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Resonantly forced surface waves in a circular cylinder

Published online by Cambridge University Press:  20 April 2006

John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093

Abstract

The weakly nonlinear, weakly damped response of the free surface of a liquid in a vertical circular cylinder that is subjected to a simple harmonic, horizontal translation is examined by extending the corresponding analysis for free oscillations. The problem is characterized by three parameters, α, β, and d/a, which measure damping, frequency offset (driving frequency–natural frequency), and depth/radius. The asymptotic (t↑∞) response may be any of: (i) harmonic (at the driving frequency) with a nodal line transverse to the plane of excitation (planar harmonic); (ii) harmonic with a rotating nodal line (non-planar harmonic); (iii) a periodically modulated sinusoid (limit cycle); (iv) a chaotically modulated sinusoid. It appears, from numerical integration of the evolution equations, that only motions of type (i) and (ii) are possible if 0.30 < d/a < 0.50, but that motions of type (iii) and (iv) are possible for all other d/a in some interval (or intervals) of β if α is sufficiently small. Only motion of type (i) is possible if α exceeds a critical value that depends on d/a.

Type
Research Article
Copyright
© 1984 Cambridge University Press

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References

Gurel, O. 1979 Poincaré's bifurcation analysis. Ann. NY Acad. Sci. 316, 526.Google Scholar
Hutton, R. E. 1963 An investigation of resonant, nonlinear, nonplanar, free surface oscillations of a fluid. NASA Tech. Note D-1870 (Washington).Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130141.Google Scholar
Miles, J. W. 1962 Stability of forced oscillations of a spherical pendulum. Q. Appl. Maths 20, 2132.Google Scholar
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419448.Google Scholar
Miles, J. W. 1984a Resonant motion of a spherical pendulum. Physica 11D, 309323.Google Scholar
Miles, J. W. 1984b Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 114.Google Scholar
Miles, J. W. 1984c Resonant non-planar motion of a stretched string. J. Acoust. Soc. Am. 75, 15051510.Google Scholar