Trapped modes in a two-layer fluid of finite depth bounded above by a rigid lid

Based on linear water wave theory, we investigate the existence of trapped modes supported by a submerged horizontal circular cylinder in a two-layer fluid of finite depth bounded above by a rigid lid and below by an impermeable horizontal bottom. The effect of surface tension at the surface of separation is neglected. Under such a situation time-harmonic waves propagate with one wavenumber only, unlike the case where the upper layer has a free surface and the waves propagate with two wavenumbers. Using a multipole expansion method, we obtain a homogeneous infinite system of linear equations. For a fixed geometrical configuration, we numerically compute the values of those frequencies for which the values of the truncated determinant become approximately zero, which confirms the existence of trapped modes. We plot the dispersion curves and observe that trapped mode frequencies always exist below a cut-off value. Trapped mode frequencies are plotted against various values of the density ratios for different depths of either of the layers. We also observe the effect of submergence depth on trapped mode frequencies. We find that only a single mode exists for a depth of submergence of the cylinder greater than about 1.05 times its radius. As the depth of submergence decreases, further trapped modes appear. The existence of trapped modes shows that, in general, a radiation condition for the waves at infinity is insufficient for the uniqueness of the scattering problem.