Title: Spatial structure of shock formation,
Abstract: The formation of a singularity in a compressible gas, as described by the
Euler equation, is characterized by the steepening, and eventual overturning of
a wave. Using a self-similar description in two space dimensions, we show that
the spatial structure of this process, which starts at a point, is equivalent
to the formation of a caustic, i.e. to a cusp catastrophe. The lines along
which the profile has infinite slope correspond to the caustic lines, from
which we construct the position of the shock. By solving the similarity
equation, we obtain a complete local description of wave steepening and of the
spreading of the shock from a point.