The problem starts with a square drum head of side length $L$. The boundaries at $x = 0, L$ and $y=0,L$ are both held at $h=0$, where $h$ is the axis perpendicular to $x$ and $y$. The drive function $d(x,y)e^{i\omega t}$ and the reaction function $h(x,y)e^{i\omega t}$ are modeled by the differential equation:
$\nabla^2h(x,y)+\frac{\omega^2}{c^2}h(x,y)=d(x,y)$,
with the corresponding Green's function:
$(\nabla^2+\frac{\omega^2}{c^2})g(x|\xi_x,y|\xi_y) = \delta(x-\xi_x)\delta(y-\xi_y)$.
I need to find the Green's function after expanding the drum head potential infinitely in the $x$ and $y$ directions.
I generalized the drive as follows:
$\sum_{n,m}[\delta(x-(2nL+\xi_x))-\delta(x-(2nL-\xi_x))][\delta(y-(2mL+\xi_y))-\delta(y-(2mL-\xi_y))]$.
I'm really at a loss as to how to proceed. I've done multivariabled Green's functions before, but not like this.
Any help or hints would be appreciated.