Sound Waves

A sound wave is a type of longitudinal wave that causes a disturbance in the pressure and density of an ideal gas through which it passes. Consider a plane sound wave propagating in the $x$-direction. Let $\xi(x,t)$ be the longitudinal displacement of the gas associated with the wave. Consider a slab of gas of cross-sectional area $A$ lying between $x-dx/2$ and $x+dx/2$. The mass of the slab is $\rho\,A\,dx$, where $\rho$ is gas's mass density. The slab's equation of longitudinal motion is

$$\rho\,A\,dx\,\frac{\partial^{2}\xi}{\partial t^{2}} = A\left[-p(x+dx/2)+p(x-dx/2)\right]=-A\,\frac{\partial p}{\partial x}\,dx,$$ (5.155)

which gives

$$\rho\,\frac{\partial^{2}\xi}{\partial t^{2}}=-\frac{\partial p}{\partial x}.$$ (5.156)

The change in volume of the slab of gas is

$$\delta V = A\left[\xi(x+dx/2)-\xi(x-dx/2)\right]=A\,\frac{\partial\xi}{\partial x}\,dx,$$ (5.157)

which yields

$$\frac{\delta V}{V} =\frac{\partial\xi}{\partial x},$$ (5.158)

because $V=A\,dx$. However,

$$\frac{\delta V}{V} = -\frac{\delta p}{\kappa},$$ (5.159)

where $\kappa$ is the bulk modulus. [See Equation (5.146).] Hence,

$$\frac{\partial \xi}{\partial x} = -\frac{\delta p}{\kappa}.$$ (5.160)

Equation (5.156) gives

$$\rho\,\frac{\partial^{2}}{\partial t^{2}}\!\left(\frac{\partial \xi}{\partial x}\right)=-\frac{\partial^{2} \delta p}{\partial x^{2}},$$ (5.161)

writing $p=p_0+\delta p(x,t)$, where $p_0$ is a constant background pressure. The previous two equations can be combined to yield

$$\frac{\partial^{2}\delta p}{\partial t^{\,2}} = v_s^{\,2}\,\frac{\partial^{2} \delta p}{\partial x^{\,2}},$$ (5.162)

where

$$v_s= \left(\frac{\kappa}{\rho}\right)^{1/2}.$$ (5.163)

Equation (5.162) is a one-dimensional wave equation that has the standard solution

$$\delta p(x, t)=\delta p_0 \,\cos[k\,(x-v_s\,t)],$$ (5.164)

where $\delta p_0$ and $k$ are constants. The previous solution corresponds to a wave-like disturbance in the gas pressure of amplitude $\delta p_0$, wavenumber ${\bf k} = k\,{\bf e}_x$, and phase velocity $v_s$. In other words, Equation (5.163) specifies the speed of sound in an ideal gas.

It remains to determine whether the compression of the gas associated with the passage of a sound wave is isothermal or isentropic. In fact, because ideal gases are relatively poor conductors of heat (see Section 5.3.10), the period of vibration of a sound wave is generally much shorter than the relaxation time necessary for a small element of the gas to exchange energy with the remainder of the gas by means of heat flow. Hence, the compression of the gas associated with the passage of a sound wave is isentropic. It follows from Equations (5.154) and (5.163) that the speed of sound in an ideal gas is

$$v_s = \left(\frac{\kappa_S}{\rho}\right)^{1/2}=\left(\frac{\gamma\,p}{\rho}\right)^{1/2}.$$ (5.165)

Making use of Equations (5.97) and (5.129), the previous equation becomes

$$v_s =\left(\frac{\gamma\,R\,T}{\mu}\right)^{1/2},$$ (5.166)

where $\mu$ is the molecular mass. Note that the speed of sound in an ideal gas only depends on the gas temperature, and is independent of the pressure.

It is a good approximation to treat the Earth's atmosphere as an ideal gas. The atmosphere is mostly diatomic, which implies that $\gamma=1.4$. [See Equation (5.122).] Moreover, the molecular weight of the atmosphere is $\mu=29\times 10^{-3}$ kg. (See Section 5.2.6.) Hence, the speed of sound in air at $15^\circ$ C is $340\,{\rm m}\,{\rm s}^{-1}$.