On the group velocity for the shallow water equations with source terms

The group velocity of the shallow water according to Saint-Venant's equations with source terms is analyzed. For these equations the classical group velocity relation describes the propagation velocity of a wave packet in normal dispersion e.g. in homogeneous form. The presence of source terms in momentum equation, such as the bottom slope and the friction of bed, gives rise to a singularity in the dispersion relation, causing an anomalous dispersion in which the standard group velocity becomes infinite. This non-physical result reveals that, for non-homogeneous shallow water equations, the classic relation is not appropriate for describing a wave packet. In order to overcome this difficulty we consider an asymptotic approximation, based on the Taylor series expansion, for the representation of the propagation velocity of a wave packet. The analysis includes the effects of the friction resistance term, Courant number and Froude number. Numerical results are discussed.