Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations

We give a complete proof of the existence of eigenmodes for a vibrating elliptic membrane: for each pair (m, n) ∈ {0, 1, 2, . . .} there exists a unique even eigenmode with m ellipses and n hyperbola branches as nodal curves and, similarly, for each (m, n) ∈ {0, 1, 2, . . .} × {1, 2, . . .} there exists a unique odd eigenmode with m ellipses and n hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters a and q cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of the modes of the elliptic membrane is presented and also approximate formulae which explain rather well the qualitative asymptotic behaviour of the eigenfrequencies.