A comprehensive theoretical treatment of the convergence of the four-point implicit numerical model of alluvial channel bed transients is presented. The dimensionless celerity and logarithmic decrement of the analytical and numerical solutions are calculated. Convergence is tested by establishing the ratio of celerities (translation convergence) and the ratio of attenuation factors (attenuation convergence). Two physical and three numerical parameters are identified: the equilibrium flow Froude number, the transient flow dimensionless wave number, the spatial resolution, the Courant number, and the weighting factor of the scheme.