Mathematics – Analysis of PDEs
Scientific paper
2011-12-06
Mathematics
Analysis of PDEs
17 pages. Submitted to Quarterly of Applied Mathematics
Scientific paper
Asymptotic decay laws for planar and nonplanar shock waves and the first order induced discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity. The asymptotic behavior of both the discontinuities is completely analyzed. It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. Indeed, the decay of the induced discontinuity is much slower than the decay of the shock. It is noticed that for the case when $[p_x] = \mathrm(\epsilon),$ the shock decays faster as compared to the case when $[p_x] = \mathrm(1).$ We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of nonlinear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.
Sharma Vishnu D.
Venkatraman Raghavendra
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