Generation of Long Water Waves by Moving Disturbances

Abstract

Several theoretical models are developed to study generation of nonlinear dispersive long waves by moving disturbances. All these models belong to the same class as the original Boussinesq or KdV model. The newly developed models, now with external forcing functions added to the KdV equation and the pair of coupled Boussinesq equations, have been chosen for numerical investigations. A predictor-corrector method is adopted to develop the numerical schemes employed here. In order to make the region of computation reasonably small for the case with moving disturbances, a pseudo-moving frame and the sufficiently transparent open boundary conditions are devised. The numerically obtained surface elevations exhibit a series of positive waves running ahead of the disturbance over a wide range of transcritical speeds of the disturbance. The numerical results show that, for speeds close to the critical value, the generation of such waves appears to continue indefinitely. The numerically obtained wave resistance coefficient is compared to the results given by linear dispersive theory. Numerical solutions have been obtained using the KdV and Boussinesq models with surface pressure and bottom bump as forcing functions. Comparisons are made between these results for various cases. Experiments were conducted for a two-dimensional bottom bump moving steadily in shallow water of a towing tank. Experimental results so attained are compared with the numerical solutions, and the agreement between them is good in terms of both the magnitude and the phase of the waves for the range of parameters used in the current study.