In Section 16.4 we used Newton’s second law to show that traveling waves can propagate on a stretched string and to predict the wave speed in terms of properties of the string. Now we wish to do the same for sound waves—longitudinal waves propagating through a fluid.

A sound wave is a sequence of compressions and rarefactions in which the fluid is alternately compressed and expanded. A substance’s compressibility is characterized by its bulk modulus B, which you met in 《 SECTION 14.7 when we looked at the elastic properties of materials. If excess pressure p is applied to an object of volume V, then the fractional change in volume—the fraction by which it’s compressed—is

The minus sign indicates that the volume decreases when pressure is applied. Gases are much more compressible than liquids, so gases have much smaller values of B than liquids.

Let’s apply this to a fluid—either a liquid or a gas. FIGURE 16.22 shows a small cylindrical piece of fluid with equilibrium pressure $p_0$ located between positions x and $x+\text{Δ}x$. The initial length and volume of this little piece of fluid are $L_{\text{i}}=\text{Δ}x$ and $V_{\text{i}}=a{L}_{\text{i}}=a\text{Δ}x$. Notice that we use a for area in this chapter so that there is no conflict with A for amplitude. Suppose the pressure changes to $p_0+p$. The volume of this little piece of fluid will either decrease (compression) or increase (expansion), depending on whether p is positive or negative.

FIGURE 16.22 An element of fluid changes volume as the pressure changes.

The volume changes only if the ends of the cylinder undergo different displacements. (Equal displacements would shift the cylinder but not change its volume.) In the bottom half of Figure 16.22 we see that the left end of the cylinder has undergone displacement $D\left(x\text{,\hspace{0.17em}}t\right)$ while the displacement at the right end is $D(x+\text{Δ}x\text{,\hspace{0.17em}}t)$. Now the cylinder has length

and consequently its volume has changed by

Substituting both the initial volume and the volume change into Equation 16.39, we have

After canceling the a, you can see that, just as in Section 16.4, we’re left—in the limit $\text{Δ}x\to 0$—with the definition of the derivative of D with respect to x. It is again a partial derivative because we’re holding the variable t constant. Thus we find that the fluid pressure (or, to be exact, the pressure deviation from $p_0$) at position x is related to the displacement of the medium by

The pressure depends on how rapidly the fluid’s displacement changes with position.

We anticipate that we’ll discover sinusoidal sound waves later in this section, so a displacement wave of amplitude A,

is associated with a pressure wave

The pressure amplitude, or maximum pressure, is

where we used $\omega =2\pi f=vk$ (Equation 16.13) in the last step to write the result in terms of the wave’s speed and frequency. In other words, a sound wave is not just a traveling wave of molecular displacement. A sound wave is also a traveling pressure wave.

As an example, a quite loud 100 decibel, 500 Hz sound wave in air has a pressure amplitude of 2 Pa. That is, the pressure varies around atmospheric pressure by $\pm 2\text{Pa}$. You can use Equation 16.46 and $B_{\text{air}}=1.42\times {10}^5\text{Pa}$ to find that the amplitude of the oscillating air molecules is a microscopic $1.5\mu \text{m}$.

FIGURE 16.23 uses Equations 16.44 and 16.45 to draw snapshot graphs of displacement and pressure for a sound wave propagating to the right. Positive displacement pushes molecules to the right while negative displacement pushes them to the left, so molecules pile up (a compression) at points where the displacement is changing from positive to negative. These are the points where the displacement has the most negative slope and thus, from Equation 16.43, the greatest pressure.

FIGURE 16.23 Snapshot graphs of the sound displacement and pressure.

In general, the pressure wave has a maximum or minimum at points where the displacement wave is zero, and vice versa. This observation will help us understand standing sound waves in Chapter 17.