Wave, motion energy and momentum

In previous post we have talked about dynamics of fluid motion, characteristics of sound, developed equation of fluid motion etc. Now we are able to develop and understand the equation of acoustic wave motion which is valid when wave amplitude is small. For now we are still working with waves that have small amplitude, but the more accurate form of the equation will be considered later.

Let’s imagine the fluid that in the absence of sound has uniform density, pressure and temperature and every particle in the fluid is at rest. We assume for now that this fluid is nonviscous and have zero heat conductivity, so that the only energy involved in the acoustic motion is mechanical, and the only forces are those of compressive elasticity, measured by the compressibility κ. In this case we assume that the compressibility is the adiabatic one which means that there is zero heat conduction.

In the presence of a sound wave the pressure becomes P+p(x,t) if the wave is one-dimensional. Since the sound is moving through the fluid in x-direction (one-dimensional) the wave front planes are perpendicular to the x axis and are parallel to the yz-plane. As pressure of the fluid changes the density of the fluid and the temperature of the fluid also changes form ρ+δ(x,t) and T+τ(x,t) respectively.  Of course the change is uniform which means that the density, pressure or temperature change in whole fluid in the presence of sound and that means that this parameters also changes over the whole plane which is at distance x from yz plane.

Waves of this sort are plane waves and for them we will develop the one dimensional wave equation which describes their motion in the fluid. As I mention in previous posts the p(x,t) is called the sound pressure or acoustic pressure. This pressure is responsible for the fluid motion and the fluid motion is in turn responsible for the change in density.