  We study an imbibition problem for porous media. When a wetted layer is above
a dry medium, gravity leads to the propagation of the water downwards into the
medium. In experiments, the occurrence of fingers was observed, a phenomenon
that can be described with models that include hysteresis. In the present
paper, we describe a single finger in a moving frame and set up a free boundary
problem to describe the shape and the motion of one finger that propagates with
a constant speed. We show the existence of solutions to the travelling wave
problem and investigate the system numerically.
