3. Acoustic Equations
Imagine taking a snapshot of u, (or of ρσ2) at time t; then at time t+dt all pixels in the image move by the vector udt. The resulting change in u, (or ρσ2) 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 ρo and σo are constants. The second approximation asssumes that ρv<< ρo and σv<< σo. We then can write
P0 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
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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.
