The complicated phenomena of refraction and diffraction of wave with the presence of currents were studied numerically. A set of governing equations was derived based on dynamic boundary condition with variation method. We can get different physical equations according to different water depth distribution assumptions. For example, we know the water depth distribution in Airy wave. If f is equal to 1, we can get shallow-water wave equation. In this paper we only checked the shallow-water wave equation including current effects. To validate the governing equation, the conditions of incident wave and current were used to calculate the wave height distributions around a circular island numerically. It was found that numerical results are in good agreement.
As waves enter into open shallow water, shoaling and refraction occur due to water depth variations. When wave propagation encounters an obstruction such as a coastal structure, reflection and diffraction will occur in front of and behind the obstruction. The presence of various currents in the ocean makes wave motion much more complicated. The current effects must be considered when treating nearshore problem. Various researchers have striven for the prediction of nearshore wave phenomena for several decades. Some past investigations are shown below. Homma (1950) firstly used shallow-water wave equation to solve wave height distribution around a circular island analytically. Vanstano & Reid (1967) proved Homma's (1950) analytic solutions using the finite-difference calculations. Berkhoff (1972) used perturbation method to derive a mild-slope equation, which can describe combined effect of wave refraction and diffraction. Smith & Sprinks (1975) also derived a mild-slope equation from classical linear theory of water waves, which approximately described the propagation of periodic surface waves in water of slowly varying depth.
