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

We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ 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 these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.