Free-surface flow over a bottom topography with an
asymptotic depth change (a ‘step’) is considered
for different ranges of Froude numbers varying from subcritical,
transcritical, to supercritical. For the subcritical case, a linear
model indicates that a train of transient waves propagates upstream
and eventually alters the conditions there. This leading-order
upstream influence is shown to have profound effects on
higher-order perturbation models as well as on the Froude
number which has been conventionally defined in terms of the
steady-state upstream depth. For the transcritical case, a
forced Korteweg–de Vries (fKdV) equation is derived, and the
numerical solution of this equation reveals a surprisingly
conspicuous distinction between positive and negative forcings.
It is shown that for a negative forcing, there exists a
physically realistic nonlinear steady state and our preliminary
results indicate that this steady state is very likely to be
stable. Clearly in contrast to previous findings associated
with other types of forcings, such a steady state in the
transcritical regime has never been reported before. For
transcritical flows with Froude number less than one,
the upstream influence discovered for the subcritical
case reappears.