Wave resonances and the time-dependent capillary gravity wave motion

The interconnection of time and frequency domains for capillary gravity wave motion in the presence of current is discussed in this article. The general time-dependent problem is solved using Green’s function technique, and the asymptotic solution is derived using the method of stationary phase for large time and space. Also, the frequency domain solution is derived as a special case using the Cauchy Residue theorem. Different types of wave resonances like Trapping, Blocking and Bragg resonances are discussed. The existence of the trapped mode below the cutoff frequency is justified theoretically, and numerical results are obtained using the multipole expansion method. The blocking and Bragg resonances are analyzed above the cutoff frequency. It is found that in the presence of current, when the ripple wavenumber of the bottom undulation equals twice the cosine angle of incidence of wave times the wavenumber of the wave, Bragg resonance occurs. It is found that three propagating modes exist in the case of wave blocking, and the trapped modes exist only for the first propagating mode. Furthermore, because of the negative group velocity inside the blocking zone, the Bragg reflection increases while decreasing outside. The effect of current on the wave energy propagation in the form of group velocity is analyzed and the same is verified in the case of time-dependent problem.