Derivation of the third-order evolution equations for weakly nonlinear water waves propagating over uneven bottoms

Using the multi-scale perturbation methods, the evolution equations for the modulated wave groups over an uneven bottom are derived. The bottom topography consists of two components: the slowly varying component whose horizontal lenght scale is longer than the carrier wave lenght, and the fast varying component with the carrier wave lenght as the horizontal lenght scale. The size of the fast varying depth component is, however, much smaller than the carrier wave amplitude. Assuming that the nonlinearity parameter ε is in the same order of magnitude as the modulation parameter δ, a detailed perturbation expansion analysis based on the carrier wave frequency and carrier wave number is presented. The third-order evolution equation is coupled with a long wave equation which describes the generation and the propagation of a second-order long wave associated with the modulated wave train. It is also shown that the real part of the evolution equation represents a modified wave action equation, while the imaginary part of the evolution equation gives the second-order dispersion relation. For quasi-steady state problems simplifications are made to derive a nonlinear parabolic wave equation.