The initial formation and structure of two-dimensional diffusive shock waves

✍ Scribed by Richard Haberman

• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 449 KB
• Volume: 8
• Category: Article
• ISSN: 0165-2125
• DOI: 10.1016/s0165-2125(86)80049-x

✦ Synopsis

Shock wave formation is described for quasi-linear partial differential equations with weak diffusion. A singularity develops from smooth two-dimensional initial conditions when characteristics intersect. Within the caustic surface for the aharacteristics, the wave folds over and becomes triple-valued. Near the first breaking, the characteristics and their singularity are described by a generic cubic equation. A transition region exists satisfying the one-dimensional Burgers' equation. The diffusion equation is obtained from the Hopf-Cole transformation. The solution, corresponding to the usual formation of a two-dimensional shock wave, is shown to be a canonical exponential integral with a quartic phase. Critical points satisfy the fundamental cubic equation. The Rankine-Hugoniot shock conditions are shown to emerge from within the caustic surface.