Gravity waves oscillations at semicircular and general 2D containers: an efficient computational approach to 2D sloshing problem

We compute the natural frequencies for the oscillations of the free boundary of gravity waves in contact with a solid container. First, we study the case of a semicircular shaped container. We deduce an integrodifferential evolutionary equation for the linearized free boundary and impose pinned-end and free-end boundary conditions. For both cases, the natural oscillations frequencies for the free surfaces are provided and compared with the frequencies in the absence of solid walls. Then, we study the effect of having an underwater rectangle-shaped bottom in a rectangular container and the corresponding frequencies. The method introduced can be applied to arbitrary 2D containers, with all the information on their geometry contained into a matrix (related to the conformal mapping into a half-plane) that appears as a factor in a linear system for the computation of eigenfrequencies.