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

机译：修正的缓坡方程对于在各种形状的沟槽上传播的波的精确解析解

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摘要

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.
机译：针对在各种形状的不对称沟槽上传播的波的泰勒级数，给出了修正的缓坡方程（MMSE）的精确解析解。由于使用了MMSE，一方面，本分析解决方案可以在从长波到短波的整个波范围内有效，这显然优于所有以前的长波分析解决方案。另一方面，当前的解析解可以摆脱“温和坡度”假设的局限性，并且适用于高达1：1的底部坡度。需要说明的是，相对于传统的匹配条件，通过使用质量守恒的匹配条件，求解精度的提高主要取决于所有共同边界处的跳变量。另外，与以前的基于近似缓坡方程的近似分析模型相比，本模型更精确，并且可以在整个沟槽区域收敛，而对沟槽深度没有任何限制。基于目前的MMSE解决方案，分析了沟槽尺寸对反射效果的影响，结果表明，当沟槽壁变陡时，全反射效应增加，对称沟槽主要发生零反射现象。

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