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On the propagation of waves exhibiting both positive and negative nonlinearity. (English) Zbl 0577.76073
(Authors’ summary.) One-dimensional small-amplitude waves in which the local value of the fundamental derivative changes sign are examined. The undisturbed medium is taken to be a Navier-Stokes fluid which is at rest and uniform with a pressure and density such that the fundamental derivative is small. A weak shock theory is developed to treat inviscid motions, and the method of multiple scales is used to derive the nonlinear parabolic equation governing the evolution of weakly dissipative waves. The latter is used to compute the viscous shock structure. New phenomena of interest include shock waves having an entropy jump of the fourth order in the shock strength, shock waves having sonic conditions either upstream or downstream of the shock, and collisions between expansion and compression shocks. When the fundamental derivative of the undisturbed media is identically zero it is shown that the ultimate decay of a one-signed pulse is proportional to the negative 1/3-power of the propagation time.
Reviewer: Z.Qian

MSC:
76Q05 Hydro- and aero-acoustics
76N15 Gas dynamics, general
76L05 Shock waves and blast waves in fluid mechanics
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