The mean horizontal momentum and surface velocity of finite-amplitude random gravity waves

Simple and exact expressions for two quantities of interest in the theory of finite-amplitude random gravity waves are presented. The first is the mean horizontal momentum of the wave motion per unit projected area. The second is the mean value of the horizontal component of the fluid velocity at the free surface, measured at a fixed location in the horizontal plane. These expressions are valid for either deep or shallow water, provided the water motion is irrotational and the surface displacement ξ(x, y, t) is either spatially periodic or a stationary random function of position (x, y). From these primary results, expressions are readily found for these quantities in terms of the spectrum of the surface displacement ξ that are correct to order .