Speed of Sound

The speed of sound is the distance traveled over time by a sound wave in an elastic medium. The SI unit of speed is meters per second or (m/s). The speed of sound in dry air at 20 ° C is 343.2 m/s. In different mediums (gas, solids, or liquids) the speed of sound is different than 343.2 m/s.

The Main Idea

The main idea of this topic is to determine the speed of sound in different mediums.

A Mathematical Model

The speed of sound, or c, is usually determined by the Newton-Laplace Equation, shown below. This equation shows the relationship between the stiffness of an object and it's density.

c is the speed of sound
K_s is the coefficient of stiffness
ρ is the density

This equation shows that the speed of sound increases as the stiffness increases or the density of a material decreases.

The general equation for the speed of sound, using classical mechanics is given as:

c is the speed of sound
p is the pressure
ρ is the density and the derivative at constant entropy s

Dependence on the Medium

A dispersive medium is a medium in which waves of different frequencies travel at different speeds and a non-dispersive medium is a medium in which the speed is independent of frequency. In non-dispersive mediums, the speed of sound is independent of sound frequency, while in dispersive mediums the speed of sound can vary greatly based on it's frequency.

Waves travelling through a non-dispersive medium, such as an ideal gas

Waves travelling through a dispersive medium, such as a metal bar

Solids

In solid objects, the speed of sound depends on the Young's Modulus and the density of the solid is known. Young's modulus is dependent on the stress and strain of the solid.

V_s is the speed of sound
p is the density
Y is Young's Modulus

For more information about the speed of sound in a solid see this Wiki page.

Fluids

In fluid mediums, the speed of sound only depends on the mediums compressibilty and density. This is because liquid cannot have a stiffness, meaning it cannot sustain shear forces. Therefore, the equation of the speed of sound in a liquid is as follows:

c_fluid is the speed of sound in fluids
ρ is the density
K is the bulk modulus of the fluid

Gases

Gaseous mediums can be both dispersive and non-dispersive. The speed of sound in gas is only dependent on density. This is because the heat capacity ratio relates compressibility to pressure, and the pressure and density of a gas are inversely related. For ideal gases, the speed of sound is only dependent on temperature. This is because the pressure and density of an ideal gas will cancel out at a constant temperature. However, in non-deal gases, the pressure of a gas does not cancel with density, so the pressure still has an effect on the speed of sound.

Sound will travel faster in lighter gases such as Helium than in heavier gases such as Xenon.

The Newton-Laplace equation can be rewritten with respect to the bulk modulus formula:

c is the speed of sound
γ is the adiabatic index (the ratio of specific heats of a gas at a constant pressure to gas at a constant volume)
p is the pressure
ρ is the density

A Computational Model WIP

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Examples

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