The analytical solutions of long waves over geometries with linear and nonlinear variations in the form of power-law nonlinearities with solid vertical wall

In harbor engineering long waves cause some engineering challenges such as avoiding harbor resonance. Due to similar problems, it is necessary to comprehend and predict their dynamics. Whilst the literature has extensively researched their dynamics for a variety of geometries, the nonlinear geometries with power-law forms are not very well studied. The analytical solutions of the long-wave equation over nonlinear depth and breadth profiles with power-law forms are described by h(x)=c₁x^a and b(x)=c₂x^c, where the parameters c₁, c₂, a, and c are some real constants, are just recently derived by one of the authors of this paper Bayındır and Farazande, (2021). With this motivation, in this paper, we extend our work and obtain the exact analytical solutions of the long-wave equation over linear and nonlinear in the power-law form depth and breadth geometries where a solid vertical wall is present. We show that the long-wave equation admits solutions in terms of Bessel-Z functions and the Cauchy–Euler series for these particular power-law forms of depth and breadth profiles in the presence of the solid vertical wall. We present our results for the general type of the geometries considered with solid vertical walls. Additionally, six various cases for the aforementioned geometries where a solid vertical wall is present are discussed.