© 2002 by Oxford University Press
Radiation Conditions for Rough Surfaces in Linear Elasticity
( 1 Division of Mathematics, Polytechnic School, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece 2 Naval Academy of Greece, Hatzikiriakio, Pireous, Greece 3 Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece )
The elastic scattering problem, as with every exterior boundary-value problem, must be accompanied by suitable radiation conditions, which reflect the asymptotic behaviour of the underlying fields. For bounded scatterers, the well-known Kupradze radiation conditions encode the outgoing character of the elastic waves. For infinite scatterers the situation changes drastically. The boundary of the problem is now in the neighbourhood of the asymptotic region and Kupradze's conditions are proved inadequate. In this work the radiation conditions for the elastic scattering by stress-free rough surfaces are constructed. The elastic scattered field is expressed through the angular spectrum representation, which contains two vector spectral amplitudes expressing the two propagating waves in elasticity. In a properly selected region, the different types of integrals emerging from the application of Betti's formula are estimated in the asymptotic realm, using the stationary phase method. Finally, the integral representation of the problem, incorporating the aforementioned radiation conditions, is constructed.
Received 12 April 2000. Revised 28 February and 14 December 2001.