Analytical Solution for Nonlinear Shallow-water Wave Equations

Majority of the hodograph transform solutions of the one-dimensional nonlinear shallow-water wave equations are obtained through integral transform techniques. This approach, however, might involve evaluation of elliptic integrals, which are highly singular. Here, we couple the hodograph transform approach with the classical eigenfunction expansion method rather than integral transform techniques and present a new analytical model for nonlinear long wave propagation over a plane beach. In contrast to classical initial or boundary value problem solutions, an initial-boundary value problem solution is formulated. In general, initial wave profile with nonzero initial velocity distribution is assumed and the flow variables are given in the form of Fourier-Bessel series. The spatial and temporal variation of the flow quantities, i.e., free-surface height and depth-averaged velocity, are estimated accurately through the developed method with much less computational effort compared to the existing integral transform techniques.