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A finite-rate theory of quadratic resonance in a closed tube

Published online by Cambridge University Press:  20 April 2006

Michael P. Mortell  and
Brian R. Seymour
Michael P. Mortell
Affiliation:
University College, Cork, Ireland
Brian R. Seymour
Affiliation:
Department of Mathematics and Institute of Applied Mathematics and Statistics, University of British Columbia, Vancouver, B.C., Canada
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Abstract

Shock waves have been observed travelling in a closed gas-filled tube when the gas is excited by a piston operating at half the fundamental frequency of the tube. Linear theory predicts a continuous periodic solution, while its first correction in a regular expansion is unbounded at such a quadratic resonant frequency. To take account of the intrinsic nonlinearity of travelling waves, a finite-rate theory of resonance is necessary. The periodic motion is then calculated from discontinuous solutions of a functional equation. Two of the three weak-shock conditions and the entropy condition are inherent in the functional equation, and hence the addition of the equal-area rule to fit shocks ensures uniqueness of the solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 112 , November 1981 , pp. 411 - 431
Copyright
© 1981 Cambridge University Press

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References

Betchov, R. 1958 Nonlinear oscillations of a column of gas. Phys. Fluids 1, 205–212.Google Scholar
Chester, W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379.Google Scholar
Galiev, S. U., Ilgamov, M. A. & Sodykov, A. V. 1970 Periodic shock waves in a gas. Isz. Akad. Nauk S.S.S.R. Mech. Zhid. i Gaza 2, 57–66.Google Scholar
Greene, J. M. 1979 A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201.Google Scholar
Keller, J. J. 1975 Subharmonic non-linear acoustic resonances in closed tubes. Z. angew. Math. Phys. 26, 395–405.Google Scholar
Mortell, M. P. & Seymour, B. R. 1979 Nonlinear forced oscillations in a closed tube: continuous solutions of a functional equation. Proc. Roy. Soc. A 367, 253–270.Google Scholar
Mortell, M. P. & Seymour, B. R. 1980 A simple approximate determination of stochastic transition for the standard mapping. J. Math. Phys. 21, 2121–2123.Google Scholar
Moser, J. 1962 Nachr. Akad. Wiss., Göttingen, Math. Phys. Kl. 1, 1.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley-Interscience.
Seymour, B. R. & Mortell, M. P. 1973 Resonant acoustic oscillations with damping: small rate theory. J. Fluid Mech. 58, 353–373.Google Scholar
Seymour, B. R. & Mortell, M. P. 1980 A finite-rate theory of resonance in a closed tube: discontinuous solutions of a functional equation. J. Fluid Mech. 99, 365–382.Google Scholar
Seymour, B. R. & Mortell, M. P. 1981 Discontinuous solutions of a measure-preserving mapping. SIAM J. Appl. Math. 40, 94–106.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Zarembo, L. K. 1967 Forced vibrations of finite amplitude in a tube. Sov. Phys. Acoust. 13, 257–258.Google Scholar
Zaripov, R. G. & Ilgamov, M. A. 1976 Nonlinear gas oscillations in a pipe. J. Sound Vib. 46, 245–257.Google Scholar