A phase function for finding normal mode eigenvalues over a layered elastic bottom

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A method is described for the determination of normal mode eigenvalues in shallow water over a layered elastic bottom. The method avoids the usual search for eigenvalues in the complex plane by deriving a complex phase function which is a multiple of π at the normal mode eigenvalues. The path in the complex plane along which the phase function is real gives a simple curve which passes systematically through all the eigenvalues including those for the leaky modes. Along this path the phase function is real and monotonic. The eigenvalues can be found by following this path.