The solution of the long-wave equation for various nonlinear depth and breadth profiles in the power-law form

Long waves bring many important challenges in the ocean and coastal engineering, including but are not limited to harbor resonance run-up. Therefore, understanding modeling their dynamics is crucially important. Although over various types of geometries well-studied literature, study with power-law variations remains an open problem this setting. With motivation, paper, we derive exact analytical solutions long-wave equation nonlinear depth breadth profiles having forms given by h ( x ) = c 1 a b 2 , where parameters some constants. We show that for these profiles, admits terms Bessel functions Cauchy–Euler series. also seiching periods conditions variations. Our results can be used investigate envelope characteristics equilibrium beach effects entrances angled seawall on dynamics, reconstruction, geomorphological changes, sedimentation, dredging resonance, shift lakes barrages. • Solutions derived. Long-wave Bessel-Z function Cauchy-Euler Analytical long-waves Resonance