Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy

Extending the results of our previous study [2], we now investigate the propagation of interior shocks corresponding to the signaling problem of small‐amplitude, high‐frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solution is valid on both sides of this front. This solution is further distinguished to a higher order in which the effects of material inhomogeneity are accounted for. Moreover, if λ = λ( u , x ) represents the eigenvalue under consideration, we show that the single‐wave‐mode boundary disturbance of [2] can lead only to a λ‐shock. We also derive an entropy condition for the shock wave. As an application of our theory, the fluid‐filled hyperelastic tube problem of [7] is further examined and an example calculation made in which we show that a compressive shock wave is generated at the shock‐initiation point. This demonstration is effected as a particular example of the solution to a general bifurcation problem.