A new set of Boussinesq equations in potential form is derived for wave propagation in water of varying depth by using a Taylor expansion procedure. The equations are expressed by the wave surface elevations and the velocity potentials at the still water surface, which is easy for the discitization of finite element methods for curved boundaries. The equations have been used to study the interaction between waves and a circular cylinder. Computation shows that the wave run-up around the cylinder from the present method is much greater than that from the linear wave theory, and close to experimental data; the wave forces from the present method and the linear theory are close to each other and both of them are close to experimental data.
Waves are characterized by their steepness and the water depth in which they are propagating. For simulating the wave propagation in shallow water, Boussinesq equations have been proposed. The standard Boussinesq equations derived by Peregrine (1967) can only be used in shallow water, as the error of its dispersion relation increases with the increase of water depth. Later a number of modified Boussinesq equations are proposed with more accurate dispersion relation in deeper water (Witting, 1984; Madsen and Sørensen, 1992; Nwogu, 1993; Beji and Nadaoka, 1996). The Boussinesq-type equations are usually solved by finite difference methods (FDM). By this method, curved boundaries are usually approximated by stepped boundaries. For wave field not very close to a boundary, good approximation can be achieved. For wave action on structures and wave run-up around structures, the boundaries around bodies have to be discretized accurately. Li, et al. (1999) used a finite element method (FEM) to discretize the mesh around structures and solved the Boussinesq equations.