The vibration of a thin membrane is often modeled by the PDE
Δu + λu = 0
where u is the height of the membrane and Δ is the Laplacian. Solutions only exist for certain values of λ, the eigenvalues of Δ. You could think of u as giving the height of a vibrating drum head and the λs as frequencies of vibration.
The λs depend on the shape of the drum head D and the boundary conditions. If we clamp down the drumhead on the rim, i.e. specify that u equals 0 on the boundary ∂D, then we call this Dirichlet boundary conditions. If the drumhead is free to vibrate, i.e. we do not specify the height on ∂D, but we do specify that the membrane is flat on ∂D, i.e. that the normal derivative ∂u/∂n equals 0, then we call this Neumann boundary conditions.
George Pólya [1] gives lower bounds on the λs under Dirichlet boundary conditions and upper bounds on the λs under Neumann boundary conditions. His theorems require that D be bounded and that it is possible to tile the plane with congruent copies of D. For example, D could be a rectangle. Or it could have curved sides, like figure in an Escher drawing.
Let A be the area of D. Under Dirichlet boundary conditions the kth eigenvalue is bounded below by
λk ≥ 4πk / A.
Under Neumann boundary conditions, the kth eigenvalue is bounded above by
λk ≤ 4π(k − 1) / A.
Update: See the next post for how the theorem in this post compares to the special case of a square.
Related posts
- Vibrating circular membranes
- Counterexample to the Dirichlet principle
- Can you hear the shape of a network?
[1] G. Pólya. On the Eigenvalues of Vibrating Membranes. Proceedings of the London Mathematical Society, Volume s3-11, Issue 1, 1961, Pages 419–433.