The system of integro-differential equations, which combines long wave nonlinearity with full linear dispersion, is presented as new model equations for studying nonlinear evolution of shallow water waves. The dispersion term is represented by an integral with the kernel of the Fourier transform of phase velocity. Among the nonlinear wave equations developed from the system for spectral-wave components, the simplified version is applied to waves over the step-type reef in two-dimensions. Comparisons between present numerical results and experimental data are made. The results show that there occurs modulation chiefly in the amplitude and the phase of the first harmonic because of nonlinear interaction between wave components and bathymetry of the reef. Consequently, there may be the resonance of incident waves on the step-type reef due to vertical bathymetric variation.41 Coast. Eng. J. 1998.40:41-60. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 08/18/15. For personal use only. Coast. Eng. J. 1998.40:41-60. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 08/18/15. For personal use only.
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