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I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab.

Specifically, I'm trying to show that the regular 1-Dimensional wave eqt $$\frac{\partial u^2}{\partial t^2} = c^2\frac{\partial u^2}{\partial x^2}$$ Looks like diffusion with some drift, when the wave speed $c$ is chosen to randomly be $0$ or $1$ on every interval of length $\epsilon$. ( The point is to let the length of the intervals $\epsilon \to0$)

Normally for a simple equation I would simply have constant boundary conditions where $$u(x,0)=1$$ and watch the wave go on an interval $[0,1]$, but obviously if this is the case, the implicit finite difference equation becomes explicit, and I get nonsense as I mess around with $c$.

My question is, what would suitable initial conditions for this problem be?

(Also, pointing me in the direction of some literature that may help with this problem would be greatly appreciated)

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