A comprehensive theoretical treatment of the convergence of the four-point implicit numerical model of shallow water waves is presented. The propagation celerity and attenuation factor of the numerical analog are derived, and convergence is tested by establishing the ratios of attenuation and translation given by the numerical and analytical solutions. Convergence is shown to be a function of the Froude number, the dimensionless wave number, the Courant number, the spatial resolution and the weighting factor of the scheme.