Despite their importance in the ocean environment, the key physical processes in wind-wave interactions are poorly understood. Using a solver developed for undulatory boundaries, we perform numerical simulations of the wind-wave system under various sea conditions, including wind over monochromatic waves, early wind-wave generation, and wind over a broadband wave field. Our results show that the wave direction and wave age can significantly change the distribution of streamwise vorticity in the wind field. Different wave patterns are observed in the process of wind-wave generation. In a broadband wave field, the wave growth rate due to wind input is found to depend on the wave steepness.