 09174 Armando G. M. Neves
 Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations
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Sep 24, 09

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Abstract. We give a complete proof of the existence of eigenmodes for a vibrating
elliptic membrane: for each pair $(m,n) \in \{0,1,2, \dots\}^2$ there
exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola
branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2,
\dots\}\times \{1,2, \dots\}$ 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.
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