Generation of solitary waves by forward- and backward-step bottom forcing

Abstract

A finite difference method based on the Euler equations is developed for computing waves and wave resistance due to different bottom topographies moving steadily at the critical velocity in shallow water. A two-dimensional symmetric and slowly varying bottom topography, as a forcing for wave generation, can be viewed as a combination of fore and aft parts. For a positive topography (a bump), the fore part is a forward-step forcing, which contributes to the generation of upstream-advancing solitary waves, whereas the aft part is a backward-step forcing to which a depressed water surface region and a trailing wavetrain are attributed. These two wave systems respectively radiate upstream and downstream without mutual interaction. For a negative topography (a hollow), the fore part is a backward step and the aft part is a forward step. The downstream-radiating waves generated by the backwardstep forcing at the fore part will interact with the upstream-running waves generated by the forward-step forcing at the aft. Therefore, the wave system generated by a negative topography is quite different from that by a positive topography. The generation period of solitary waves is slightly longer and the instantaneous drag fluctuation is skewed for a negative topography. When the length of the negative topography increases, the oscillation of the wave-resistance coeffcient with time does not coincide with the period of solitary wave emission.