I am not sure why I am having a difficult time answering this seemingly simple question. (Perhaps I learned this topic only very superficially.)
Let us assume that we have a wave equation defined for some time $t\in [0,\infty)$ on some domain $D$:
$\partial_t^2 \phi = \nabla^2 \phi$,
with $\phi(x,y,t)$.
The go-to method to solve this system is to use separation of variables: $\phi = X(x)Y(y)T(t)$.
This makes sense if the boundaries of the domain are constants, but what if they are allowed to vary (let's only vary in space for now)?
For example, let $D$ be a box with vertical sides, and some curve $y_1(x)$ for the top, and another curve $y_2(x)$ for the bottom.
This seems like a silly question. Of course we cannot separate the solutions. We will find that $Y(y)$ is somehow dependent on $x$.
But then, we assume spatially dependent initial values in the wave equation all the time...
What am I overlooking here? I feel like there is something subtle happening and I really can't put my finger on it.
Is there a proof that anybody can provide which goes one way or another?
Thanks.