| Issue |
ESAIM: M2AN
Volume 45, Number 5, September-October 2011
|
|
|---|---|---|
| Page(s) | 947 - 979 | |
| DOI | https://doi.org/10.1051/m2an/2011004 | |
| Published online | 27 May 2011 | |
Radiation conditions at the top of a rotational cusp in the theory of water-waves
1
Institute of Mechanical Engineering Problems,
V.O., Bolshoi pr., 61, 199178, St. Petersburg, Russia.
2
University of Helsinki, Department of
Mathematics and Statistics,
P.O. Box 68, 00014 Helsinki, Finland. taskinen@cc.helsenki.fi
Received:
11
January
2010
Revised:
30
September
2010
We study the linearized water-wave problem in a bounded domain (e.g. a
finite pond of water) of 
, 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 
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 ) waves
and the unitary scattering matrix are introduced.
Mathematics Subject Classification: 76B15 / 35J25 / 35P99
Key words: Linear water-wave problem / cuspidal domain / radiation condition / scattering matrix
© EDP Sciences, SMAI, 2011
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