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Consider a wave equation $$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\partial^2 u}{\partial x^2} \tag{1}$$ In frequency domain this becomes an ODE: $$-\omega^2 u = c(x)^2 \frac{\partial^2 u}{\partial x^2} \tag{2}$$ We can solve (2) analytically when $c(x)$ is constant, then the solution is $\exp\left(\pm\frac{i \omega}{c} x\right)$. We can also solve it analytically if $c(x)$ is piecewise constant: write in every region-of-constant-$c$ a linear combination of leftward and rightward propagating waves, impose the appropriate continuity conditions on the interfaces between regions with different values of $c$, solve for the coefficients of the linear combinations.

Suppose now that $c(x)$ is some arbitrary function. We can still write it as a limit of piecewise constant functions: $$c(x)=\lim_{\Delta\rightarrow 0^+} \sum_{i=-\infty}^{\infty} \left\{\begin{matrix}c(i\Delta) & x\in [i\Delta,(i+1)\Delta[ \\ 0 & \text{otherwise}\end{matrix}\right.$$

Let us call these piecewise constant approximations to $c$, $c_{\Delta}$ $$c_{\Delta}(x)=\sum_{i=-\infty}^{\infty} \left\{\begin{matrix}c(i\Delta) & x\in [i\Delta,(i+1)\Delta[ \\ 0 & \text{otherwise}\end{matrix}\right.$$ We can analytically solve, for any strictly positive $\Delta$, $$-\omega^2 u_{\Delta}= c_{\Delta}(x)^2 \frac{\partial^2 u_{\Delta}}{\partial x^2}$$

Question

Is $\lim_{\Delta\rightarrow 0^+} u_{\Delta} = u$?

In other words, can we approximate general solutions of (2) by substituting in a piecewise constant approximation of $c(x)$, and then solving analytically?