Analytical solutions of first-mode sloshing in new containers

Three new families of vessel geometries for the fundamental mode of two-dimensional sloshing are reported. These families, here denoted C1, C2 and C3, are parameterized by separation constant B. Containers C1 exist for 1≤B≤∞, containers C2 exist for 0≤B≤1, and containers C3 exist for 0≤B≤∞. The container shapes and their depth-dependent frequencies are found in analytical form. There is an affinity between the C1 and C3 families in that their limiting container profiles, found as B→∞, take the shape of identical isochronous containers. Also, there is an affinity between the C1 and C2 families in that their limiting B=1 shapes asymptotically tend to a 90∘ wedge at low depth. The shapes of the containers are compared with the classic geometries for a rectangular box, a 90∘ wedge, and the planar isochronous container. The variation of sloshing frequencies with liquid depth is also compared with those of the three classic geometries.