In this paper, the extended mild-slope equation was adopted to be the governing equation and solved by dual reciprocity boundary element method. This model improves the accuracy of former papers caused by neglecting the curvature and slope-squared terms in their models. Due to the former works focusing on the long wave, the extended terms become very small and were omitted. Recently, the extended terms was considered that cannot be omitted even the long wave. A series of numerical experiments were conducted including conical island, Homma's island. The Homma's island was the case studies and the results were compared with the analytical results derived by Homma (1950) and/or Jonsson et al. (1976). Excellent agreements were obtained.
To simulate linear wave propagation from deep to shallow water, the use of the mild-slope equation derived by Berkhoff (1972) can be a proper and valid method. The classical mildslope equation is based on the assumption of a slowing varying bathymetry, linearzing the scattering of surface waves on variable water depth by approximating the vertical structure of motion and averaging over depth. Thus, the dimension of the problem can be reduced by one. Due to the assumption of ∇h / kh << 1 on mild-slope equation, the higher-order bottom effect terms are neglected during the original derivation procedure of Berkhoff. Motivated in part by the significant engineering applications, much of the relevant existing literature has concentrated on rapidly-varying or steep topography. Booij (1983) compared the numerical results of mild-slope equation with the finite element model results in terms of the reflection coefficients for the case of monochromatic wave propagation over a plane slope. He made a conclusion that the mild-slope equation is sufficiently accurate up to a bottom slope of 1:3. However, it has been pointed out in a lot of investigations that the classical mild-slope equation fails to produce adequate approximations for certain type of bathymetry, such as off-shore reef or bars.
