The sound of drums that tile the plane

The vibration of a thin membrane is often modeled by the PDE

u + u = 0

whereu is the height of the membrane and is the Laplacian. Solutions only exist for certain values of , the eigenvalues of . You could think of u as giving the height of a vibrating drum head and the s as frequencies of vibration.

The s depend on the shape of the drum headD 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 Polya [1] gives lower bounds on the s under Dirichlet boundary conditions and upper bounds on the s under Neumann boundary conditions. His theorems require thatD 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

_k >= 4k / A.

Under Neumann boundary conditions, the kth eigenvalue is bounded above by

_k 4(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. Polya. On the Eigenvalues of Vibrating Membranes. Proceedings of the London Mathematical Society, Volume s3-11, Issue 1, 1961, Pages 419-433.

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