  Unforced shoaling waves experience growth and changes to wave shape.
Similarly, wind-forced waves on a flat-bottom likewise experience growth/decay
and changes to wave shape. However, the combined effect of shoaling and
wind-forcing on wave shape, particularly relevant in the near-shore
environment, has not yet been investigated theoretically. Here, we consider
small-amplitude, shallow-water solitary waves propagating up a gentle, planar
bathymetry forced by a weak, Jeffreys-type wind-induced surface pressure. We
derive a variable-coefficient Korteweg-de Vries-Burgers (vKdV-B) equation
governing the wave profile's evolution and solve it numerically using a
Runge-Kutta third-order finite difference solver. The simulations run until
convective prebreaking -- a Froude number limit appropriate to the order of the
vKdV-B equation. Offshore winds weakly enhance the ratio of prebreaking height
to depth as well as prebreaking wave slope. Onshore winds have a strong impact
on narrowing the wave peak, and wind also modulates the rear shelf formed
behind the wave. Furthermore, wind strongly affects the width of the
prebreaking zone, with larger effects for smaller beach slopes. After
converting our pressure magnitudes to physically realistic wind speeds, we
observe qualitative agreement with prior laboratory and numerical experiments
on breakpoint location. Additionally, our numerical results have qualitatively
similar temporal wave shape to shoaling and wind-forced laboratory
observations, suggesting that the vKdV-B equation captures the essential
aspects of wind-induced effects on shoaling wave shape. Finally, we isolate the
wind's effect by comparing the wave profiles to the unforced case. This reveals
that the numerical results are approximately a superposition of a solitary
wave, a shoaling-induced shelf, and a wind-induced, bound, dispersive, and
decaying tail.
