Sound Waves in a Gas

A sound wave in a gas is a longitudinal disturbance of the gas's pressure and density that propagates at the fixed speed

$$c =\sqrt{ \frac{\gamma\,p}{\rho}},$$ (3.1)

(See Section 5.2.9.) Here, $\gamma$ is the gas's ratio of specific heats (which is approximately 1.4 for the atmosphere), $p$ the gas's undisturbed pressure, and $\rho$ the gas's undisturbed mass density. Note that a sound wave is a non-dispersive wave, which means that a transient wave pulse propagates at the same speed as an infinite wave train. (See Section 4.2.6.) However, a sound wave only propagates at the speed (3.1) in the rest frame of the gas.

Figure 3.1: Sound waves propagating in a stationary and in a moving reference frame.

Let ${\bf c}$ be the phase velocity of a sound wave in a stationary frame of reference in which the gas is at rest. Thus, $\vert{\bf c}\vert=c$, is the speed of sound, (3.1). Consider a moving frame of reference that moves at constant velocity ${\bf v}$, where $v<c$, with respect to the stationary frame. Incidentally, if the stationary frame is inertial then so is the moving frame. (See Section 1.5.4.) The gas appears to flow with uniform velocity $-{\bf v}$ in the moving reference frame. Furthermore, it is an experimentally verified fact that the sound wave appears to propagate with the phase velocity

$${\bf c}' = {\bf c}- {\bf v}$$ (3.2)

in the moving frame. Note that, in general, both the speed and the direction of the sound wave are different in the stationary and the moving frames. Note, further, that the previous equation is a direct consequence of the Galilean transformation, (1.106)–(1.108). (See Section 3.2.6.) The previous equation yields

$$c' = \left(c^2-2\,{\bf c}'\cdot{\bf v} - v^2\right)^{1/2}.$$ (3.3)

Hence, we deduce, from the previous two equations, that if the sound wave propagates in the same direction as ${\bf v}$ in the moving frame then it propagates at the speed

$$u_- = c-v,$$ (3.4)

but if the sound wave propagates in the opposite direction to ${\bf v}$ in the moving frame then it propagates at the speed

$$u_+=c+v$$ (3.5)

and, finally, if the sound wave propagates in a direction perpendicular to ${\bf v}$ in the moving frame then it propagates at the speed

$$u_\perp = \left(c^2-v^2\right)^{1/2}.$$ (3.6)

Of course, the sound wave propagates at the speed $c$ in all directions in the stationary frame. These ideas are illustrated in Figure 3.1.

Figure 3.2: Experiment to detect motion of moving inertial reference frame.

We could imagine performing an experiment in the moving reference frame in order to measure its velocity, ${\bf v}$, with respect to the stationary frame. See Figure 3.2. Suppose that we have a sound wave source that emits a transient sound wave pulse isotropically in all directions. Suppose that we place two small sound-wave reflectors (which are stationary in the moving frame) at equal distances $l_0$ from the source. The displacement of the first reflector from the source is in the direction of ${\bf v}$, whereas the displacement of the second reflector is in a direction that is perpendicular to ${\bf v}$. The travel time of the pulse from the source to the first reflector, and back again, is

$$t_1 = \frac{l_0}{u_-} + \frac{l_0}{u_+}= \frac{l_0}{c-v}+\frac{l_... ...\frac{2\,l_0\,c}{c^2-v^2}\simeq \frac{2\,l_0}{c}\left(1+\frac{v^2}{c^2}\right),$$ (3.7)

where we have assumed that $v\ll c$. The travel time of the pulse from the source to the second reflector, and back again, is

$$t_2 = \frac{2\,l_0}{u_\perp}= \frac{2\,l_0}{(c^2-v^2)^{1/2}}\simeq \frac{2\,l_0}{c}\left(1+\frac{v^2}{2\,c^2}\right).$$ (3.8)

Thus, if we measure the two travel times, and take the difference between them, then we obtain

$$t_1-t_2 = \frac{l_0\,v^2}{c^3}.$$ (3.9)

Hence, assuming that we know $l_0$ and $c$, we can determine $v$. We can also determine the direction of ${\bf v}$ because the time difference is maximized when the two legs of the apparatus shown in Figure 3.2 are aligned parallel and perpendicular to this direction.