Time-dependent capillary-gravity waves in the presence of current

Combined effects of current and surface tension on time-dependent transient boundary value problems (BVPs) associated with capillary-gravity wave motion in a two-layer fluid are analyzed under the assumption of small amplitude water wave theory in three dimensions. Plane wave solutions are discussed by analyzing the roots of the dispersion relation and phase velocity. Using the Laplace transform technique and Green's function method, the time-dependent initial boundary value problem (IBVP) for the capillary-gravity wave motion are analyzed in both the cases of finite and infinite water depths. Using the method of stationary phase, the time-dependent Green's function and surface and interface profiles are computed and analyzed for various values of physical parameters in two different cases. The time harmonic solutions are obtained as one of the special cases of the time-dependent IBVP in both the cases of finite and infinite water depths. It is observed that the direction of current and wave changes the phase velocity of the capillary-gravity wave motion along with the free surface and interface elevations. The position of the source significantly changes the amplitude of the source potentials in surface and internal modes. Further, the location of the interface has a significant influence on the pattern of surface and interface elevations.