# Green's function solution to 2D boundary value problem.

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.

• Do you know about the method of images? – Christopher A. Wong Mar 3 '17 at 22:45
• @ChristopherA.Wong Unfortunately yes, although not in this class so this problem should be doable without the method of images. – Spuds Mar 3 '17 at 22:51
• In your question you've made the issue ambiguous. First, you claim that the boundary conditions are Dirichlet, but then you seem to "expand the drum head potential infinitely" by turning the problem into one with periodic boundary conditions. – Christopher A. Wong Mar 3 '17 at 22:59
• @ChristopherA.Wong The first part was to give an idea of the potential. It is period though in both directions. Sorry for not making it clear. – Spuds Mar 3 '17 at 23:07
• In that case, what you're doing is literally method of images. – Christopher A. Wong Mar 3 '17 at 23:11