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

In this paper, we derive the exact solutions of the long-wave equation over nonlinear depth and 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. 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.