On the propagation of waves exhibiting both positive and negative nonlinearity
- 1 May 1984
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Fluid Mechanics
- Vol. 142, 9-37
- https://doi.org/10.1017/s0022112084000975
Abstract
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.Keywords
This publication has 11 references indexed in Scilit:
- Nonlinear distortion of 4th sound in superfluid 3He-BThe Journal of the Acoustical Society of America, 1981
- Kinematische WellenActa Mechanica, 1977
- Negative shock wavesJournal of Fluid Mechanics, 1973
- A Fundamental Derivative in GasdynamicsPhysics of Fluids, 1971
- Shock-Wave Studies of PMMA, Fused Silica, and SapphireJournal of Applied Physics, 1970
- Generalized Thermodynamic Excess Functions for Gases and LiquidsIndustrial & Engineering Chemistry, 1958
- Amplitude Dependence of the Velocity of Second SoundPhysical Review B, 1956
- Development of an equation of state for gasesAIChE Journal, 1955
- Second Sound in Liquid Helium IIProceedings of the Physical Society. Section A, 1951
- The Theory of the Propagation in Liquid Helium II of `Temperature-Waves' of Finite AmplitudeProceedings of the Physical Society. Section A, 1951