Nonlinear evolution and runup-rundown of long waves over a sloping beach

The initial value problem of the nonlinear evolution, shoreline motion and flow velocities of long waves climbing sloping beaches is solved analytically for different initial waveforms. A major difficulty in earlier work utilizing hodograph-type transformation when solving either boundary value or initial value problems has been the specification of equivalent boundary or initial condition in the transformed space. Here, in solving the initial value problem, the transformation is linearized in space at t = 0, then the full nonlinear transformation is used to solve the initial value problem of the nonlinear shallow-water wave equations. A solution method is presented to describe the most physically realistic initial waveforms and simplified equations for the runup-rundown motions and shoreline velocities. This linearization of the initial condition does not appear to affect the subsequent nonlinear evolution, as shown through comparisons with earlier studies. Comparisons with runup results from solutions of the boundary value problem suggest the same variation with the runup laws. The methodology presented here appears simpler than earlier work as it does not involve the numerical calculation of singular elliptic integrals.