**3. Acoustic Equations**

Imagine taking a snapshot of **u**, (or of ρσ^{2})
at time t; then at time t+dt all pixels in the image move by the vector **u**dt.
The resulting change in **u**, (or ρσ^{2}) is given by multiplying
the far-right hand terms in Eulers equation (17), or equation (19), by dt. So
these terms reflect the first-order changes directly caused by the average molecular
velocity **u**. Call them the "drift-velocity" terms. The remaining right-hand
term in equation (17), and in equation (19), reflect the effects of molecular
diffusion. As we will verify
later, for (almost) all of the situations we are interested in, the diffusion
terms are much larger than the drift velocity terms. So the first approximation
towards obtaining the acoustic equations is to drop the drift velocity terms.

Next, write

where ρ_{o} and σ_{o} are
constants. The second approximation asssumes that ρ_{v}<<
ρ_{o} and σ_{v}<< σ_{o}. We then
can write

P_{0} is the static pressure and p is the
pressure variation due to the sound wave,

Then the approximate forms of equations (17) and (19) are the acoustic equations

These equations are the sound analogue of Maxwell's
equations for electromagnetics. Frequently, even in textbooks, sound is represented
as a scalar pressure wave, and the vector part **u **of the wave is ignored.
One notable exception is the rigorous text by Landau
and Lifshitz. Most of my career was spent working with Maxwell's equations,
and despite the obvious differences, the parallels with sound theory fascinate
me.

The final goal of this section are the sound wave equations, obtained by differentiating equations (24) and (25) with respect to time, and substituting appropriately

Having finally obtained the primary equations governing sound waves, the next step is to investigate solutions of the equations, which is the subject of the next section.