Long waves bring many important challenges in the ocean and coastal engineering, including but are not limited to harbor resonance and run-up. Therefore, understanding and modeling their dynamics is crucially important. Although their dynamics over various types of geometries are well-studied in the literature, the study of the geometries with power-law variations remains an open problem in this setting. With this motivation, in this paper, we derive the exact analytical solutions of the long-wave equation over nonlinear depth and breadth profiles having power-law forms given by $h\left(x\right)=c_1{x}^a$ and $b\left(x\right)=c_2{x}^c$, where the parameters $c_1,c_2,a,c$ are some constants. We show that for these types of power-law forms of depth and breadth profiles, the long-wave equation admits solutions in terms of Bessel functions and Cauchy–Euler series. We also derive the seiching periods and resonance conditions for these forms of depth and breadth variations. Our results can be used to investigate the long-wave dynamics and their envelope characteristics over equilibrium beach profiles, the effects of nonlinear harbor entrances and angled nonlinear seawall breadth variations in the power-law forms on these dynamics, and the effects of reconstruction, geomorphological changes, sedimentation, and dredging to harbor resonance, to the shift in resonance periods and to the seiching characteristics in lakes and barrages.

长波给海洋和海岸工程带来了许多重要挑战，包括但不限于港口共振和爬升。因此，理解和建模它们的动力学至关重要。尽管文献中对它们在各种几何类型上的动力学进行了充分研究，但在这种情况下，对具有幂律变化的几何的研究仍然是一个悬而未决的问题。出于这个动机，在本文中，我们推导出长波方程在非线性深度和宽度剖面上的精确解析解，其幂律形式由下式给出$H\left(X\right)=C_1{X}^{\mathrm{一种}}$ 和 $乙\left(X\right)=C_2{X}^C$, 其中参数 $C_1,C_2,\mathrm{一种},C$是一些常数。我们表明，对于这些类型的深度和宽度剖面的幂律形式，长波方程承认贝塞尔函数和柯西-欧拉级数的解。我们还推导出这些深度和宽度变化形式的捕获周期和共振条件。我们的结果可用于研究平衡海滩剖面上的长波动力学及其包络特性、非线性港口入口和有角度的非线性海堤宽度变化的幂律形式对这些动力学的影响，以及重建、地貌学的影响变化、沉积和疏浚以产生共振，共振周期的变化以及湖泊和拦河坝的捕获特征。