Spatial structure of shock formation

TL;DR

This paper analyzes the spatial structure of shock formation in a compressible gas using self-similar solutions, revealing that the process is equivalent to a cusp catastrophe and providing a detailed local description of wave steepening and shock spreading.

Contribution

It introduces a self-similar framework to describe shock formation in two dimensions, linking it to caustic and cusp catastrophe theory, and offers a complete local analysis of wave steepening.

Findings

Shock formation corresponds to a cusp catastrophe.

The spatial structure of the shock is characterized by caustic lines.

A complete local description of wave steepening and shock spreading is provided.

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.