  Wave shape (i.e. skewness or asymmetry) plays an important role in beach
morphology evolution, remote sensing, and ship safety. Wind's influence on
ocean waves has been extensively studied theoretically in the context of
growth, but most theories are phase averaged and cannot predict wave shape.
Most laboratory and numerical studies similarly focus on wave growth. A few
laboratory experiments have demonstrated that wind can change wave shape, and
two-phase numerical simulations have also noted wind-induced wave shape
changes. However, wind's effect on wave shape is poorly understood, and no
theory for it exists. For weakly nonlinear waves, wave shape parameters are the
phase of the harmonic relative to the primary frequency (or harmonic phase HP,
zero for a Stokes wave) and relative amplitude of the harmonic to the primary.
Here, surface pressure profiles (denoted Jeffreys, Miles, and Generalized
Miles) are prescribed based on wind-wave generation theories. Theoretical
solutions are derived for quasi-periodic progressive waves and the wind-induced
changes to HP, relative harmonic amplitude, as well as already known phase
speed changes and growth rates. The wave shape parameters depend upon the
chosen surface pressure profile, pressure magnitude and phase relative to the
wave profile, and the nondimensional depth. Wave asymmetry is linked to the
nondimensional growth rate. Atmospheric large eddy simulations constrain
pressure profile parameters. HP predictions are qualitatively consistent with
laboratory observations. This theory, together with the observables of HP and
relative harmonic amplitude, can provide insight into the actual wave surface
pressure profile.
