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

Sudi Mungkasi*, Stephen G. Roberts

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    15 Citations (Scopus)

    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.

    Original languageEnglish
    Pages (from-to)763-780
    Number of pages18
    JournalInternational Journal for Numerical Methods in Fluids
    Volume69
    Issue number4
    DOIs
    Publication statusPublished - 10 Jun 2012

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