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

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. Copyright © 2011 John Wiley & Sons, Ltd.