# Geometrical acoustics derivation

In acoustics for non-uniform media (speed of sound is $$c(\mathbf{r}, t)$$, dependent on space and time), I want to solve $$\frac{\partial^2\phi}{\partial t^2}-\nabla^2(c^2\phi)=0$$, where $$\phi$$ is the density field due to sound. To get geometrical acoustics, I follow the derivation in my acoustics lecture, let $$\phi=\rho(\mathbf{r}, t)\mathrm{e}^{i\theta(\mathbf{r}, t)}$$, and define fields $$\omega = -\frac{\partial \theta}{\partial t}$$ and $$\mathbf{k}=\nabla\theta$$ (which implies $$-\nabla\omega=\frac{\partial\mathbf{k}}{\partial t}$$). Plugging in gives me

\begin{align*} \frac{\partial^2\rho}{\partial t^2}-\omega^2\rho-\rho\nabla^2c^2-2\nabla(c^2)\cdot\nabla(\rho)-c^2\nabla^2\rho+k^2c^2\rho &= 0 \\ -2\omega\frac{\partial\rho}{\partial t}-\rho\frac{\partial\omega}{\partial t}-2\rho\mathbf{k}\cdot\nabla c^2-2c^2\mathbf{k}\cdot\nabla\rho-c^2\rho\nabla\cdot\mathbf{k} &= 0 \end{align*}

where the two equations are the real and imaginary parts. Then I want to assume slowly varying $$\rho$$ and $$c$$ so I only keep the following terms

\begin{align*} -\omega^2\rho+k^2c^2\rho &= 0 \\ -\rho\frac{\partial\omega}{\partial t}-c^2\rho\nabla\cdot\mathbf{k} &= 0 \end{align*}

after which dividing by $$\rho$$ gives

\begin{align*} -\omega^2+k^2c^2 &= 0 \\ -\frac{\partial\omega}{\partial t}-c^2\nabla\cdot\mathbf{k} &= 0 \end{align*}

Question: In QM perturbation theory, we let $$H = H_0+\epsilon H_1$$ and we plug in a series solution $$\psi = \sum_{k=0}\epsilon^{k}\psi^{(k)}$$, then separate the equations in orders of $$\epsilon$$. Is there a similarly systematic way to find approximate solutions for geometrical acoustics, so that I can justify throwing away the terms I did? I looked at WKB approximation and Multiple-scale analysis, but I don't know how to apply ODE techniques to PDEs, and I also don't know where the small parameter $$\delta$$ is.