What is the equation of motion for multiple simultaneous pressure waves in a medium? (In the context of stimulated Brillouin scattering) My overall motivation is to derive the behavior of Brillouin scattering in a birefringent fiber. Brillouin scattering is a nonlinear interaction between light and sound.
In classic Brillouin scattering, the optical fiber has a single refractive index and hence the resulting acoustic wave has a single wavenumber:
$$
\rho = \rho_0 + [\rho(z,t)e^{i(qz - \Omega t)} + c.c]
$$
It is governed by the classic acoustic equation of motion:
$$
\frac{\partial ^2\rho}{\partial t^2} - \Gamma' \nabla^2\frac{\partial\rho}{\partial t} - \nabla^2\nu^2\rho = \nabla \cdot f
$$
where $\rho$ is the acoustic wave, $\Gamma'$ is a dampening parameter $\nu$ is the speed of sound and $f$ is the force per unit volume. In Stimulated Brillouin Scattering, the force is created by the interaction of counter propagating light waves.
For a birefringent fiber, the acoustic wave now has the form:
$$
\rho = \rho_0 + [\rho_1(z,t)e^{i(q_1z - \Omega t)} + \rho_2(z,t)e^{i(q_2z - \Omega t)} + c.c]
$$
Now here is my question, given that the optical fiber is now birefringent and has two optical axes, how will the equation of motion vary?
Specifically, is it valid to say that the dampening parameter and speed of sound are the same as they were?
If you are aware of any research papers in this area, I will appreciate referencing them as well. Thank you for your time and insight.
 A: Ok,
So after alot of research I am attempting to answer my own question.
Current state of the art research into this exact question was published in a paper by Daisy Williams, Xiaoyi Bao, and Liang Chen under the name: 
"Effects of polarization on stimulated Brillouin scattering in a birefringent optical fiber"
Those who want to delve in, should with regard to the intricates of the solution but here are the main points:


*

*The dampening factor and speed of sound in the fiber do not change with regard to the principal axes of the polarization according to the paper. I will double check this and update because I believe they should according to the lorenz lorenz derivation in Boyd implies that the dampening factor should be dependent on the refractive index.

*The solution of the acoustic equation is a super position of the solutions. This is given since the acoustic equation is a linear wave equation. Hence the exact solution is the addition of the classic solutions for each wave.

*For a linear polarization along the principal axes, we will obtain 2 acoustic waves. For a generalized elliptic polarization, we will obtain 4 acoustic waves.

