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This paper studies water wave diffraction and radiation by a submerged sphere in front of a vertical wall. Based on the image principle, the physical problem is transformed into an imaginary problem of two spheres, which are symmetric with respect to the original vertical wall, in a horizontally open water domain. Then, an analytical solution for the imaginary problem is developed in the context of linear wave theory. The velocity potential of fluid motion is obtained by adopting the method of multipole expansions combined with the shift of local spherical coordinate systems. The wave exciting forces in surge, sway and heave directions, and added mass and radiation damping due to the sphere oscillation are calculated. The convergence of the series solution is very rapid, and the results of analytical solution and numerical solution based on boundary element method are highly consistent. Calculation examples are presented to examine the effects of wave/oscillation frequency, spacing between sphere and wall, submergence depth and wave incident angle on various hydrodynamic quantities of the sphere. Results show that because the submerged sphere is subjected to not only incident/radiated waves but also reflected waves from the wall, the hydrodynamic quantities noticeably differ from those in an open water domain.

This study develops new analytical solutions to oblique wave scattering by submerged porous/perforated structures on the basis of linear potential theory. Two cases of a porous (rubble mound) breakwater and a horizontal perforated plate are considered. The new solutions have the novelty of using a contour integral technique to avoid finding the complex roots (wave numbers) of complex dispersion equations for water wave motion over porous/perforated structures. In the solution procedure, the linear equation systems for the expansion coefficients in velocity potentials are obtained by using the conventional mode-matching method. However, the matrix elements involving the complex wave numbers are recast by using the contour integral technique, and then the expansion coefficients are determined without knowing the explicit knowledge of complex wave numbers. As a result, the difficulties arising in solving the complex dispersion equations in the traditional solutions are avoided completely. The calculation results of the new solutions agree well with known results of analytical approaches with complex wave numbers, analytical solutions based on the velocity potential decomposition method, and numerical solutions based on the multi-domain boundary element method. This study gives a simple and elegant procedure to tackle water wave interactions with submerged porous/perforated structures, and the solutions can be used as a reliable alternative tool for fast engineering preliminary analysis.

This article develops novel analytical solutions to oblique wave scattering by porous rubble-mound structures based on linear potential theory. The novelty of the solutions is that the usual complex root finding algorithm is avoided by applying a contour integral technique. In the solving procedure, the whole fluid domain is divided into multiple regions, and the series solution of the velocity potential in each region is obtained using the separation of variables. A linear equation system for the expansion coefficients is achieved by matching the interface conditions between adjacent regions. The matrix elements, which initially include the complex roots of complex dispersion relations, are evaluated by applying the contour integral technique. The evaluation process is free of complex roots, and thus the difficulties arising in solving the complex dispersion relations in traditional solutions are avoided. Case studies show that the series solution converge rapidly, and the calculation results of the new solutions agree well with those of previous analytical solutions with complex roots and multi-domain boundary element method (BEM) solutions. Besides providing a simple and elegant approach for analyzing the hydrodynamic performance of porous structures, the new solutions can be extended to water wave problems involving other dissipative coastal structures.

Water wave interaction with a submerged perforated quarter-circular caisson breakwater is investigated in this study. Based on linear potential theory, analytical solutions of the problem are developed for normally and obliquely incident waves, respectively. In the solving procedure, the entire fluid domain is divided into an external region and two quarter-circular inner regions. The series solutions of the velocity potential in the external region are given through multipole expansions, and those in the quarter-circular regions are developed by the separation of variables. A quadratic pressure drop condition is imposed on the perforated quarter-circular arc to model the wave energy dissipation. The expansion coefficients in velocity potentials are determined by matching the conditions on the boundaries between adjacent regions. The calculation methods of the reflection, transmission and energy loss coefficients as well as the horizontal and vertical wave forces on the breakwater are given. Laboratory experiments are carried out to measure the reflection and transmission coefficients of the breakwater. The convergence of the analytical solutions is good, and the calculation results with at least three-figure effective accuracy are obtained. The analytical results agree reasonably with the experimental data. The hydrodynamic quantities of the breakwater are carefully examined by presenting calculation examples. The analysis results can provide valuable references for the application of perforated quarter-circular caisson breakwaters.