ABSTRACT For the 3-D system of wave motion consisted of a free-surface progressive gravity wave train propagating over a two side ripple bottom , a perturbation expansion procedure is developed to second order. The mechanism and the properity of the induced wave motion by the effect of the ripple bottom on the free-surface progressive gravity waves , including the resonance and non-resonance case , is described. To the induced resonance wave motion to the second order of the solution , the growth of its energy flux with time , and consequently the growth for the transmission speed of its energy flux are vividly demonstrated. Furthermore , the relation between the transmission speed in time and space is also established. Finally , the growth of the induced resonance wave motion with its propagating distance in the region of the ripple bottom located is emphasized. The variation and growth of the whole flow field of the wave motion system in time and space is therefore presented. To ascertain the accuracy and the generality of the analytical result , it was verified not only to by reducing which the solution of the one side wavy bottom, but also conform to the three kinds of the actual physical situation.

For gravity wave trains propagating over an arbitrary wavy bottom, a perturbation expansion is developed to the third-order by employing three small perturbation parameters. Both the resonant and non-resonant cases are treated and the singular behavior at resonance is treated separately. All the theoretical results are presented in explicit forms and easy to apply. The bottom effects of different mean water depths and different degrees of undulation, as well as the steepness of incident waves, are clearly described by the theoretical results. In general non-resonant cases, the surface fluctuations deduced from undulated bottom topography decrease as the relative water depth increases and vice versa. The theory can be applied to the cases for wave trains propagating over wavy bottom topography with any arbitrary incident angles which are closer to natural phenomenon in coastal zone. Not only the well-known Bragg resonance but also the higher-order Bragg resonances are included in resonant cases. Unlike previous studies that analyze specific bottom topographies based on prescribed resonant conditions, both Bragg and higher-order Bragg resonances are revealed through the perturbation procedure step by step. For the resonant wave field, the amplification with propagating distance and time is revealed with the aid of the growth of energy flux. This theory is successfully verified by reducing to simpler situations. Also, the analytical results for the special case of two-dimensional wavy bottom are compared with experimental data for validation.