ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - September 2011 - Volume 45, Issue 05  

Research Article

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

Nazarov, Sergey A.a1 and Taskinen, Jaria2

a1 Institute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178, St. Petersburg, Russia.

a2 University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 Helsinki, Finland. taskinen@cc.helsenki.fi

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.

(Received January 11 2010)

(Revised September 30 2010)

(Online publication May 27 2011)

Key Words:

  • Linear water-wave problem;
  • cuspidal domain;
  • radiation condition;
  • scattering matrix

Mathematics Subject Classification:

  • 76B15;
  • 35J25;
  • 35P99