# Finding the frequencies of vibration of a drum; PDE

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 want to find approximate eigenvalues using a numerical method (like power method) but not sure if it is possible.

Any information would be appreciated

• Once you are familiar with the method of separation of variable, find the frequencies of vibration of a circular and square drum is straightforward. In both cases you have to solve the wave equation with Dirichlet's boundary conditions. With this type of boundary conditions, there are some theorems that give the eigenfunctions of the laplacian. Then you have to integrate w.r.t time and you will get your frequencies. – Joelafrite May 31 '15 at 2:06
• Thanks. Is it possible to get the laplacian matrix and find the eigenvalues that way? This would be ideal, but I'm not sure how to construct that matrix (or even if it's possible) – user444444242 May 31 '15 at 3:03
• The method of separation of variables let you to find the eigenfunctions and eigenvalues of the laplacian. Here the laplacian is a hermitian operator onto a Hilbert space (the space of eigenfunctions) and can be viewed as a matrix in an infinite dimensional vector space. See this link to solve the wave equation analytically. – Joelafrite May 31 '15 at 3:20

## 1 Answer

If you are looking for an analytic solution (i.e., some formula for eigenvalues), then separation of variables is the way to go. It will work only in domains of special geometry, which fortunately include disks and squares. For an arbitrary domain, there is no practical analytic way to get the eigenvalues, and one uses numerical methods instead.

Since a numerical method is what you want, I recommend the site Computational Science, and especially the questions Library to compute eigenvalues of the Laplace operator in a polyhedral domain and Solving PDE or eigenvalue problems without FEM.

• Thank you for the websites – user444444242 May 31 '15 at 22:13