Effect of wind on a soliton's shape shoaling up a planar slope

Wave shape influences sediment transport, beach morphology, and ship safety. Previously, shoaling and wind forcing have been shown to cause wave growth and change wave shape separately. However, the combined influence of these two phenomena has not been studied theoretically despite their joint prevalence in the nearshore environment. Here, we force small-amplitude, shallow-water solitary waves with a Jeffreys-type wind-induced surface pressure and allow them to propagate up a gentle, planar bathymetry. The resulting variable-coefficient KdV--Burgers equation is solved numerically. The resulting wave consists of a solitary wave and bound shelf as described by Miles in combination with a wind-induced, bound, dispersive tail. The wind-induced tail is consistent with prior studies of wind-forced shallow-water waves on flat bottoms. Additionally, onshore winds cause a narrowing of the wave peak and depress the wave's rear shelf. Finally, the wind influences the width of the pre-breaking zone in qualitative agreement with prior laboratory measurements and numerical experiments.