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Radiation conditions at the top of a rotationalcusp in the theory of water-waves

Published online by Cambridge University Press:  27 May 2011

Sergey A. Nazarov  and
Jari Taskinen
Sergey A. Nazarov
Affiliation:
Institute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178, St. Petersburg, Russia.
Jari Taskinen
Affiliation:
University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 Helsinki, Finland. taskinen@cc.helsenki.fi
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Abstract

We study the linearized water-wave problem in a bounded domain (e.g. a finite pond of water) of ${\mathbb R}^3$, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point ${\mathcal O}$ of the water surface, where a submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement that the linear operator associated to the problem be Fredholm of index zero in relevant weighted function spaces with separated asymptotics. The classification of incoming and outgoing (seen from ${\mathcal O}$) waves and the unitary scattering matrix are introduced.

Keywords

Linear water-wave problemcuspidal domainradiation conditionscattering matrix
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 947 - 979
Copyright
© EDP Sciences, SMAI, 2011

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