Trapped flexural waves supported by a pair of identical cylinders in a two-layer fluid

Abstract

Existence of trapped waves in ocean shows the presence of discrete wave frequencies in the continuous spectrum. In the present work, we compute those trapped mode frequencies due to a pair of identical horizontal circular cylinders submerged in one of the layers of a two-layer fluid with a thin ice-cover at the upper layer and an infinite depth for the lower layer. Theory is developed for multiple cylinders but numerical computation is carried out only for a pair of cylinders. Due to a thin ice-cover replacing the free surface, a fifth-order boundary condition is to be considered in the upper layer that makes the problem complex and challenging but more practical. Considering linear water wave theory, the boundary value problem is developed through modified Helmholtz equation and associated conditions. Subsequently, applying multipole expansion method, an infinite system of homogeneous linear equations with complex coefficients is obtained and solved. By fixing the geometrical parameters and density ratio, the trapped mode frequencies are computed numerically by tracing the zeros of the determinant generated from the truncated system of the above mentioned equations. In the first instance, the cylinders are placed in the lower layer and the variation of trapped modes is examined by varying different parameters such as upper layer depth, submergence depth, ice-cover thickness etc. In the second instance, the same conditions and configurations are considered in the upper layer and existence of trapped modes is looked into in a similar manner. For the considered parameter values, the number of trapped modes enclosed in the continuous spectrum decreases corresponding to an increase in the flexural rigidity of the ice-cover. The trapped mode frequency decreases when either the the upper layer depth or the submergence depth increases. Further, corresponding to a small change in the separation parameter, embedded trapped modes are observed to cease to exist for the free surface and also for a very thin ice-cover. Our findings are supported by graphs depicting various modes. Further, comparison of present result with an established result shows excellent agreement.