  We study the fronts that appear when a shear-thickening suspension is
submitted to a sudden driving force at a boundary. Using a
quasi-one-dimensional experimental geometry, we extract the front shape and the
propagation speed from the suspension flow field and map out their dependence
on applied shear. We find that the relation between stress and velocity is
quadratic, as is generally true for inertial effects in liquids, but with a
pre-factor that can be much larger than the material density. We show that
these experimental findings can be explained by an extension of the Wyart-Cates
model, which was originally developed to describe steady-state
shear-thickening. This is achieved by introducing a sole additional parameter:
the characteristic strain scale that controls the crossover from start-up
response to steady-state behavior. The theoretical framework we obtain unifies
both transient and steady-state properties of shear-thickening materials.
