The properties of weakly nonlinear kinematic waves in suspensions are inves
tigated under the assumption that the particle concentration deviates only
slightly from the value at the inflexion point of the drift flux curve. Spe
cial emphasis is placed on the conditions for the existence of an internal
dissipative-dispersive shock structure. The resulting shock admissibility c
riteria are found to be significantly different from those following from s
tandard theories of kinematic waves. Most interesting, the analysis shows t
hat non-classical shocks which emanate rather than absorb characteristics m
ay be admissible under certain conditions.