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

$$\frac{\partial ^2Ψ}{\partial t^2}=c^2\frac{\partial ^2Ψ}{\partial x^2}\frac{\partial ^2Ψ}{\partial y^2}-\gammaΨ.$$

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!

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!

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

$\frac{\partial ^2Ψ}{\partial t^2}=c^2\frac{\partial ^2Ψ}{\partial x^2}\frac{\partial ^2Ψ}{\partial y^2}-\gammaΨ$

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!

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

$$\frac{\partial ^2Ψ}{\partial t^2}=c^2\frac{\partial ^2Ψ}{\partial x^2}\frac{\partial ^2Ψ}{\partial y^2}-\gammaΨ.$$

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!