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Faraday waves: rolls versus squares

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093-0225, USA
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Abstract

Weakly nonlinear capillary—gravity waves of frequency ω and wavenumber k that are induced on the surface of a liquid in a square cylinder that is subjected to the vertical displacement a0 cos 2ωt are studied on the assumptions that: 0 < δ < ka0, where δ is the linear damping ratio; the dominant modes are cos kx and cos ky, where x and y are Cartesian coordinates in a horizontal plane. The formulation extends those of Simonelli & Gollub (1989), Feng & Sethna (1989), Nagata (1989, 1991) and Umeki (1991) by incorporating capillarity, cubic forcing and cubic damping. The results are also applicable to a laterally unbounded fluid, but the basic symmetry then is hypothetical rather than imposed by the boundaries. Canonical evolution equations for the modal amplitudes are determined from an average Lagrangian. The fixed points of the evolution equations comprise: (i) the null solution; (ii) an orthogonal pair of rolls described by either cos kx or cos ky; (iii) an orthogonal pair of squares described by either cos kx + cos ky or cos kx – cos ky; (iv) coupled-mode solutions for which both modes are active and neither in phase nor in antiphase. The solutions for squares are isomorphic to those for rolls through a linear transformation of the coefficients in the Hamiltonian. The fixed points for rolls and squares lie on separate loci in an energy–frequency plane that intersect the null solution at a pair of pitchfork bifurcations, one of which is definitely supercritical and the other of which may be either subcritical or supercritical. The parametric domain of the various solutions includes subdomains in which squares/rolls are stable/unstable and conversely. In the limiting case of deep-water capillary waves in the threshold domain 0 < ka0 — δ < 8δ3/9 all of the rolls and coupled-mode solutions are unstable, while squares are stable except for fixed points between the subcritical bifurcation (if it exists) and the corresponding turning point.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 353 - 371
Copyright
© 1994 Cambridge University Press

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