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

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

The *fractional change in volume* of the cylinder
due to the *difference* between the displacements at the
two ends is

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

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

so that the

We now use
on the mass element, giving

where we have noted that the acceleration of all the particles in the volume element (assuming ) is just .

If we cancel out of Eq. (35), divide through by
and collect terms, we get

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

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

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 .
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.

- . . . shakes.
^{14.10} - Note also that any of , , or can be either positive or negative; we merely illustrate the math using an example in which they are all positive.

Jess H. Brewer - Last modified: Sun Nov 15 21:26:25 PST 2015