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I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched to solve this uses separation of variables to generate two ODE's, and this is referred to as the "eigenvalue problem". My question is, is there any other way to do this that does not use separation of variables? I am looking for a numerical method.

I want to find approximate eigenvalues using a numerical method (like finite differences in 2-dimensions) but not sure if it is possible.

Any information would be appreciated

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You can of course solve the partial differential eigenvalue problem that describes the shape of the eigenmode numerically, using finite differences of finite elements. For the square and the circle that is pointless since the solution is well known, but it allows you to generalize the approach to other domains.

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