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Surface waves in basins of variable depth. (English) Zbl 0588.76021

The linearized boundary-value problem for surface waves of frequency \(\omega\) in a closed basin of variable depth is reduced to a non-self- adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue \(\kappa =\omega^ 2/g\) is constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to \(\kappa)\), also is constructed; it offers significant advantages vis- à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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