Approximations of the Carrier-Greenspan periodic solution to the shallow water wave equations for flows on a sloping beach

Abstract

The Carrier-Greenspan solutions to the shallow water wave equations for flows on a sloping beach are of two types, periodic and transient. This paper focuses only on periodic-type waves. We review an exact solution over the whole domain presented by Johns ['Numerical integration of the shallow water equations over a sloping shelf', Int. J. Numer. Meth. Fluids, 2(3): 253-261, 1982] and its approximate solution (the Johns prescription) prescribed at the zero point of the spatial domain. A new simple formula for the shoreline velocity is presented. We also present new higher order approximations of the Carrier-Greenspan solution at the zero point of the spatial domain. Furthermore, we compare numerical solutions obtained using a finite volume method to simulate the periodic waves generated by the Johns prescription with those found using the same method to simulate the periodic waves generated by the Carrier-Greenspan exact prescription and with those found using the same method to simulate the periodic waves generated by the new approximations. We find that the Johns prescription may lead to a large error. In contrast, the new approximations presented in this paper produce a significantly smaller error.