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Nonlinear waves near a cut-off frequency in an acoustic duct – a numerical study

Published online by Cambridge University Press:  20 April 2006

J. A. Aranha ,
D. K. P. Yue  and
C. C. Mei
J. A. Aranha
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
D. K. P. Yue
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
C. C. Mei
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Abstract

Cut-off frequencies are well known in acoustic ducts to be the thresholds of propagation and evanescence. If at one end of a duct the piston oscillates at very near the cut-off frequency, cross-duct resonance occurs and the linearized theory breaks down. This paper studies the nonlinear response, near a cut-off frequency of a guided wave, as an initial-boundary-value problem. The asymptotic state is shown to be governed by a modified cubic Schrodinger equation. Numerical solutions are then obtained for inputs of finite and long duration. In addition to the characteristics of the input envelope, two quantities control the transient phenomenon: frequency detuning and nonlinearity. Under certain circumstances, energy can be trapped near the piston long after a short-lived input has expired, while for a sustained input there is no sign of a steady state. Dissipation is not considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 121 , August 1982 , pp. 465 - 485
Copyright
© 1982 Cambridge University Press

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References

Aranha, J. J. 1978 Nonlinear Resonance in Wave Guides. Part I of Ph.D. thesis, Department of Civil Engineering, Massachusetts Institute of Technology.
Barnard, B. J. S., Mahony, J. J. & Pritchard, W. G. 1977 The excitation of surface waves near a cut-off frequency. Phil. Trans. R. Soc. Lond. A 286, 87–123.Google Scholar
Karpman, V. I. 1975 Nonlinear Waves in Dispersive Media. Pergamon.
Keller, J. B. & Millman, M. H. 1971 Finite amplitude sound wave propagation in a wave guide. J. Acoust. Soc. Am. 49, 329–342.Google Scholar
Mahony, J. J. 1971 Excitation of surface waves near to cut-off frequency. University of Essex Fluid Mech. Res. Inst. Rep. no. 7.Google Scholar
Mei, C. C. & Liu, P. L. F. 1973 The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59, 239–256.Google Scholar
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. McGraw-Hill.
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397–413.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 79–97.Google Scholar