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On the propagation of waves exhibiting both positive and negative nonlinearity

Published online by Cambridge University Press:  20 April 2006

M. S. Cramer  and
A. Kluwick
M. S. Cramer
Affiliation:
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.
A. Kluwick
Affiliation:
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A. Permanent Address: Institut für Strömungslehre und Warmeübertragung, Technische Universität Wien, Wiedner Hauptstrasse 7, A-1040 Vienna, Austria.
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 142 , May 1984 , pp. 9 - 37
Copyright
© 1984 Cambridge University Press

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References

Barker, L. M. & Hollenbach, R. E. 1970 Shock-wave studies of PMMA, fused silica, and sapphire. J. Appl. Phys. 41, 4208–4226.Google Scholar
Bethe, H. A. 1942 The theory of shock waves for an arbitrary equation of state. Office Sci. Res. & Dev. Rep. 545.Google Scholar
Bezzerides, B., Forslund, D. W. & Lindman, E. L. 1978 Existence of rarefaction shocks in a laser-plasma corona. Phys. Fluids 21, 2179–2185.Google Scholar
Borisov, A. A., Borisov, Al. A., Kutateladze, S. S. & Nakorykov, V. E. 1983 Rarefaction shock wave near the critical liquid—vapour point. J. Fluid Mech. 126, 59–73.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Springer.
Crighton, D. G. 1982 Propagation of non-uniform shock waves over large distances. In Proc. IUTAM Symp. on Nonlinear Deformation Waves (ed. U. Nigul & J. Engelbrecht). Springer.
Dessler, A. J. & Fairbank, W. M. 1956 Amplitude dependence of the velocity of second sound. Phys. Rev. 104, 6–10.Google Scholar
Garrett, S. 1981 Nonlinear distortion of 4th sound in superfluid 3He-B. J. Acoust. Soc. Am. 60, 139–144.Google Scholar
Germain, P. 1972 Shock waves, jump relations and structure. Adv. Appl. Mech. 12, 131–134.Google Scholar
Hayes, W. D. 1960 Gasdynamic discontinuities. Princeton Series on High Speed Aerodynamics and Jet Propulsion. Princeton University Press.
Hirschfelder, J. O., Buehler, R. J., McGee, H. A. & Sutton, J. R. 1958 Generalized thermodynamic excess functions for gases and liquids. Ind. Engng Chem. 50, 386.Google Scholar
Khalatnikov, I. M. 1952 Discontinuities and large amplitude sound waves in helium II. Zh. Eksp. Teor. Fiz. 23, 253.Google Scholar
Khalatnikov, I. M. 1965 Introduction to the Theory of Superfluidity. Benjamin.
Kluwick, A. 1977 Kinematische Wellen. Acta Mech. 26, 15–46.Google Scholar
Lax, P. D. 1971 Shock waves and entropy. In Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello). Academic.
Lee-Bapty, I. P. 1981 Ph.D. dissertation, Leeds University, England.
Leibovich, S. & Seebass, A. R. 1974 Nonlinear Waves. Cornell University Press.
Lighthill, M. J. 1957 River waves. Nav. Hydrodyn. Publ. 515. Natl Acad. Sci., Natl Res. Council.Google Scholar
Lighthill, M. J. & Whitham, G. B. 1955 On kinematic waves. I. Flood movement in long rivers. Proc. R. Soc. Lond. A 229, 281–316.Google Scholar
Martin, J. J. & Hou, Y. C. 1955 Development of an equation of state for gases. AIChE J. 1, 142.Google Scholar
Nimmo, J. J. C. & Crighton, D. G. 1982 Bäcklund transformations for nonlinear parabolic equations: the general results. Proc. R. Soc. Lond. A 384, 381–401.Google Scholar
Osborne, D. V. 1951 Second sound in liquid helium II. Proc. Phys. Soc. Lond. A 64, 114–123.Google Scholar
Taylor, G. I. 1910 The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84, 371–377.Google Scholar
Temperley, H. N. V. 1951 The theory of propagation in liquid helium II of ‘temperature waves’ of finite amplitude. Proc. Phys. Soc. Lond. A 64, 105–114.Google Scholar
Thompson, P. A. 1971 A fundamental derivative in gasdynamics. Phys. Fluids 14, 1843–1849.Google Scholar
Thompson, P. A. & Lambrakis, K. C. 1973 Negative shock waves. J. Fluid Mech. 60, 187–208.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Zel'dovich, Ya. B. 1946 On the possibility of rarefaction shock waves. Zh. Eksp. Teor. Fiz. 4, 363–364.Google Scholar