SIAM Journal on Applied Mathematics, Volume 81, Issue 2, Page 285-303, January 2021.
The present study was originally motivated by some intriguing exact solutions for waves propagating in nonuniform media. In particular, for special depth profiles reflected waves did not appear and ray theory became exact. Herein, geometrical optics is employed to obtain asymptotic series for waves of general shapes in nonuniform narrow channels, within the framework of linear shallow water theory. While being kept simple, the series incorporate higher order contributions that may describe the evolution of waves with high accuracy. The higher orders also contain a secondary wave system. For selected classes of geometries and wave shapes explicit solutions are calculated and compared to numerical solutions. Apart from the vicinity of shorelines, say, higher order expansions generally may provide very accurate approximations to the full solutions. The asymptotic series are analyzed for different wave shapes and are found to be convergent for cases where the basic wave profiles have compact support (finite length). A number of new, closed form, exact solutions are also found. The asymptotic expansion is put into a context by employing it for the transmission of waves from a uniform channel section into a nonuniform one. Additional results and side topics are presented in a supplement.

中文翻译：

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具有幂律几何的通道的线性浅水方程的渐近，收敛和精确截断级数解

SIAM应用数学杂志，第81卷，第2期，第285-303页，2021年1月。
本研究最初是由一些有趣的，针对非均匀介质中传播的波的精确解所激发的。特别是，对于特殊的深度剖面，没有出现反射波，并且射线理论变得精确。这里，在线性浅水理论的框架内，采用几何光学来获得非均匀窄通道中一般形状的波的渐近级数。在保持简单性的同时，该系列结合了更高阶的贡献，这些贡献可以高精度地描述波的演化。高阶还包含次级波系统。对于选定的几何形状和波形类别，将计算显式解并将其与数值解进行比较。除了海岸线附近，例如，高阶展开通常可以为整个解决方案提供非常准确的近似值。分析了渐近级数的不同波形，发现在基本波形具有紧凑支撑（有限长度）的情况下收敛。还找到了许多新的封闭式精确解决方案。通过将渐进式扩展用于将波从均匀的通道段传输到不均匀的通道段中，从而使渐进式扩展处于上下文中。补充中提供了其他结果和附带主题。