Calculating vibrational mode coefficients of a drum for an arbitrary excitation

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.

• If $\omega_{min}=\omega_{01}=2 \text{rad/s}$, how can I make these discrete frequency components add up to $f(t)=e^{i \omega_{forced} t}$ where $\omega_{forced}=\sqrt2 \text{rad/s}$? $e^{i\omega t}$ are orthogonal for different $\omega$. I don't think I can add up bunch discrete of $e^{i\omega t}$ and end up with an $f(t)=e^{i \omega_{forced} t}$ that is orthogonal to every frequency component we just added up. Commented Jun 6, 2023 at 20:34