Scattering and radiation of water waves by a submerged rigid disc in a two-layer fluid

Abstract

Interaction of water waves with a horizontal rigid disc submerged in the lower layer of a two-layer fluid is studied in three dimensions using linear theory. The governing boundary value problem is reduced to a two-dimensional hypersingular integral equation. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind in terms of a newly defined function. The solution to the latter integral equation is used to compute the total scattering cross section and the hydrodynamic force for the scattering problem and the added mass and the damping coefficient for the radiation problem. Haskind relations connecting the solutions of the radiation and the scattering problems are also derived. The effects of variations of the submergence depth of the disc and the depth of the upper layer on different physical quantities are investigated. We observe amplification of the added mass and the damping coefficient, the total scattering cross section and the hydrodynamic force when the disc goes near the interface or when the height of the upper layer decreases. Known results for a horizontal disc submerged in a single-layer fluid of infinite depth are recovered from the present analysis.