A large class of problems in the field of fluid–structure interaction involves higher-order boundary conditions for the governing partial differential equation and the eigenfunctions associated with these problems are not orthogonal in the usual sense. In the present study, mode-coupling relations are derived by utilizing the Fourier integral theorem for the solutions of the Laplace equation with higher-order derivatives in the boundary conditions in both the cases of a semi-infinite strip and a semi-infinite domain in two dimensions. The expansion for the velocity potential is derived in terms of the corresponding eigenfunctions of the boundary-value problem. Utilizing such an expansion of the velocity potential, the symmetric wave source potentials or the so-called Green's function for the boundary-value problem of the flexural gravity wave maker is derived. Alternatively, utilizing the integral form of the wave source potential, the expansion formulae for the velocity potentials are recovered, which justifies the completeness of the eigenfunctions involved. As an application of the wave maker problem, oblique water wave scattering caused by cracks in a floating ice-sheet is analysed in the case of infinite depth.

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a b s t r a c tThe effect on the efficiency of the device of implementing a dual-chamber oscillating water column (OWC) placed over stepped bottom is analysed. The mathematical problem is formulated in the two dimensional Cartesian coordinate system under the linear water wave theory. Two different mathematical approaches are adopted to solve the associated boundary value problem (BVP), one is the method of matched eigenfunction expansion and the other is the boundary integral equation method (BIEM). The numerical results show good agreement with the analytical results. The performance of the proposed device is analysed and compared with the typical single chamber OWCs with/without step and with dual-chamber OWCs over uniform bottom profile. The present investigation shows that by considering dual-chamber OWC device on the stepped sea bottom the performance of the device can be improved significantly in wide range of frequencies, as compared with the single chamber case.

The behavior of flexural gravity waves propagating over a semi-infinite floating ice sheet is studied under the assumptions of small amplitude linear wave theory. The vertical wall is assumed to be either fixed or harmonically oscillating with constant horizontal displacement, in which case the problem is analogous with a harmonically oscillating plane vertical wavemaker. The potential flow approach is adhered to and the higher-order mode-coupling relations are applied to determine the unknown coefficients present in the Fourier expansion formula of the potential functions. The ice sheet is modeled as a thin semi-infinite elastic beam. Three different edge conditions are applied at the finite edge of the floating ice sheet. The effects of different edge conditions, the thickness of the ice sheet and the water depth on the surface strain, the shear force along the ice sheet, the horizontal force on the vertical wall, and the flexural gravity wave profile are analyzed in detail.

Oblique flexural gravity-wave scattering due to an abrupt change in water depth in the presence of a compressive force is investigated based on the linearized water-wave theory in the case of finite water depth and shallow-water approximation. Using the results for a single step, wide-spacing approximation is used to analyze wave transformation by multiple steps and submerged block. An energy relation for oblique flexural gravity-wave scattering due to a change in bottom topography is derived using the argument of wave energy flux and is used to check the accuracy of the computation. The changes in water depth significantly contribute to the change in the scattering coefficients. In the case of oblique wave scattering, critical angles are observed in certain cases. Further, a resonating pattern in the reflection coefficients is observed due to change in the water depth irrespective of the presence of a compressive force in the case of a submerged block.

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