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A finite-rate theory of resonance in a closed tube: discontinuous solutions of a functional equation

Published online by Cambridge University Press:  19 April 2006

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

The only solutions to date which describe nonlinear resonant acoustic oscillations are those for which the distortion of the travelling waves is negligible. Many experiments do not comply with this small-rate restriction. A finite-rate theory of resonance for an inviscid gas, in which the intrinsic nonlinearity of the waves is taken into account, necessitates the construction of periodic solutions of a nonlinear functional equation. This is achieved by introducing the notion of a critical point of the functional equation, which corresponds physically to a resonating wavelet. In a finite-rate theory a wave may break in a single cycle in the tube, and thus there may be more than one shock present even at fundamental resonance. Discontinuous solutions of the functional equation are constructed which satisfy the weak shock conditions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 99 , Issue 2 , 25 July 1980 , pp. 365 - 382
Copyright
© 1980 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
Brocher, E. 1977 Oscillatory flows in ducts: a report on Euromech 73. J. Fluid Mech. 79, 113–126.Google Scholar
Chester, W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Galiev, S. U., Ilgamov, M. A. & Sadykov, A. V. 1970 Periodic shock waves in a gas. Isv. Akad. Nauk S.S.S.R. Mech. Zhid. i Gaza 2, 57–66.Google Scholar
Gorkov, L. P. 1963 Nonlinear acoustic oscillations of a gas column within an enclosed duct. Ingenernij J. 3, 246–250.Google Scholar
Gulyayev, A. I. & Kusnetsov, W. T. 1963 Gas oscillations with finite amplitude in a closed tube. Ingenernij J. 3, 216–245.Google Scholar
Kuczma, M. 1968 Functional Equations in a Single Variable. Warsaw: PWN.
Lettau, E. 1939 Messungen an Gasschwingungen grosser Amplitude in Rohrleitungen. Deutsche Kraftfahrforschung 39, 1–17.Google Scholar
Mortell, M. P. 1971 Resonant thermal-acoustic oscillations. Int. J. Engng Sci. 9, 175–192.Google Scholar
Mortell, M. P. & Seymour, B. R. 1976 Exact solutions of a functional equation arising in nonlinear wave propagation. SIAM J. Appl. Math. 30, 587–596.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 finite rate theory of quadratic resonance. In preparation.
Saenger, R. A. & Hudson, G. E. 1960 Periodic shock waves in resonating gas columns. J. Acoust. Soc. Am. 32, 961–970.Google Scholar
Seymour, B. R. & Mortell, M. P. 1973 Resonant acoustic oscillations with damping: small rate theory. J. Fluid Mech. 58, 353–373.Google Scholar
Sturtevant, B. 1974 Nonlinear gas oscillations in pipes. Part 2. Experiment. J. Fluid Mec. 63, 97–120.Google Scholar
Whitham, G. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Zapirov, R. G. & Ilgamov, M. A. 1976 Nonlinear gas oscillations in a pipe. J. Sound Vib. 46, 245–257.Google Scholar