The nonlinear interactions between long surface waves and interfacial waves in a two-layer fluid are studied theoretically. The fluid is density-stratified and the thicknesses of the top and bottom layers are both assumed to be shallow relative to the length of a typical surface wave and interfacial wave, respectively. A set of Boussinesq-type equations are derived for potential flow in this system. The equations are then analyzed for the dynamics of the nonlinear resonant interactions between a monochromatic surface wave and two oblique interfacial waves. The analysis uses a second order perturbation approach. Consequently, a set of coupled transient evolution equations of wave amplitudes is derived. Moreover, the effect of weak viscosity of the lower layer is incorporated in the problem and the influences of important parameters on surface and interfacial wave evolution (namely the directional angle of interfacial waves, density ratio of the layers, thickness of the fluid layers, surface wave frequency, surface wave amplitude, and lower layer viscosity) are investigated. The results of the parametric study are discussed and are generally in qualitative agreement with previous studies. In shallow water, a triad formed of surface waves (or interfacial waves) can be considered in near-resonant interaction. In contrast to the previous studies which limited the study to a triad (one surface wave and two interfacial waves or one interfacial and two surface waves), the problem is generalized by considering the nonlinear interactions between a triad of surface waves and three oblique pairs of interfacial waves. In this system, each surface wave is in near-resonance interaction with other surface waves and in exact resonance with a pair of oblique interfacial waves. Similarly, each interfacial wave is in near-resonance interaction with other interfacial waves which are propagating in the same direction. Inclusion of all the interactions considerably changes the pattern of evolution of waves and highlights the necessity of accounting for several wave harmonics. Effects of density ratio, depth ratio, and surface wave frequency on the evolution of waves are discussed. Finally, a formulation is derived for spatial evolution of one surface wave spectrum in nonlinear interaction with two oblique interfacial wave spectra. The two-layer Boussinesq-type equations are treated in frequency domain to study the nonlinear interactions of time-harmonic waves. Based on weakly two-dimensional propagation of each wave train, a parabolic approximation is applied to derive the formulation.
The nonlinear interactions between long surface waves and interfacial waves in a two-layer fluid are studied theoretically. The fluid is density-stratified and the thicknesses of the top and bottom layers are both assumed to be shallow relative to the length of a typical surface wave and interfacial wave, respectively. A set of Boussinesq-type equations are derived for potential flow in this system. The equations are then analyzed for the dynamics of the nonlinear resonant interactions between a monochromatic surface wave and two oblique interfacial waves. The analysis uses a second order perturbation approach. Consequently, a set of coupled transient evolution equations of wave amplitudes is derived. Moreover, the effect of weak viscosity of the lower layer is incorporated in the problem and the influences of important parameters on surface and interfacial wave evolution (namely the directional angle of interfacial waves, density ratio of the layers, thickness of the fluid layers, surface wave frequency, surface wave amplitude, and lower layer viscosity) are investigated. The results of the parametric study are discussed and are generally in qualitative agreement with previous studies.
In shallow water, a triad formed of surface waves (or interfacial waves) can be considered in near-resonant interaction. In contrast to the previous studies which limited the study to a triad (one surface wave and two interfacial waves or one interfacial and two surface waves), the problem is generalized by considering the nonlinear interactions between a triad of surface waves and three oblique pairs of interfacial waves. In this system, each surface wave is in near-resonance interaction with other surface waves and in exact resonance with a pair of oblique interfacial waves. Similarly, each interfacial wave is in near-resonance interaction with other interfacial waves which are propagating in the same direction. Inclusion of all the interactions considerably changes the pattern of evolution of waves and highlights the necessity of accounting for several wave harmonics. Effects of density ratio, depth ratio, and surface wave frequency on the evolution of waves are discussed.
Finally, a formulation is derived for spatial evolution of one surface wave spectrum in nonlinear interaction with two oblique interfacial wave spectra. The two-layer Boussinesq-type equations are treated in frequency domain to study the nonlinear interactions of time-harmonic waves. Based on weakly two-dimensional propagation of each wave train, a parabolic approximation is applied to derive the formulation.