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I was given the following wave function, related to an unspecified drum:

$$\frac{\partial ^2Ψ}{\partial t^2}=c^2 \left ( \frac{\partial ^2Ψ}{\partial x^2} + \frac{\partial ^2Ψ}{\partial y^2} \right )-\gammaΨ.$$

I was asked to find an expression for the frequency, which led me to: $$ω=\sqrt{c^2(k_x^2+k_y^2)-γ}$$

I was then asked for the minimal frequency of the drum.

Now here is my question: had there been any information as to the shape of the drum (we have studied rectangular, round and square drums), I would know how to proceed. But this information is purposefully left out, and I have no idea how to approach the question without it. How can one come up with the minimum frequency without the basic geometric data + boundary conditions leading to the normal modes, substituting for $k_x$ and $k_y$ accordingly?

Any insights are very much appreciated!

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  • $\begingroup$ Hint: the general solution must include all harmonics indicated with $\omega_n$ and the question is asking about $\omega_1$. $\endgroup$
    – John Alexiou
    Commented Dec 8, 2021 at 23:49
  • $\begingroup$ Hi John! Sure, but I don't see how I can avoid having to define $k_x$ or $ky$ via boundary conditions, which purposefully aren't given. I mean, otherwise how do you get to $ω1$? $\endgroup$
    – dalta
    Commented Dec 9, 2021 at 6:34
  • $\begingroup$ Oh yeah, $k_{ii}$ need to be quantized to the BC as only integer multiples fit on the drum. $\endgroup$
    – John Alexiou
    Commented Dec 9, 2021 at 13:05
  • $\begingroup$ so how do you even begin to analyse that if the shape of the drum isn't given? $\endgroup$
    – dalta
    Commented Dec 9, 2021 at 15:18
  • $\begingroup$ IDK. There is a minimum tension that keeps the square root real though. $\endgroup$
    – John Alexiou
    Commented Dec 9, 2021 at 15:46

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