Gravity Forcing of Surface Waves

Surface waves in deep water are forced entirely by gravity at the air-sea interface when no other forces act tangentially to the surface. Then, according to Newton's second law, the fluid acceleration parallel to the surface must equal the component of gravity parallel to the surface. Between crest and trough the fluid accelerates; between trough and crest the fluid decelerates. By replacing Bernoulli's law, gravity forcing becomes the dynamic boundary condition needed to solve the mathematical problem of these waves. Irrotational waves with a sinusoidal profile satisfy the gravity forcing condition, with the usual dispersion relation, provided the slope is small compared to one, as is true also of the Stokes development. However, the exact wave shape can be calculated using the gravity forcing method in a way that is less complex and less time-consuming than that of the Stokes perturbation expansion. To the second order the surface elevation is the same as the Stokes result; the third-order calculation has not been made. Extensions of the gravity forcing concept can easily be carried out for multiple wave trains, solitary waves and bores, waves in finite constant mean depths, and internal waves in a two-layer system. For shoaling surface waves that encounter gradually decreasing mean depths of water gravity forcing provides a physical understanding of the progressive steepening often observed near shore. The initial wave steepening, which might lead to breaking, is caused by a decrease in wavelength rather than an increase in wave height, and therefore this process does not behave like a normal instability mechanism.