Abstract A physically realistic mathematical problem has been modelled to discuss the reflection and refraction phenomena of three-dimensional (3-D) plane waves. The considered composite structure comprises of water layer of finite width lying between isotropic and ice substrate has been considered. The mathematical analysis pertaining to this problem has been addressed analytically. The closed form expressions for reflection and refraction coefficients of different reflected and refracted waves are derived. Mathematical expressions for the energy share associated with the various waves are also enlisted in a very concise form in connection with the reflection and refraction coefficients. The effects of various polar and azimuthal angles have been exhibited on the reflection and refraction coefficients. The energy conservation law is established to validate this model. Numerical examples and computations have been performed to illustrate the results of this model graphically. Further, as a particular case of the present problem, the deduced results are compared and validated with the pre-established classical results.

中文翻译：

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浮冻层结构三维平面波反射和折射现象的数学研究

摘要 为了讨论三维 (3-D) 平面波的反射和折射现象，我们对一个物理现实的数学问题进行了建模。所考虑的复合结构包括位于各向同性和冰基底之间的有限宽度的水层。与此问题有关的数学分析已通过分析解决。推导出不同反射波和折射波的反射系数和折射系数的闭式表达式。与各种波相关的能量份额的数学表达式也以非常简洁的形式与反射和折射系数相关联。各种极角和方位角对反射系数和折射系数的影响已经表现出来。建立能量守恒定律来验证该模型。已经进行了数值示例和计算，以图形方式说明该模型的结果。此外，作为当前问题的一个特殊情况，将推导出的结果与预先建立的经典结果进行比较和验证。