7.7 Sound Waves

So far we have been looking at transverse waves, where the displacement is perpendicular to the direction of wave travel. Examples of transverse waves in real life are waves on a string and light waves. Waves can also be longitudinal, where the displacement is along the direction of travel. Sound waves are longitudinal, with regions of high density along the direction of travel (compression) and low density (rarefaction).

Let’s look at some snapshots of longitudinal waves. We denote the displacement of a particle from its equilibrium position by $s(x,t)$ . Then the particles oscillate back and forth along the direction of motion with SHM. Where $s$ is positive, the particles are displaced to the right, and where $s$ is negative they are displaced to the left. The points of zero displacement represent pressure maxima and minima.

It is a general rule for all mechanical waves that the wave speed is related to the ratio of restoring force to inertia. For liquids and gases, this is

v=\sqrt{B}{\rho},

(7.131)

where $\rho$ is the density and $B$ is the bulk modulus, which is a measure of how easy it is to compress the medium, given by

B=-V\frac{\operatorname{\mathrm{d}}\!P}{\operatorname{\mathrm{d}}\!V}.

(7.132)

In everday situations, we can approximate $\frac{\operatorname{\mathrm{d}}\!P}{\operatorname{\mathrm{d}}\!V}$ as $\frac{\Delta P}{\Delta V}$ i.e. the pressure change that accompanies a small volume change. We find that for air $B_{\text{air}}=$1.42\text{\times}{10}^{5}\text{\,}\mathrm{N}\text{\,}{\mathrm{% m}}^{-2}$$ , and for water $B_{\text{water}}=$2.2\text{\times}{10}^{9}\text{\,}\mathrm{N}\text{\,}{\mathrm% {m}}^{-2}$$ . The density of air will change in everyday situations depending on the temperature. At room temperature ( $293\text{\,}\mathrm{K}$ ), we have $v=$343\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ . More generally, we have $v\propto T$ .

7.7.1 Mathematical Description of Pressure Waves

The equations of pressure waves have the same form as transverse waves:

s(x,t)=A\cos(kx-\omega t+\phi_{0}).

(7.133)

Can we get a formula for the variation of pressure across a sound wave? Along the direction of propogation, volume elements oscillate in SHM and pressure variations cause the volumes to change slightly. This is because the left and right-hand sides of undergo slightly different displacements. Thus by the definition of the bulk modulus, we get

$\displaystyle P(x,t)$	$\displaystyle=-B\frac{\partial s(x,t)}{\partial x}$	(7.134)
	$\displaystyle=BAk\sin(kx-\omega t+\phi_{0})$	(7.135)
	$\displaystyle=P_{\text{max}}\sin(kx-\omega t+\phi_{0}).$	(7.136)

Note that this pressure is the excess pressure deviation from equilibrium, not the absolute pressure in the fluid. From this, we can see that on the plot of $s(x,t)$ , points of zero displacement with positive derivative are the pressure maxima, and those with negative derivative are the pressure minima.

7.7.2 Power Transmitted through a Pressure Wave

When we calculated the power transmitted by a transverse wave on a string, we could neglect the other directions and consider the problem in 1D. For a sound wave this is not possible, so we have to consider plane waves (waves that have constant value on a plane perpendicular to the direction of motion). Instead of power, we look at intensity which is defined as power per unit area, and we measure it across surfaces perpendicular to the propagation of the wave. We have a force $F$ which does work along a distance $\operatorname{\mathrm{d}}\!s$ , so $\operatorname{\mathrm{d}}\!W=F\operatorname{\mathrm{d}}\!s$ , then the intensity is the power $\frac{\operatorname{\mathrm{d}}\!W}{\operatorname{\mathrm{d}}\!t}$ divided by the area which the force acts on $S$ :

$\displaystyle I_{\text{inst}}$	$\displaystyle=\frac{\operatorname{\mathrm{d}}\!}{\operatorname{\mathrm{d}}\!t}% \left(\frac{F}{S}\operatorname{\mathrm{d}}\!s\right)$	(7.137)
	$\displaystyle=\frac{F}{S}\frac{\partial s}{\partial t}$	(7.138)
	$\displaystyle=P(x,t)\frac{\partial s}{\partial t}$	(7.139)
	$\displaystyle=-B\frac{\partial s}{\partial x}\frac{\partial s}{\partial t}$	(7.140)
	$\displaystyle=BA^{2}k\omega\sin^{2}(kx-\omega t+\phi_{0}).$	(7.141)

Note how similar the second last line looks to the equation for instantaneous power of a transverse wave on a string. Taking the time average, we get

	$\displaystyle I_{\text{avg}}$	$\displaystyle=\frac{1}{2}BA^{2}k\omega$		(7.142)
		$\displaystyle=\frac{1}{2}\frac{P_{\text{max}}^{2}}{\sqrt{\rho B}},$		(7.143)

where in the second line we have substituted more physical quantities. The denominator $\sqrt{\rho B}$ is known as the specific acoustic impedance $Z$ , and this can be used to define transmission and reflection coefficients as we did before.