A complementary mild-slope equation derived using higher-order depth function for waves obliquely propagating on sloping bottom

Abstract

In this paper a complementary mild-slope equation (CMSE) is derived in order to investigate the transformation of progressive waves obliquely propagating over the sloping bottom more realistically. We introduce a new depth function which includes the wave refraction and the influence of the bottom slope α, perturbed to the second-order in the integral equation. A new depth-integrated mild-slope equation is derived, by using the above mentioned depth function, to model a time-harmonic motion of small amplitude waves in varying water depth. The simulated results reveal that the proposed model provides a significant improvement in the calculation of the wavenumber and the group velocity at different bottom slopes. With the increasing bottom slope, the discrepancies in the reflection coefficient of Bragg scattering between the analytical solution and the one calculated from the conventional mild-slope equation (MSE) and the modified MSE (MMSE) are seen to steadily increase. The group velocity of the waves, when compared with the conventional MSE and MMSE, also shows its dependence on the bottom slope and wave propagating angle. The present model is observed to be quite efficient in taking into account the effect of steeper bottom slope.