  We consider a dissipative, dispersive system of Boussinesq type, describing
wave phenomena in settings where dissipation has an effect. Examples include
undular bores in rivers or oceans where dissipation due to turbulence is
important for their description. We show that the model system admits traveling
wave solutions known as diffusive-dispersive shock waves, and we categorize
them into oscillatory and regularized shock waves depending on the relationship
between dispersion and dissipation. Comparison of numerically computed
solutions with laboratory data suggests that undular bores are accurately
described in a wide range of phase speeds. Undular bores are often described
using the original Peregrine system which, even if not possessing traveling
waves tends to provide accurate approximations for appropriate time scales. To
explain this phenomenon, we show that the error between the solutions of the
dissipative versus the non-dissipative Peregrine systems are proportional to
the dissipation times the observational time.
