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A Sufficient Condition for the Existence of Trapped Modes for Oblique Waves in a Two-Layer Fluid

Nazarov, Sergey A.; Videman, J. H.

Proceedings of the Royal Society A-Mathematical Physical and Engineering, 465(2112) (2009), 3799-3816

The interaction of linear water waves with totally or partially submerged obstacles is considered in a two-layer fluid consisting of two immiscible liquid layers of different densities. A sufficient condition for the existence of trapped modes is established by introducing a trace operator that restricts the solutions to the free surface and the interface. The modes correspond to localized solutions of a spectral problem, decaying at large distances from the obstacles and belonging to the discrete spectrum below a positive cut-off value of the continuous spectrum. The sufficient condition is a simple relation between the cut-off 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 and the interface pierced by the obstacles.