  Long waves in shallow water propagating over a background shear flow towards
a sloping beach are being investigated. The classical shallow-water equations
are extended to incorporate both a background shear flow and a linear beach
profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is
shown how several changes of variables based on the hodograph transform may be
used to transform the system into a linear equation which may be solved exactly
using the method of separation of variables. This method can be used to
investigate the run-up of a long wave on a planar beach including the
development of the shoreline.
