An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.
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Agnon et al. (1999) studied this question.