SHOCK-WAVE TRACKING FOR HYPERBOLIC SYSTEMS EXHIBITING LOCAL LINEAR DEGENERACY

Extending the results of our previous study [2], we now investigate th e propagation of interior shocks corresponding to the signaling proble m of small-amplitude, high-frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solut ion is valid on both sides of this front. This solution is further dis tinguished to a higher order in which the effects of material inhomoge neity are accounted for. Moreover, if lambda = lambda(u, x) represents the eigenvalue under consideration, we show that the single-wave-mode boundary disturbance of [2] can lead only to a lambda-shock. We also derive an entropy condition for the shock wave. As an application of o ur theory, the fluid-filled hyperelastic tube problem of [7] is furthe r examined and an example calculation made in which we show that a com pressive shock wave is generated at the shock-initiation point. This d emonstration is effected as a particular example of the solution to a general bifurcation problem.