  Wave front propagation with non-trivial bottom topography is studied within
the formalism of hyperbolic long wave models. Evolution of non-smooth initial
data is examined, and in particular the splitting of singular points and their
short time behaviour is described. In the opposite limit of longer times, the
local analysis of wavefronts is used to estimate the gradient catastrophe
formation and how this is influenced by the topography. The limiting cases when
the free surface intersects the bottom boundary, belonging to the so-called
"physical" and "non-physical" vacuum classes, are examined. Solutions expressed
by power series in the spatial variable lead to a hierarchy of ordinary
differential equations for the time-dependent series coefficients, which are
shown to reveal basic differences between the two vacuum cases: for
non-physical vacuums, the equations of the hierarchy are recursive and linear
past the first two pairs, while for physical vacuums the hierarchy is
non-recursive, fully coupled and nonlinear. The former case may admit solutions
that are free of singularities for nonzero time intervals, while the latter is
shown to develop non-standard velocity shocks instantaneously. We show that
truncation to finite dimensional systems and polynomial solutions is in general
only possible for the case of a quadratic bottom profile. In this case the
system's evolution can reduce to, and is completely described by, a low
dimensional dynamical system for the time-dependent coefficients. This system
encapsulates all the nonlinear properties of the solution for general power
series initial data, and in particular governs the loss of regularity in finite
times at the dry point. For the special case of parabolic bottom topographies,
an exact, self-similar solution class is introduced and studied to illustrate
via closed form expressions the general results.
