Surface Capillary-Gravity Short-Crested Waves with a Current in Water of Finite Depth

Abstract

Firstly, it is found that the third-order, or most probably higher-order, solution for the standing waves and short-crested waves of finite amplitude in water of finite depth cannot completely match the required amplitude equations. The solution to this incompatibility, involving both the amplitude equations and, probably, the other required phase equations for higher-order solution, is found entirely. Secondly, a linear dynamical system of surface capillary-gravity short-crested waves is developed by including a uniform current, thus leading to analytical expressions for the kinematic and dynamic variables in which a number of the classical, typical and latest special wave cases are involved. Thirdly, a second-order solution for surface capillary-gravity short-crested waves with a uniform current in finite depth is presented, depicting a series of typical three-dimensional wave patterns for comparison, concerning currents, shallow and deep water waves, and surface capillary waves. Fourthly, a third-order solution for surface gravity short-crested waves with a uniform current in finite depth is obtained, giving exhaustively explicit formulas for the wave force and moment exerted on a vertical breakwater, and showing a number of surprising features on the maximum load varying respectively with wave phase and incident angle. Finally, a third-order solution for surface capillary-gravity short-crested waves in finite depth is derived with graphs, showing the surface profiles, patterns and their projections, and the locations of the zeros and poles of the third-order angular frequency involving the second- and third-order surface elevations and velocity potentials.