An exact analytic solution to the modified mild-slope equation for waves propagating over a trench with various shapes

Abstract

An exact analytic solution to the modified mild-slope equation (MMSE) in terms of Taylor series for waves propagating over an asymmetrical trench with various shapes is given. Because of the use of the MMSE, on one hand, the present analytic solution can be valid in the whole wave range from long waves to short waves, which is clearly superior to all previous long-wave analytic solutions; on the other hand, the present analytic solution can get rid of the limitation of the ‘mild slope’ assumption and be valid for bottom slope as high as 1:1. It is clarified that the improvement in solution accuracy by using the mass-conserving matching condition against the conventional matching condition mainly depends upon the jump quantities at all common boundaries. In addition, in comparison with previous approximate analytic model based on the approximate mild-slope equation, the present model is more accurate and can converge in the whole trench region without any restriction to trench depth. Based on the present MMSE solution, influence of trench dimensions to reflection effect is analyzed, which shows that total reflection effect increases when trench wall becomes steep and the phenomenon of zero reflection mainly occurs for symmetrical trenches.