), radius=0.7*r)
left = curve(color=vector(1,0.95,0.95), radius=r)
right = curve(color=vector(0.9,1,1), radius=r)
wave = curve(color=color.blue, radius=r)
N = 0
for x in arange(0,12*lamb+dx/2,dx):
wave.append(pos=vector(x,0,0))
left.append(pos=vector(x,A*sin(k*x),r))
right.append(pos=vector(x,A*sin(k*x),r))
N += 1

xleft = 0
xright = 0

##################################################################################
# The following code has dependencies on wxPython (classic) or jquery (GlowScript)

def reset(evt):
global xleft, xright
xleft = -12*lamb
xright = 10*lamb
for i in range(N):
wave.modify(i, y=0)

def B_Showbutton(evt):
left.visible = not left.visible
right.visible = not right.visible
if left.visible:
Showbutton.text("Hide traveling waves")
else:
Showbutton.text("Show traveling waves")

Resetbutton = $('<button/>').text("Restart").appendTo(scene.caption).click(reset)

scene.caption.append(" ")

Showbutton = $('<button/>').text("Hide traveling waves").appendTo(scene.caption).click(B_Showbutton)

scene.caption.append(" A standing wave can result from the interference of two traveling waves.")

######################################################################################################

reset(None)

while True:
left.origin.x = xleft
right.origin.x = xright
xleft += dx
xright -= dx
if xleft > 0:
xleft -= lamb
if xright < -lamb:
xright += lamb
i = 0
for x in arange(0,10*lamb+3*dx/2,dx):
yleft = yright = 0
if x < xleft+12*lamb:
yleft = left.pos[int(round((x-xleft)/dx))].y
if x > xright:
yright = right.pos[int(round((x-xright)/dx))].y
wave.modify(i, y=yleft+yright)
i += 1
rate(100)

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:57:50Z

Anvikshabusa: /* Mathematical model: Standing waves at different harmonics */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... (the nodes) the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:56:18Z

Anvikshabusa: /* Mathematical model: Standing waves at different harmonics */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the ''amplitude'' of the wave,
*''ω'' (called ''angular frequency'', measured in ''radians per second'') is ''2π'' times the frequency (in ''hertz''),
*''k'' (called the ''wave number'' and measured in ''radians per metre'') is ''2π'' divided by the wavelength ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the [[Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity]] for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... called the [[node (physics)|node]]s the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the [[anti-node]]s, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:54:38Z

Anvikshabusa: /* 2 fixed ends */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves at different harmonics==

===2 fixed ends===

===Mathematical description===

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become nodes. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the '''Fundamental frequency'''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called '''Harmonic''' or overtone.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y<sub>0</sub>'' is the [[amplitude]] of the wave,
*''ω'' (called [[angular frequency]], measured in ''[[radian]]s per second'') is ''2π'' times the [[frequency]] (in ''[[hertz]]''),
*''k'' (called the [[wave number]] and measured in ''radians per metre'') is ''2π'' divided by the [[wavelength]] ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y<sub>1</sub>'' and ''y<sub>2</sub>'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the [[Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity]] for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... called the [[node (physics)|node]]s the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the [[anti-node]]s, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:45:25Z

Anvikshabusa: /* Mathematical model: Standing waves with different ends */

Standing waves

2015-12-05T23:43:02Z

Anvikshabusa: /* Nodes and Antinodes */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave .
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:42:18Z

Anvikshabusa: /* Nodes and Antinodes */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.

2. There are either a half-number or a whole number of waves within the pattern established on the string.

3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.

4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:41:52Z

Anvikshabusa: /* Nodes and Antinodes */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

There are a variety of patterns by strings could naturally vibrate. Each pattern is associated with one its natural frequencies. Each standing wave pattern is referred to as a harmonic. There are a variety of other low energy vibrational patterns that could be established by some basic traits:

1. There is an alternating pattern of nodes and antinodes.
2. There are either a half-number or a whole number of waves within the pattern established on the string.
3. Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
4. One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:36:55Z

Anvikshabusa: /* Standing Wave Patterns */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|500px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:36:31Z

Anvikshabusa: /* Standing Wave Patterns */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|300px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:35:53Z

Anvikshabusa: /* Standing Wave Patterns */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====[[File:h1.png|200px|thumb|right|First Harmonic.]]

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====[[File:h2.png|200px|thumb|right|Second Harmonic.]]

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====[[File:h3.png|200px|thumb|right|Third Harmonic.]]

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====[[File:h44.png|300px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:34:51Z

Anvikshabusa: /* Standing Wave Patterns */

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|200px|thumb|right|First Harmonic.]]
The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|200px|thumb|right|Second Harmonic.]]
Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|200px|thumb|right|Third Harmonic.]]
If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====
[[File:h44.png|300px|thumb|right|The features of the standing wave patterns for ''n'' harmonics.]]
As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

File:H44.png

2015-12-05T23:33:34Z

Anvikshabusa:

Standing waves

2015-12-05T23:32:17Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====
[[File:h1.png|400px|thumb|right|First Harmonic.]]
The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====
[[File:h2.png|400px|thumb|right|Second Harmonic.]]
Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====
[[File:h3.png|400px|thumb|right|Third Harmonic.]]
If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

File:H3.png

2015-12-05T23:29:55Z

Anvikshabusa:

File:H1.png

2015-12-05T23:29:39Z

Anvikshabusa:

File:H2.png

2015-12-05T23:29:25Z

Anvikshabusa:

Standing waves

2015-12-05T23:24:57Z

Anvikshabusa:

Claimed by Anviksha Busa

[[File:Resultant_Standing_Wave.png|400px|thumb|right|Standing Wave as a result of wave interference.]]

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:23:59Z

Anvikshabusa:

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

[[File:Resultant_Standing_Wave.png|400px|thumb|right|alt text]]

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:20:10Z

Anvikshabusa: /* Nodes and Antinodes */

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

pic 1

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

pic 2

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

===Standing Wave Patterns===

====First Harmonic====

The simplest standing wave pattern has points of no displacement (nodes) at the two ends and one point of maximum displacement (antinode) in the middle. This is the first harmonic. It is the simplest wave pattern produced and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

====Second Harmonic====

Other wave patterns can be observed when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends and an additional node in the exact center. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic.

====Third Harmonic====

If the frequency is increased even more, then the third harmonic wave pattern can be produced. The standing wave pattern for the third harmonic has an additional node and antinode between its ends.

====Other Harmonics====

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. Each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:12:16Z

Anvikshabusa: /* Application */

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

pic 1

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

pic 2

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:12:00Z

Anvikshabusa: /* Application */

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

pic 1

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

pic 2

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:11:43Z

Anvikshabusa: /* Application */

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

pic 1

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

pic 2

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.

-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)

-Waves on the beach

-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Standing waves

2015-12-05T23:10:59Z

Anvikshabusa: /* Connectedness */

Claimed by Anviksha Busa

The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

pic 1

The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. The illustration above involves the transverse waves on a string, but standing waves also occur with the longitudinal waves in an air column. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns.

pic 2

==Characteristics of a Standing Wave==

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Nodes and Antinodes==
Standing waves on a string are a result of traveling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference, and the antinodes (places of maximum amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. When the conditions of the tension in the string, the linear density and the frequency of oscillations are just right, standing waves appear.

<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>

The pattern above is not the only pattern of vibration for a guitar string. There are a variety of patterns by which the guitar string could naturally vibrate. Each pattern is associated with one of the natural frequencies of the guitar strings. Three other patterns are shown in the diagrams at the right. Each standing wave pattern is referred to as a harmonic of the instrument (in this case, the guitar string). The three diagrams at the right represent the standing wave patterns for the first, second, and third harmonics of a guitar string. (Harmonics will be discussed in more detail in the next section of this lesson.)

There are a variety of other low energy vibrational patterns that could be established in the string. For guitar strings, each pattern is characterized by some basic traits:

There is an alternating pattern of nodes and antinodes.
There are either a half-number or a whole number of waves within the pattern established on the string.
Nodal positions (points of no displacement) are established at the ends of the string where the string is clamped down in a fixed position.
One pattern is related to the next pattern by the addition (or subtraction) of one or more nodes (and antinodes).

==Mathematical model: Standing waves with different ends==

Be sure to show all steps in your solution and include diagrams whenever possible

===2 fixed ends===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

A wave has both a frequency and a wavelength that are related by the following equation:
<math>{λf = v}</math>

Where λ is the wavelength, f the frequency, and v the velocity of the wave on the string. The wavelengths of the standing waves are fixed by the length of the string. As both ends are fixed, there must be a node at each end of the string. As the standing waves on the string are sinusoidal, the allowed number of waves on the string will be an integral number of half wavelengths, or:

nλ/2 = L

<math>{\frac{nλ}{2}} = L}</math>

where n is a positive integer, λ the wavelength, and L the length of the string.
===1 fixed end===

==Application==

Standing waves are used and seen in multiple real life examples. It is a major contributor in quantum mechanics. It can also be found in the following:

-In string instruments, a plucked or bowed string makes the note is does because only certain frequencies of stationary (or "standing") waves are able to form on that string under those conditions (e.g., a finger holding down the string at a certain position). Any vibrations that aren't at the right frequencies to make standing waves quickly cancel themselves out, and so it's the standing wave frequencies that we actually hear.
-Similarly, for woodwind instruments, we get the notes we get because of what standing waves are able to form within the tube of air inside the instrument. (Strings and woodwinds sound different from one another because they allow different combinations of overtones (higher-frequency standing waves) to form.)
-waves on the beach
-When you tune in a radio station, you are making the antenna circuit resonate with a particular carrier wave. The radio wave induces a standing wave of alternating current in the antenna circuit.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

http://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf
https://en.wikipedia.org/wiki/Standing_wave
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html#c4
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c3

[[Category: Waves]]

Main Page

2015-12-05T22:48:24Z

Anvikshabusa: /* Waves */

__NOTOC__
Welcome to the Georgia Tech Wiki for Intro Physics. This resources was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it!

Looking to make a contribution?
#Pick a specific topic from intro physics
#Add that topic, as a link to a new page, under the appropriate category listed below by editing this page.
#Copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]

== Organizing Categories ==
These are the broad, overarching categories, that we cover in two semester of introductory physics. You can add subcategories or make a new category as needed. A single topic should direct readers to a page in one of these catagories.

<div class="toccolours mw-collapsible mw-collapsed">
===Interactions===
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
**[[Ball and Spring Model of Matter]]
*[[Detecting Interactions]]
*[[Escape Velocity]]
*[[Fundamental Interactions]]
*[[Determinism]]
*[[System & Surroundings]]
*[[Free Body Diagram]]
*[[Newton's First Law of Motion]]
*[[Newton's Second Law of Motion]]
*[[Newton's Third Law of Motion]]
*[[Gravitational Force]]
*[[Electric Force]]
*[[Conservation of Energy]]
*[[Conservation of Charge]]
*[[Terminal Speed]]
*[[Simple Harmonic Motion]]
*[[Speed and Velocity]]
*[[Electric Polarization]]
*[[Perpetual Freefall (Orbit)]]
*[[2-Dimensional Motion]]
*[[Center of Mass]]
*[[Reaction Time]]
*[[Time Dilation]]
*[[Pauli exclusion principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Modeling with VPython===
<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython 3D Objects]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Theory===
<div class="mw-collapsible-content">
*[[Einstein's Theory of Special Relativity]]
*[[Einstein's Theory of General Relativity]]
*[[Quantum Theory]]
*[[Maxwell's Electromagnetic Theory]]
*[[Atomic Theory]]
*[[String Theory]]
*[[Elementary Particles and Particle Physics Theory]]
*[[Law of Gravitation]]
*[[Newton's Laws]]
*[[Higgs field]]
*[[Supersymmetry]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Notable Scientists===
<div class="mw-collapsible-content">
*[[Alexei Alexeyevich Abrikosov]]
*[[Christian Doppler]]
*[[Albert Einstein]]
*[[Ernest Rutherford]]
*[[Joseph Henry]]
*[[Michael Faraday]]
*[[J.J. Thomson]]
*[[James Maxwell]]
*[[Robert Hooke]]
*[[Carl Friedrich Gauss]]
*[[Nikola Tesla]]
*[[Andre Marie Ampere]]
*[[Sir Isaac Newton]]
*[[J. Robert Oppenheimer]]
*[[Oliver Heaviside]]
*[[Rosalind Franklin]]
*[[Enrico Fermi]]
*[[Robert J. Van de Graaff]]
*[[Charles de Coulomb]]
*[[Hans Christian Ørsted]]
*[[Philo Farnsworth]]
*[[Niels Bohr]]
*[[Georg Ohm]]
*[[Galileo Galilei]]
*[[Gustav Kirchhoff]]
*[[Max Planck]]
*[[Heinrich Hertz]]
*[[Edwin Hall]]
*[[James Watt]]
*[[Count Alessandro Volta]]
*[[Josiah Willard Gibbs]]
*[[Richard Phillips Feynman]]
*[[Sir David Brewster]]
*[[Daniel Bernoulli]]
*[[William Thomson]]
*[[Leonhard Euler]]
*[[Robert Fox Bacher]]
*[[Stephen Hawking]]
*[[Amedeo Avogadro]]
*[[Wilhelm Conrad Roentgen]]
*[[Pierre Laplace]]
*[[Thomas Edison]]
*[[Hendrik Lorentz]]
*[[Jean-Baptiste Biot]]
*[[Lise Meitner]]
*[[Lisa Randall]]
*[[Felix Savart]]
*[[Heinrich Lenz]]
*[[Max Born]]
*[[Archimedes]]
*[[Jean Baptiste Biot]]
*[[Carl Sagan]]
*[[Eugene Wigner]]
*[[Marie Curie]]
*[[Pierre Curie]]
*[[Werner Heisenberg]]
*[[Johannes Diderik van der Waals]]
*[[Louis de Broglie]]
*[[Aristotle]]
*[[Émilie du Châtelet]]
*[[Blaise Pascal]]
*[[Siméon Denis Poisson]]
*[[Benjamin Franklin]]
*[[James Chadwick]]
*[[Henry Cavendish]]
*[[Thomas Young]]
*[[James Prescott Joule]]
*[[John Bardeen]]
*[[Leo Baekeland]]
*[[Alhazen]]
*[[Willebrord Snell]]
*[[Fritz Walther Meissner]]
*[[Johannes Kepler]]
*[[Johann Wilhelm Ritter]]
*[[Philipp Lenard]]
*[[Robert A. Millikan]]
*[[Joseph Louis Gay-Lussac]]
*[[Guglielmo Marconi]]
*[[William Lawrence Bragg]]
*[[Robert Goddard]]
*[[Léon Foucault]]
*[[Henri Poincaré]]
*[[Steven Weinberg]]
*[[Arthur Compton]]
*[[Pythagoras of Samos]]
*[[Subrahmanyan Chandrasekhar]]
*[[Wilhelm Eduard Weber]]
*[[Edmond Becquerel]]
*[[Joseph Rotblat]]
*[[Carl David Anderson]]
*[[Hermann von Helmholtz]]
*[[Nicolas Leonard Sadi Carnot]]
*[[Wallace Carothers]]
*[[David J. Wineland]]
*[[Rudolf Clausius]]
*[[Edward L. Norton]]
*[[Shuji Nakamura]]
*[[Pierre Laplace Pt. 2]]
*[[William B. Shockley]]
*[[Osborne Reynolds]]
*[[Alexander Graham Bell]]
*[[Hans Bethe]]
*[[Erwin Schrodinger]]
*[[Wolfgang Pauli]]
*[[Paul Dirac]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Properties of Matter===
<div class="mw-collapsible-content">
*[[Mass]]
*[[Velocity]]
*[[Relative Velocity]]
*[[Density]]
*[[Charge]]
*[[Spin]]
*[[SI Units]]
*[[Heat Capacity]]
*[[Specific Heat]]
*[[Wavelength]]
*[[Conductivity]]
*[[Malleability]]
*[[Weight]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Inertia]]
*[[Non-Newtonian Fluids]]
*[[Ferrofluids]]
*[[Color]]
*[[Temperature]]
*[[Plasma]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Contact Interactions===
<div class="mw-collapsible-content">
* [[Young's Modulus]]
* [[Friction]]
* [[Tension]]
* [[Hooke's Law]]
*[[Centripetal Force and Curving Motion]]
*[[Compression or Normal Force]]
* [[Length and Stiffness of an Interatomic Bond]]
* [[Speed of Sound in Solids]]
* [[Iterative Prediction of Spring-Mass System]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Momentum===
<div class="mw-collapsible-content">
* [[Vectors]]
* [[Kinematics]]
* [[Conservation of Momentum]]
* [[Predicting Change in multiple dimensions]]
* [[Derivation of the Momentum Principle]]
* [[Momentum Principle]]
* [[Impulse Momentum]]
* [[Curving Motion]]
* [[Projectile Motion]]
* [[Multi-particle Analysis of Momentum]]
* [[Iterative Prediction]]
* [[Analytical Prediction]]
* [[Newton's Laws and Linear Momentum]]
* [[Net Force]]
* [[Center of Mass]]
* [[Momentum at High Speeds]]
* [[Change in Momentum in Time for Curving Motion]]
* [[Momentum with respect to external Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Angular Momentum===
<div class="mw-collapsible-content">
* [[The Moments of Inertia]]
* [[Moment of Inertia for a cylinder]]
* [[Rotation]]
* [[Torque]]
* [[Systems with Zero Torque]]
* [[Systems with Nonzero Torque]]
* [[Torque vs Work]]
* [[Angular Impulse]]
* [[Right Hand Rule]]
* [[Angular Velocity]]
* [[Predicting the Position of a Rotating System]]
* [[Translational Angular Momentum]]
* [[The Angular Momentum Principle]]
* [[Angular Momentum of Multiparticle Systems]]
* [[Rotational Angular Momentum]]
* [[Total Angular Momentum]]
* [[Gyroscopes]]
* [[Angular Momentum Compared to Linear Momentum]]
*[[Torque 2]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Energy===
<div class="mw-collapsible-content">
*[[The Photoelectric Effect]]
*[[Photons]]
*[[The Energy Principle]]
*[[Predicting Change]]
*[[Rest Mass Energy]]
*[[Kinetic Energy]]
*[[Potential Energy]]
**[[Potential Energy for a Magnetic Dipole]]
**[[Potential Energy of a Multiparticle System]]
*[[Work]]
**[[Work Done By A Nonconstant Force]]
*[[Work and Energy for an Extended System]]
*[[Thermal Energy]]
*[[Conservation of Energy]]
*[[Electric Potential]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Gravitational Potential Energy]]
*[[Point Particle Systems]]
*[[Real Systems]]
*[[Spring Potential Energy]]
**[[Ball and Spring Model]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
*[[Translational, Rotational and Vibrational Energy]]
*[[Franck-Hertz Experiment]]
*[[Power (Mechanical)]]
*[[Transformation of Energy]]

*[[Energy Graphs]]
**[[Energy graphs and the Bohr model]]
*[[Air Resistance]]
*[[Electronic Energy Levels]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Specific Heat Capacity]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Electronic Energy Levels and Photons]]
*[[Energy Density]]
*[[Bohr Model]]
*[[Quantized energy levels]]
**[[Spontaneous Photon Emission]]
*[[Path Independence of Electric Potential]]
*[[Energy in a Circuit]]
*[[The Photovoltaic Effect]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Collisions===
<div class="mw-collapsible-content">
[[File:opener.png]]

*[[Collisions]]
Collisions are events that happen very frequently in our day-to-day world. In the realm of Physics, a collision is defined as any sort of process in which before and after a short time interval there is little interaction, but during that short time interval there are large interactions. When looking at collisions, it is first important to understand two very important principles: the Momentum Principle and the Energy Principle. Both principles serve use when talking of collisions because they provide a way in which to analyze these collisions. Collisions themselves can be categorized into 3 main different types: elastic collisions, inelastic collisions, maximally inelastic collisions. All 3 collisions will get touched on in more detail further on.
[[File:pe.png]]

*[[Elastic Collisions]]
A collision is deemed "elastic" when the internal energy of the objects in the system does not change (in other words, change in internal energy equals 0). Because in an elastic collision no kinetic energy is converted over to internal energy, in any elastic collision Kfinal always equals Kinitial.
[[File:Elco.png]]

*[[Inelastic Collisions]]
A collision is said to be "inelastic" when it is not elastic; therefore, an inelastic collision is an interaction in which some change in internal energy occurs between the colliding objects (in other words, change in internal energy does not equal 0). Examples of such changes that occur between colliding objects include, but are not limited to, things like they get hot, or they vibrate/rotate, or they deform. Because some of the kinetic energy is converted to internal energy during an inelastic collision, Kfinal does not equal Kinitial.
There are a few characteristics that one can search for when identifying inelasticity. These indications include things such as:
*Objects stick together after the collision
*An object is in an excited state after the collision
*An object becomes deformed after the collision
*The objects become hotter after the collision
*There exists more vibration or rotation after the collision
[[File:inve.gif]]

*[[Maximally Inelastic Collision]]
Maximally inelastic collisions, also known as "sticking collisions", are the most extreme kinds of inelastic collisions. Just as its secondary name implies, a maximally inelastic collision is one in which the colliding objects stick together creating maximum dissipation. This does not automatically mean that the colliding objects stop dead because the law of conservation of momentum. In a maximally inelastic collision, the remaining kinetic energy is present only because total momentum can't change and must be conserved.
[[File:inel.gif]]

*[[Head-on Collision of Equal Masses]]
The easiest way to understand this phenomenon is to look at it through an example. In this case, we can analyze it through the common game of billiards. Taking the two, equally massed billiard balls as the system, we can neglect the small frictional force exerted on the balls by the billiard table. The Momentum Principle states that in this head-on collision of billiard balls the total final momentum in the x direction must equal the total initial momentum. However, this alone does not give us the knowledge to know how the momentum will be divided up between the two balls. Considering the law of conservation of energy, we can more accurately depict what will happen. This will also allow for one to identify what kind of collision occurs (elastic, inelastic, or maximally inelastic). It is important to know that head-on collisions of equal masses do not have a definite type of collision associated with it.
[[File:momentum-real-life-applications-2895.jpg]] [[File:8ball.gif]]

*[[Head-on Collision of Unequal Masses]]
Just as with head-on collisions of equal masses, it is easy to understand head-on collisions of unequal masses by viewing it through an example. Let's take for example two balls of unequal masses like a ping-pong ball and a bowling ball. For the purpose of this example (so as to allow for no friction and no other significant external forces), let's imagine these objects collide in outer space inside an orbiting spacecraft. If there were to be a collision between the two, what would one expect to happen? One could expect to see the ping-pong ball collide with the bowling ball and bounce straight back with a very small change of speed. What one might not expect as much is that the bowling ball also moves, just very slowly. Again, this can all be explained through the conservation of momentum and the conservation of energy.
[[File:mi3e.jpg]]

*[[Frame of Reference]]
In the world of Physics, a frame of reference is the perspective from which a system is observed. It can be stationary or sometimes it can even be moving at a constant velocity. In some rare cases, the frame of reference moves at an nonconstant velocity and is deemed "noninertial" meaning the basic laws of physics do not apply. Continuing with the trend of examples, pretend you are at a train station observing trains as they pass by. From your stationary frame of reference, you observe that the passenger on the train is moving at the same velocity as the train. However, from a moving frame of reference, say from the eyes of the train conductor, he would view the train passengers as "anchored" to the train.
[[File:train.png]]

*[[Scattering: Collisions in 2D and 3D]]
Experiments that involve scattering are often used to study the structure and behavior of atoms, nuclei, as well as of other small particles. In an experiment like such, a beam of particles collides with other particles. If it is an atomic or nuclear collision, we are unable to observe the curving trajectories inside the tiny region of interaction. Instead, we can only truly observe the trajectories before and after the collision. This is only possible because the particles are at a farther distance apart and have a very weak mutual interaction; this essentially means that the particles are moving almost in a straight line. A good example which demonstrates scattering is the collision between an alpha particle (the nucleus of a helium atom) and the nucleus of a gold atom. One will understand this phenomenon more in depth after first understanding the Rutherford Experiment which will get touched on later.

*[[Rutherford Experiment and Atomic Collisions]]
In England in 1911, a famous experiment was performed by a group of scientists led by Mr. Ernest Rutherford. This experiment, later known as "The Rutherford Experiment", was a tremendous breakthrough for its time because it led to the discovery of the nucleus inside the atom. Rutherford's experiment involved the scattering of a high-speed alpha particle (now known as a helium nuclei - 2 protons and 2 neutrons) as it was shot at a thin gold foil (consisting of a nuclei with 79 protons and 118 neutrons). In the experiment, Rutherford and his team discovered that the velocity of the alpha particles was not high enough to allow the particles to make actual contact with the gold nucleus. Although they never actually made contact, it is still deemed a collision because there exists a sizable force between the alpha particle and the gold nucleus over a very short period of time. In conclusion, we say the alpha particle is "scattered" by its interaction with the nucleus of a gold atom and experiments like such are called "scattering" experiments.
[[File:ruthef.jpg]]

*[[Coefficient of Restitution]]
The coefficient of restitution is a measure of the elasticity in a collision. It is the ratio of the differences in velocities before and after the collision. The coefficient is evaluated by taking the difference in the velocities of the colliding objects after the collision and dividing by the difference in the velocities of the colliding objects before the collision.

All of the following information was collected from the Matter and Interactions 4th Edition physics textbook. The book is cited as follows...

Chabay, Ruth W., and Bruce A. Sherwood. "Chapter 10: Collisions." Matter & Interactions. Fourth Edition ed. Wiley, 2015. 383-409. Print.

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</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Fields===
<div class="mw-collapsible-content">
* [[Electric Field]] of a
** [[Point Charge]]
** [[Electric Dipole]]
** [[Capacitor]]
** [[Charged Rod]]
** [[Charged Ring]]
** [[Charged Disk]]
** [[Charged Spherical Shell]]
** [[Charged Cylinder]]
**[[A Solid Sphere Charged Throughout Its Volume]]
*[[Charge Density]]
*[[Superposition Principle]]
*[[Electric Potential]]
**[[Potential Difference Path Independence]]
**[[Potential Difference in a Uniform Field]]
**[[Potential Difference of point charge in a non-Uniform Field]]
**[[Potential Difference at One Location]]
**[[Sign of Potential Difference]]
**[[Potential Difference in an Insulator]]
**[[Energy Density and Electric Field]]
** [[Systems of Charged Objects]]
*[[Electric Force]]
*[[Polarization]]
**[[Polarization of an Atom]]
**[[Charged Conductor and Charged Insulator]]
*[[Charge Motion in Metals]]
*[[Charge Transfer]]
*[[Magnetic Field]]
**[[Right-Hand Rule]]
**[[Direction of Magnetic Field]]
**[[Magnetic Field of a Long Straight Wire]]
**[[Magnetic Field of a Loop]]
**[[Magnetic Field of a Solenoid]]
**[[Bar Magnet]]
**[[Magnetic Dipole Moment]]
***[[Stern-Gerlach Experiment]]
**[[Magnetic Torque]]
**[[Magnetic Force]]
**[[Earth's Magnetic Field]]
**[[Atomic Structure of Magnets]]
*[[Combining Electric and Magnetic Forces]]
**[[Hall Effect]]
**[[Lorentz Force]]
**[[Biot-Savart Law]]
**[[Biot-Savart Law for Currents]]
**[[Integration Techniques for Magnetic Field]]
**[[Sparks in Air]]
**[[Motional Emf]]
**[[Detecting a Magnetic Field]]
**[[Moving Point Charge]]
**[[Non-Coulomb Electric Field]]
**[[Electric Motors]]
**[[Solenoid Applications]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Simple Circuits===
<div class="mw-collapsible-content">
*[[Components]]
*[[Steady State]]
*[[Non Steady State]]
*[[Charging and Discharging a Capacitor]]
*[[Work and Power In A Circuit]]
*[[Thin and Thick Wires]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Resistivity]]
*[[Power in a circuit]]
*[[Ammeters,Voltmeters,Ohmmeters]]
*[[Current]]
**[[AC]]
*[[Ohm's Law]]
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[RC]]
*[[AC vs DC]]
*[[Charge in a RC Circuit]]
*[[Current in a RC circuit]]
*[[Circular Loop of Wire]]
*[[Current in a RL Circuit]]
*[[RL Circuit]]
*[[Feedback]]
*[[Transformers (Circuits)]]
*[[Resistors and Conductivity]]
*[[Semiconductor Devices]]
*[[Insulators]]
*[[Voltage]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Maxwell's Equations===
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
**[[Electric Fields]]
***[[Examples of Flux Through Surfaces and Objects]]
**[[Magnetic Fields]]
**[[Proof of Gauss's Law]]
*[[Ampere's Law]]
**[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
**[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
**[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Faraday's Law]]
**[[Curly Electric Fields]]
**[[Inductance]]
***[[Transformers (Physics)]]
***[[Energy Density]]
**[[Lenz's Law]]
***[[Lenz Effect and the Jumping Ring]]
**[[Lenz's Rule]]
**[[Motional Emf using Faraday's Law]]
*[[Ampere-Maxwell Law]]
*[[Superconductors]]
**[[Meissner effect]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Radiation===
<div class="mw-collapsible-content">
*[[Producing a Radiative Electric Field]]
*[[Sinusoidal Electromagnetic Radiaton]]
*[[Lenses]]
*[[Energy and Momentum Analysis in Radiation]]
**[[Poynting Vector]]
*[[Electromagnetic Propagation]]
**[[Wavelength and Frequency]]
*[[Snell's Law]]
*[[Effects of Radiation on Matter]]
*[[Light Propagation Through a Medium]]
*[[Light Scaterring: Why is the Sky Blue]]
*[[Light Refraction: Bending of light]]
*[[Cherenkov Radiation]]
*[[Rayleigh Effect]]
*[[Image Formation]]
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Sound===
<div class="mw-collapsible-content">
*[[Doppler Effect]]
*[[Nature, Behavior, and Properties of Sound]]
*[[Speed of Sound]]
*[[Resonance]]
*[[Sound Barrier]]
*[[Sound Rarefaction]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Waves===
<div class="mw-collapsible-content">
*[[Bragg's Law]]
*[[Standing waves]]
*[[Gravitational waves]]
*[[Plasma waves]]
*[[Wave-Particle Duality]]
*[[Electromagnetic Spectrum]]
*[[Color Light Wave]]
*[[The Wave Equation]]
*[[Pendulum Motion]]
*[[Transverse and Longitudinal Waves]]
*[[Planck's Relation]]
*[[interference]]
*[[Polarization of Waves]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Real Life Applications of Electromagnetic Principles===
<div class="mw-collapsible-content">
*[[Electromagnetic Junkyard Cranes]]
*[[Maglev Trains]]
*[[Spark Plugs]]
*[[Metal Detectors]]
*[[Speakers]]
*[[Radios]]
*[[Ampullae of Lorenzini]]
*[[Electrocytes]]
*[[Generator]]
*[[Using Capacitance to Measure Fluid Level]]
*[[Cyclotron]]
*[[Railgun]]
*[[Magnetic Resonance Imaging]]
*[[Electric Eels]]
*[[Windshield Wipers]]
*[[Galvanic Cells]]
*[[Magnetoreception]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

===Optics===
<div class="mw-collapsible-content">
*[[Mirrors]]
*[[Refraction]]
*[[Quantum Properties of Light]]
*[[Lasers]]

</div>
</div>

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A page for review of [[Vectors]] and vector operations

File:Swfe.png

2015-12-05T22:47:25Z

Anvikshabusa:

File:Pulse.png

2015-12-05T22:46:20Z

Anvikshabusa: Anvikshabusa uploaded a new version of "File:Pulse.png"

File:Pulse.png

2015-12-05T22:45:37Z

Anvikshabusa:

Standing waves

2015-12-05T22:41:18Z

Anvikshabusa: