Normal surface water waves, as generated by wind, do not have sine form but wave peak is higher and shorter than wave trough with different wave steepness. What parameters characterize such a surface water wave and how can one predict amplitude of water for given waves as function of time?
Deep water waves are often described as "cnoidal", with a mathematical description involving the Jacobian elliptic function cn(). This is an exact solution to the nonlinear Korteweg–de Vries differential equation. A more accurate equation is the Boussinesq. These are the basis for describing all water waves, whether stirred up by wind or otherwise. The basic parameters for a particular solution are wave height, period (or wavelength), depth of the water, and acceleration of gravity.
I hate to cite Wikipedia due to its propensity to change, but the best explanations I could find, including math, are there.
As for the details of wind pushing on the wave peaks, and the peaks disturbing the air flow, and big waves breaking over in ways that excite surfers, there are no nice mathematical forms I know of, but then I'm not expert on this. Numerical modeling is king in the area. Some original research was done for the movie "The Perfect Storm" on how to do better simulations and crunch the numbers faster.