The problem of propagation of obliquely incident surface water waves involving an asymmetric rectangular trench in a channel of finite depth is examined for its solution. The problem under consideration leads to multiple series relations involving trigonometric functions. Instead of converting these relations into systems of integral equations, direct algebraic approaches are utilized and the corresponding solutions are obtained approximately, by way of solving the reduced system of over-determined algebraic equations. The system of over-determined algebraic equations are solved with the aid of the well-known algebraic approaches “Least square” (LS) and “Singular value decomposition” (SVD) methods. The presently determined results involving the reflection and transmission coefficients related to the scattering problem under consideration, agree very well, with the known results obtained recently, where integral equations and their approximate numerical solutions were determined. The effect of system of parameters on hydrodynamics quantities such as reflection and transmission coefficients, wave elevation are analysed and shown through tables and graphs. The present algebraic methods appear to be very direct and quick. A similar techniques are found to be applicable to the problem of scattering of surface water waves by any finite number of asymmetrical placed rectangular trenches. The problem of a pair of asymmetrical trenches is under progress. The energy balance relation for the given problem is derived and used to check the accuracy of numerical results. Some important results such as wave elevation profiles, the singularity behaviour of the flow near trench edges are investigated and analyzed through graphs.

中文翻译：

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解决不对称沟槽的水波散射问题的代数方法

为了解决该问题，研究了在有限深度的通道中涉及非对称矩形沟槽的斜入射水波传播问题。考虑中的问题导致涉及三角函数的多个级数关系。不用将这些关系转换成积分方程组，而是利用直接代数方法，并通过求解超定代数方程的简化系统来近似获得相应的解。借助已知的代数方法“最小二乘”（LS）和“奇异值分解”（SVD）方法来解决超定代数方程组。当前确定的结果涉及与所考虑的散射问题有关的反射系数和透射系数，与最近获得的已知结果非常吻合，确定了积分方程及其近似数值解。分析并通过表格和曲线图分析了参数系统对流体力学量的影响，例如反射系数和透射系数，波高。当前的代数方法似乎是非常直接和快速的。发现类似的技术适用于通过任意数量的不对称放置的矩形沟槽来散射表面水波的问题。一对不对称沟槽的问题正在研究中。导出给定问题的能量平衡关系，并将其用于检查数值结果的准确性。一些重要的结果，例如波高剖面，