I am interested in the modes of a rectangular drum of size WxL that is fixed on 4 edges. The differential equation that governs the vibrations of a 2D membrane is $\nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = 0 $. Coordinate origin is on one of the corners of the rectangle. The modes are proportional to
$$\phi_{mn}(x,y)=\sin(\frac{\pi x m}{W})\sin(\frac{\pi y m}{L})$$
For mathematical simplicity, I will assume time dependency to be a complex exponential.
$$\phi_{mn}(x,y,t)=\sin(\frac{\pi x m}{W})\sin(\frac{\pi y m}{L})e^{i \omega_{mn} t}$$
$$(\frac{\omega_{mn}}{c})^2=(\frac{\pi m}W)^2+(\frac{\pi n}L)^2$$
I think following summation should be able to account for all physically meaningful solutions to the problem. (Please let me know if I am wrong, or if there is more to know.)
$$\phi(x,y,t)=\sum_m\sum_n c_{mn}\sin(\frac{\pi x m}{W})\sin(\frac{\pi y m}{L})e^{i \omega_{mn} t}$$
I am confused about what to do if we force an excitation $f(t)$ at the point $(x',y')$. I think the following summation should give the expansion
$$f(t)=\sum_m \sum_n c_{mn}\sin(\frac{\pi x' m}{W})\sin(\frac{\pi y' m}{L})e^{i \omega_{mn} t}$$
However, I can't wrap my head around how this expression can account for all possible $f(t)$. What I recall from Fourier transform is that an arbitrary function $f(t)$ requires continuous frequency components (ie $\omega \in \mathbb R$). Since the values of $\omega_{mn}$ are discrete, I don't see how this can be the general solution. Yet, it doesn't feel like it would violate any law of physics to force an arbitrary $f(t)$ at the point $(x',y')$.
One possibility is that instead of enforcing the solution equals $f(t)$ at $(x',y')$, I could put $f(t)$ as a forcing term in the equation $\nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = \delta(x-x')\delta(y-y')f(t) $. The more I think about it, the more reasonable this approach seems. I am not sure how to solve this either.
In summary, the question is: Solve $\nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = 0 $ for a rectangular region of size WxL that is fixed ($\phi=0$) at 4 edges and is "excited" by the arbitrary function $f(t)$ at the point $(x',y')$.
Also please let me know if I likely have misunderstood something about the subject and which detailed source I should read. Thanks.
