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Radiation conditions at the top of a rotationalcusp in the theory of water-waves
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- Journal:
- ESAIM: Mathematical Modelling and Numerical Analysis / Volume 45 / Issue 5 / September 2011
- Published online by Cambridge University Press:
- 27 May 2011, pp. 947-979
- Print publication:
- September 2011
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- Article
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We study the linearized water-wave problem in a bounded domain (e.g. afinite pond of water) of ${\mathbb R}^3$
, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered thatin 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, wherea submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement thatthe linear operator associated to the problem be Fredholm of index zeroin 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.
