The Existence of Generalized Eigenfunctions in Underwater Acoustics
NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON CT
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Non-self-adjoint problems occur in underwater acoustics when the square of the wavenumber is given a complex value to allow for volume attenuation. This report shows that complex wavenumbers can give rise to multiple eigenvalues and generalized eigenfunctions in the same way as complex impedance and admittance boundary conditions give rise to multiple eigenvalues. 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 a multiplicity greater than one. Both eigenfunctions and generalized eigenfunctions are required to provide a complete representation of some functions. The separation of variables solution for a point source in a waveguide, based on the usual eigenfunctions or normal modes, is not valid for this example.
- Theoretical Mathematics