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On step approximations for water-wave problems. (English) Zbl 0832.76007

Authors discuss the scattering of two-dimensional linear water waves by a varying bottom topography. A new method is developed for the determination of the reflection and transmission coefficients for the two-dimensional problem of the scattering of an incident wave of small amplitude. It is shown that the wave problem can be reduced to a problem in a uniform strip with a variable free surface condition. The new problem is then discretized, and the transition matrix formulation is made which is used to relate the wave amplitudes at \(\pm \infty\).
The authors claim that this new method is applicable to a wide class of problems when the domain can be discretized into regions each supporting waves of a different wave number. The method is verified with examples for which the solution is known. Results indicate that the present method is a simple accurate technique for different scattering wave problems. In the opinion of the reviewer, this is a very interesting and important contribution to the theory of approximations for water wave problems. This work seems to be useful for other wave problems.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
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