Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy
Extending the results of our previous study , 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  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  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.
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Document Type: Research Article
Affiliations: University of Alberta
Publication date: August 1, 1993