Behavior of long waves associated with the reflection of shortwave groups due to an impermeable wall is numerically investigated using the Boussinesq equations. The sea bottom is assumed to be horizontal and of variable deptE Two sinusoidal waves of slightly different frequency and their sub-harmonic component are given in various water depths at the incident boundary. The numerical results indicate that the temporally long wave components of the bottom velocity, bottom pressure and surface elevation differ from the spatial ones in their amplitudes. The wave number-frequency spectra are also calculated to examine nonlinear interaction terms between the incident and reflected short-wave components. It will be shown that the temporal components become smaller than the spatial ones due to the nonlinear interaction terms with the increase of relative water depth.
Progressive short waves with a narrow frequency band are known to be accompanied by long set-down waves traveling with the groups. The reflection of wave groups by coastal and offshore structures may generate the long waves locked to the envelopes of the incident and reflected short waves. The long waves induced by the reflection of short-wave groups are, however, not fully understood, although these are closely connected with the problems of wave run-up and slow-drift oscillation of a moored body near reflective boundaries such as seawalls and breakwaters. In this study, the behavior of long waves associated with the reflection of short-wave groups due to an impermeable wall is numerically investigated using the Boussinesq equations. The temporal and spatial long-wave components are computed for the surface elevation, the bottom velocity and the bottom pressure. The standard Boussinesq equations in terms of the depth averaged velocity is used as a numerical model, which can incorporate the fully reflective boundary condition in the numerical scheme with accuracy.
