NONLINEAR EVOLUTION OF SURFACE GRAVITY-WAVES OVER AN UNEVEN BOTTOM

A unified theory is developed which describes nonlinear evolution of s urface gravity waves propagating over an uneven bottom in the case of two-dimensional incompressible and inviscid fluid of arbitrary depth. Under the assumptions that the bottom of the fluid has a slowly varyin g profile and the wave steepness is small, a system of approximate non linear evolution equations (NEEs) for the surface elevation and the ho rizontal component of surface velocity is derived on the basis of a sy stematic perturbation method with respect to the steepness parameter. A single NEE for the surface elevation is also presented. These equati ons are expressed in terms of original coordinate variables and theref ore they have a direct relevance to physical systems. Since the formal ism does not rely on the often used assumptions of shallow water and l ong waves, the NEEs obtained are uniformly valid from shallow water to deep water and have wide applications in various wave phenomena of ph ysical and engineering importance, The shallow- and deep-water limits of the equations are discussed and the results are compared with exist ing theories. It is found that our theory includes as specific cases a lmost all approximate theories known at present.