  During motion from deep to shallow water, multiple equilibria may emerge,
each with identical drag - a phenomenon that can be explained by a localised
amplification of the wave drag near the shallow wave speed. The implication of
this is the emergence of several previously unstudied bifurcation patterns and
hysteresis routes. Here, we address these nonlinear dynamics by considering the
quasi-steady motion of a body between deep and shallow water, where the depth
is slowly varying. We survey several theoretical models for the drag, compare
these against our tow-tank experimental measurements, and then use the
validated theory to explore the bifurcation patterns using two parameters: the
depth of motion and the forcing. In particular, using a case study of a lake
with a sinusoidal depth profile, we illustrate that hysteresis effects can play
a significant role on the speed of motion and journey time, presenting
interesting implications for naval and racing applications.
