Standing waves

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.

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Characteristics of a Standing Wave

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A Mathematical Model

A Computational Model

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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.

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  Standing wave in stationary medium. The red dots represent the wave nodes.

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  A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

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

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2 fixed ends

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A wave has both a frequency and a wavelength that are related by the following equation:

[math]\displaystyle{ {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]\displaystyle{ {\frac{nλ}{2}} = L} }[/math]

where n is a positive integer, λ the wavelength, and L the length of the string.

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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