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

Sergey A. Nazarov; Jari Taskinen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 5, page 947-979
  • ISSN: 0764-583X

Abstract

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We study the linearized water-wave problem in a bounded domain (e.g.a finite pond of water) of 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 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 𝒪 ) waves and the unitary scattering matrix are introduced.

How to cite

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Nazarov, Sergey A., and Taskinen, Jari. "Radiation conditions at the top of a rotational cusp in the theory of water-waves." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.5 (2011): 947-979. <http://eudml.org/doc/273340>.

@article{Nazarov2011,
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.},
author = {Nazarov, Sergey A., Taskinen, Jari},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {linear water-wave problem; cuspidal domain; radiation condition; scattering matrix},
language = {eng},
number = {5},
pages = {947-979},
publisher = {EDP-Sciences},
title = {Radiation conditions at the top of a rotational cusp in the theory of water-waves},
url = {http://eudml.org/doc/273340},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Nazarov, Sergey A.
AU - Taskinen, Jari
TI - Radiation conditions at the top of a rotational cusp in the theory of water-waves
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 5
SP - 947
EP - 979
AB - 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.
LA - eng
KW - linear water-wave problem; cuspidal domain; radiation condition; scattering matrix
UR - http://eudml.org/doc/273340
ER -

References

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