Exact Analytic Solutions To The Mild-slope Equation And Modified Mild-slope Equation For Wave Reflection By One-dimensional Piecewisely Smooth Topographies

In this paper, by using the real analytic implicit function theorem, existence and an-alyticity of the wave number function k(h) defined by the implicit linear dispersion equa-tion for nonzero water depth h is proved. Then both existence and convergency of seriessolutions to the mild-slope equation (MSE) and modified mild-slope equation (MMSE) areproved for smooth bathymetry. In order to conveniently construct series solutions to the MSEand MMSE, two recursive formulae for calculating arbitrary order derivatives of two im-plicit coe?cients in the MSE or MMSE are derived which permit us to expand these twoimplicit coe?cients into Taylor series till to any order. Finally, exact analytic solutions tothe MSE and MMSE in terms of Taylor series are developed for wave re?ection by variousone-dimensional piecewisely smooth topographies, including a single slope, a trench and ashoal of general trapezoidal shape, a symmetric parabolic bed and single periodic and doubleperiodic sinusoidal ripple beds. It is shown that the present exact analytic solution can notonly re-produce the results by the analytic long-wave model in the shallow water range, butalso coincides with numerical results based on the MSE or MMSE in the whole wave range.