An analytic solution to the modified mild-slope equation for wave propagation over one-dimensional piecewise smooth topographies

In this paper, several recursive formulae for calculating arbitrary order derivatives of the bottom curvature term and the slope-square term in the modified mild-slope equation (MMSE) are derived and an analytic model based on Taylor series to solve the MMSE for linear waves propagating over one-dimensional piecewise smooth topographies is proposed. By using this analytic model, wave reflections by four bathymetries, i.e., a single linear slope, a parabolic hump, a cosine hump and a singly periodic sinusoidal ripple bed, are studied and explicit formulae to calculate the related reflection coefficients are established. Excellent agreements between the present analytic solution and the numerical solution based on the same MMSE for the four bathymetries are obtained which show the correctness of the present analytic model. It is also shown that, in comparison with experimental data, ‘exact’ full linear solutions or approximate analytic solutions based on the Laplace equation, the present analytic model produces much more accurate results than its traditional MSE (mild-slope equation) based predecessor does no matter the bottom slopes of the topographies are ‘mild’ or not ‘mild’. Based on the present analytic solution to the MMSE, the influence of the number of sinusoidal ripples to wave reflection is investigated.

► Recursive formulae for calculating derivatives of additional terms are derived.
► An analytical solution to the modified mild-slope equation is obtained.
► Analytic formulae of reflections by one-dimensional piecewise smooth seabeds are given.
► The influence of the number of sinusoidal ripples to Bragg resonant reflection is investigated.