Caustics of weak shock waves

The caustics of weak shock waves are studied through matched asymptotic expansions. It is shown that these caustics are thinner and more intense than those of smooth waves with a comparable amplitude. This difference in scalings solves a paradox that would have the caustics of weak shock waves behave linearly, even though linear theory for discontinuous fronts predicts infinite amplitudes near the caustic and in the reflected wave. With the new scalings, the behavior of shocks both near the caustic and in the far field is described by nonlinear equations. The new scales are robust, in the sense that they survive the addition of a small amount of viscosity to the equations. As a viscous shock approaches the caustic, its intensity amplifies and its width decreases in such a way that the new scalings are actually reinforced. A new paradox arises, however: The nonlinear Tricomi equation which describes the behavior of the fronts near caustics does not appear to admit the triple shock intersections which have been observed experimentally. This new open problem is closely related to the von Neumann paradox of oblique shock reflection. © 1998 American Institute of Physics. @S1070-6631~97!03912-3#