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Long waves bring many important challenges in the ocean and coastalengineering, 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 inthe literature, the study of the geometries with power-law variations remainsan open problem in this setting. With this motivation, in this paper, we derivethe exact analytical solutions of the long-wave equation over nonlinear depthand breadth profiles having power-law forms given by $h(x)=c_1 x^a$ and$b(x)=c_2 x^c$, where the parameters $c_1, c_2, a, c$ are some constants. Weshow that for these types of power-law forms of depth and breadth profiles, thelong-wave equation admits solutions in terms of Bessel functions andCauchy-Euler series. We also derive the seiching periods and resonanceconditions for these forms of depth and breadth variations. Our results can beused to investigate the long-wave dynamics and their envelope characteristicsover equilibrium beach profiles, the effects of nonlinear harbor entrances andangled nonlinear seawall breadth variations in the power-law forms on thesedynamics, and the effects of reconstruction, geomorphological changes,sedimentation, and dredging to harbor resonance, to the shift in resonanceperiods and to the seiching characteristics in lakes and barrages.

The Solution of the Long-Wave Equation for Various Nonlinear Depth and Breadth Profiles in the Power-Law Form