### ASA 124th Meeting New Orleans 1992 October

## 3aUW5. The existence of generalized eigenfunctions and multiple
eigenvalues in underwater acoustics.

**Richard B. Evans
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*Sci. Appl. Internatl. Corp., New London, CT 06320
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An example is given of a horizontally stratified acoustic waveguide that
supports generalized eigenfunctions in addition to the usual eigenfunctions or
normal modes. Generalized eigenfunctions occur when the characteristic equation
has a zero with multiplicity greater than one. Both eigenfunctions and
generalized eigenfunctions are required to provide a complete representation of
a wide class of functions. The separation of variables solution for a point
source in a waveguide, based only on the usual eigenfunctions or normal modes,
is not valid for the example presented here. The correct solution, based on a
combination of eigenfunctions and generalized eigenfunctions, contains modes
that are simultaneously exponentially damped and algebraically amplified in the
direction of propagation.