Juha Videman
Departamento de Matemática
Instituto Superior Técnico
Abstract
The interaction of linear water waves with totally or partially submerged obstacles is considered in a homogeneous, inviscid, incompressible liquid. A sufficient condition for the existence of localized trapped modes is established by introducing a trace operator that restricts the solutions to the free surface. The modes correspond to localised solutions of a spectral problem, decaying at large distances from the obstacles and belonging to the discrete spectrum below a positive cut-o ff value of the continuous spectrum.
The sufficient condition is a simple relation between the cut-of f value and some geometrical constants, namely the surface integrals taken over the cross-sections of the submerged parts of the obstacles and the line integrals along the parts of the free surface pierced by the obstacles.
Several particular cases are considered and the results are extended to a two-layer fluid consisting of two immiscible liquid layers of diff erent densities and to the problem of existence of Rayleigh-Bloch surface waves traveling along a periodic family of obstacles and edge waves guided by, and propagating along, a periodic seashore.
A new and simple proof for the comparison principle, a method often used in proving existence of trapped modes, is also presented.
Jornada SPM/CIM/CMAT - "Dia das Equações"