Acoustic theory

Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have
In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as
Where formula_3 is the perturbed velocity of the fluid, formula_4 is the pressure of the fluid at rest, formula_5 is the perturbed pressure of the system as a function of space and time, formula_6 is the density of the fluid at rest, and formula_7 is the variance in the density of the fluid over space and time.
In the case that the velocity is irrotational (formula_8), we then have the acoustic wave equation that describes the system:
Where we have
Derivation for a medium at rest.
Starting with the Continuity Equation and the Euler Equation:
If we take small perturbations of a constant pressure and density:
Then the equations of the system are
Noting that the equilibrium pressures and densities are constant, this simplifies to
A Moving Medium.
Starting with
We can have these equations work for a moving medium by setting formula_16, where formula_17 is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and formula_18 is the fluid velocity.
In this case the equations look very similar:
Note that setting formula_20 returns the equations at rest.
Linearized Waves.
Starting with the above given equations of motion for a medium at rest:
Let us now take formula_22 to all be small quantities.
In the case that we keep terms to first order, for the continuity equation, we have the formula_23 term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density:
Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by
Under this condition, we see that we now have
Defining the speed of sound of the system:
Everything becomes
For Irrotational Fluids.
In the case that the fluid is irrotational, that is formula_29, we can then write formula_30 and thus write our equations of motion as
The second equation tells us that
And the use of this equation in the continuity equation tells us that
This simplifies to
Thus the velocity potential formula_35 obeys the wave equation in the limit of small disturbances. The boundary conditions required to solve for the potential come from the fact that the velocity of the fluid must be 0 normal to the fixed surfaces of the system.
Taking the time derivative of this wave equation and multiplying all sides by the unperturbed density, and then using the fact that formula_36 tells us that
Similarly, we saw that formula_38. Thus we can multiply the above equation appropriately and see that
Thus, the velocity potential, pressure, and density all obey the wave equation. Moreover, we only need to solve one such equation to determine all other three. In particular, we have
For a moving medium.
Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with
We can linearize these into
For Irrotational Fluids in a Moving Medium.
Given that we saw that
If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have
Under these assumptions, our linearized sound equations become
Importantly, since formula_17 is a constant, we have formula_47, and then the second equation tells us that
Or just that
Now, when we use this relation with the fact that formula_50, alongside cancelling and rearranging terms, we arrive at
We can write this in a familiar form as
This differential equation must be solved with the appropriate boundary conditions. Note that setting formula_53 returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have