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Sound Waves

Figure: Cylindrical element of a compressible medium.

Picture a "snapshot" (holding time $t$ fixed) of a small cylindrical section of an elastic medium, shown in Fig. 14.6: the cross-sectional area is  $A$  and the length is  $dx$. An excess pressure  $P$  (over and above the ambient pressure existing in the medium at equilibrium) is exerted on the left side and a slightly different pressure  $P+dP$  on the right. The resulting volume element  $dV = A \, dx$  has a mass   $dm = \rho \, dV = \rho \, A \, dx$,  where  $\rho$  is the mass density of the medium. If we choose the positive $x$ direction to the right, the net force acting on  $dm$  in the $x$ direction is   $dF_x = P A - (P+dP) A = - A \, dP$.

Now let $s$ denote the displacement of particles of the medium from their equilibrium positions. (I didn't use $A$ here because I am using that symbol for the area. This may also differ between one end of the cylindrical element and the other:  $s$  on the left vs.  $s+ds$  on the right. We assume the displacements to be in the $x$ direction but very small compared to  $dx$,  which is itself no great shakes.^14.10

The fractional change in volume  $dV/V$  of the cylinder due to the difference between the displacements at the two ends is

$${dV \over V} \; = \; {(s+ds)\, A - s \, A \over A \, dx} \; = \; {ds \over dx}$$

$$= \; \left(\partial s \over \partial x \right)_t$$ (14.31)

where the rightmost expression reminds us explicitly that this description is being constructed around a "snapshot" with $t$ held fixed.

Now, any elastic medium is by definition compressible but "fights back" when compressed ($dV < 0$) by exerting a pressure in the direction of increasing volume. The BULK MODULUS $B$ is a constant characterizing how hard the medium fights back - a sort of 3-dimensional analogue of the SPRING CONSTANT. It is defined by

$$P \; = \; - B \, {dV \over V} .$$ (14.32)

Combining Eqs. (31) and (32) gives

$$P \; = \; - B \; \left(\partial s \over \partial x \right)_t$$ (14.33)

so that the difference in pressure between the two ends is

$$dP \; = \; \left(\partial P \over \partial x \right)_t \, d . . . . . . B \; \left(\partial^2 s \over \partial x^2 \right)_t \, dx .$$ (14.34)

We now use   $\sum F_x = m \, a_x$  on the mass element, giving

$$-A \, dP \; = \; A B \; \left(\partial^2 s \over \partial x^2 \right)_t \, dx$$

$$= \; dm \, a_x \; = \; \rho \, A \, dx \, \left(\partial^2 s \over \partial t^2 \right)_x$$ (14.35)

where we have noted that the acceleration of all the particles in the volume element (assuming $ds \ll s$) is just   $a_x \equiv (\partial^2 s / \partial t^2 )_x$.

If we cancel  $A \, dx$  out of Eq. (35), divide through by  $B$  and collect terms, we get

$$\left(\partial^2 s \over \partial x^2 \right)_t \; - \; { . . . . . . \over \partial t^2 \right)_x \; = \; 0 \qquad \hbox{\rm or}$$

$$\left(\partial^2 s \over \partial x^2 \right)_t \; - \; { . . . . . . } \, \left(\partial^2 s \over \partial t^2 \right)_x \; = \; 0$$ (14.36)

which the acute reader will recognize as the WAVE EQUATION in one dimension ($x$), provided

$$c \; = \; \sqrt{B \over \rho}$$ (14.37)

is the velocity of propagation.

The fact that disturbances in an elastic medium obey the WAVE EQUATION guarantees that such disturbances will propagate as simple waves with phase velocity  $c$  given by Eq. (37).

We have now progressed from the strictly one-dimensional propagation of a wave in a taut string to the two-dimensional propagation of waves on the surface of water to the three-dimensional propagation of pressure waves in an elastic medium (i.e. sound waves); yet we have continued to pretend that the only simple type of traveling wave is a plane wave with constant $\Vec{k}$. This will never do; we will need to treat all sorts of wave phenomena, and although in general we can treat most types of waves as local approximations to plane waves (in the same way that we treat the Earth's surface as a flat plane in most mechanics problems), it is important to recognize the most important features of at least one other common idealization - the SPHERICAL WAVE.

Footnotes

. . . shakes.^14.10

Note also that any of $s$, $ds$, $P$ or $dP$ can be either positive or negative; we merely illustrate the math using an example in which they are all positive.