# What causes an increase in sound speed in a medium?

Its an established fact that increase in the temperature causes increase in speed of sound waves but what is the property which is changed by changing temperature ? Does frequency and wavelength get affected by temperature?

The speed of sound is given by the Newton-Laplace equation:

$$v = \sqrt{\frac{K}{\rho}}$$

where $K$ is the bulk modulus (i.e. a measure of stiffness) and $\rho$ is the density. The physical interpretation of this is fairly obvious. Stiffer substances recoil faster from a displacement so increasing the stiffness increases the speed of sound. Heavier substances recoil more slowly from a displacement so increasing the density decreases the speed of sound.

The effect of temperature lies in how it changes $K$ and $\rho$, but the effect will vary for different materials. For an ideal gas the the bulk modulus P is simply the gas pressure multiplied by the adiabatic index, $\gamma$, so the speed is given by:

$$v = \sqrt{\frac{\gamma P}{\rho}} \tag{1}$$

We can manipulate this equation using the ideal gas formula:

$$PV = nRT$$

For example the density is $nM/V$, where $M$ is the molar mass of the gas, so:

$$\rho = \frac{nM}{V} = \frac{PM}{RT}$$

If we make this substitution in equation (1) we get:

$$v = \sqrt{\frac{\gamma RT}{M}}$$

giving us the result that the speed of sound increases with temperature as you said in your question.

• But does increasing temperature increases its wavelength and frequency? – user60180 Oct 10 '14 at 14:42
• @user60180: that depends on what you're asking. The frequency of a sound wave depends on what is generating it. If I play a 1kHz tone on my HiFi then the frequency doesn't change with temperature. However because the speed of sound changes the wavelength will also change in proportion. – John Rennie Oct 10 '14 at 15:24