# 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.

• Welcome to Physics.SE! Let me just say this is a great question. – Jim Oct 1 '15 at 16:48