Water wave diffraction and radiation by a submerged sphere in front of a vertical wall

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.