Model Equations Combining Full Linear Dispersion With Long Wave Nonlinearity, Part.2

For estimation of spectral deformation of irregular waves in shallow water, analyses for waves with wide range of wave periods are necessary. A system of integro-differential equations for nonlinear wave propagation is therefore developed for representing wave fields with incident and reflected waves. The model equations combine long wave nonlinearity with full linear dispersion of short-period waves. The dispersion term is described as the integral where the kernel is the Fourier transform of the dimensionless local wave speed. Although the Fourier transform for the tree-dimensional model is unknown. the kernel for two-dimensional one acts as the logarithmic potential, as is usual in the potential theory in hydrodynamics. An approximate equation for two-dimensional. unidirectional propagation of waves is derived from the system. Numerical simulation with this equation verifies validity of the proposed system of equations, by comparing with the results of the KdV equation