Regular by Pieces Kinematic Waves: Obtaining Maslov Chains

Abstract

The Maslov chains method is applied to the study of smooth by pieces kinematics waves. For any structured time interval and any associated singular path, we have an upper and a lower (physically meaningful) smooth piece of the solution. The method presupposed that they admit power series expansions centered at the singular path; in consequence, both pieces of the physical solution can be -mathematically- continued to the other side of that path. Then, the difference (jump function) between these mathematically extended pieces has a similar expansion. By imposing the differential form of the conservation law -at both sides of the singular path- and the Hugoniot jump condition, the Maslov chain conditions (in a preliminary -non standard- form) are obtained. They are a set of infinitely many differential equations with infinitely many unknowns: the singular path and the coefficients of the expansions of the two associated smooth pieces. Recursive relations involving the coefficients of the expansion of the jump function are also derived. These relations help us to resolve the indetermination who is present in the Hugoniot jump condition when the rupture order is not cero. The Maslov chains are then rewritten in an equivalent but standard form.