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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.

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