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 backward-step
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 coefficient with time does
not coincide with the period of solitary wave emission.