Wave scattering by a horizontal circular cylinder in a three-layer fluid

Using linear water-wave theory, wave scattering by a horizontal circular cylinder submerged in a three-layer ocean consisting of a layer of finite depth bounded above by finite depth water with free surface and below by an infinite layer of fluid of greater density is considered. The cylinder is submerged in either of the three layers. In such a situation time-harmonic waves with given frequency can propagate with three different wave numbers. Employing the method of multipoles the problem is reduced to an infinite system of linear equations which are solved numerically by standard technique after truncation. The transmission and reflection coefficients are obtained and depicted graphically against the wave number for all cases. In a two-layer fluid there are energy identities that exist connecting the transmission and reflection coefficients that arise. These energy identities are systematically extended to the three-fluid cases which are obtained