Analytical Solution for Bichromatic Waves on Linearly Varying Currents

It is well-known that the bound long waves play an important role on beach morphology. The existence of the current will modify the intensity of the waves. To examine the more realistic problem where the current is non-uniform, a third-order analytical solution for bichromatic waves on currents with constant vorticity is derived using a perturbation method. Earlier derivations of theories for interactions between waves and currents have been limited to cases of bichromatic waves on depth-uniform currents and cases of monochromatic waves on linear shear currents. Unlike the derivations of monochromatic waves, the moving-frame method cannot be used in the case of bichromatic waves because there are multiple waves with different celerities, and the flow can in no way be treated as steady-state. Moreover, for shear currents where the flows are rotational, the velocity potential cannot be simply defined. For these reasons, it is difficult to carry out mathematical operations when the dynamic free surface boundary condition is applied. However, with the consideration of the wave part of the fluid motion remaining irrotational, some of the terms in the expanded boundary conditions can be ignored; thus, the derivations can be further processed. As a result, the third-order explicit expressions for the stream function, the velocity potential, and the surface elevation can be solved. The nonlinear dispersion relation is also derived to account for interacting wave components with different frequencies and amplitudes which can be used to solved for wavenumbers of both wave trains. The obtained solutions are verified by reducing to those of previous results for monochromatic waves and uniform currents. Furthermore, the influence of constant vorticity on the wave kinematics is illustrated. A comparison between following/opposing currents is also carried out. Finally, the effects of the shear current on the strength of the bound long waves are also illustrated.