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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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