Standing wave patterns may be parametrically excited on the free surface of a
fluid layer when the gravitational acceleration is periodically modulated. In
the classic Faraday wave experiment sinusoidal forcing is employed, leading to
excitation of subharmonic standing waves. However, over the last decade, it has
been demonstrated that harmonic response is also possible; for example, when
two-frequency periodic forcing is used [11, or when the fluid layer is
sufficiently thin [21, or for certain viscoelastic fluids [3]. For each of these
systems it is possible to tune the periodic forcing function in order to access
a transition between harmonic and subharmonic response. At the bicritical point,
both instabilities set in simultaneously but with different spatial wavenumbers.
Near this codimension-two point, a variety of unusual patterns have been
observed, including quasipatterns [11 and superlattice patterns [4, 51.
Superlattice patterns have also been observed recently in Rayleigh Bernard
convection subjected to vibration, near the harmonic/subharmonic transition
point [6].
Much of the theoretical work on the Faraday wave pattern formation problem has
focused on the role that resonant triads play in the pattern selection process
(see for example [71). Resonant triads are comprised of three critical wave
vectors kl, k2, and k3 = k, ± k2,
where k1 = k2 is the wavenumber of one critical mode
and k3 is the wavenumber of the other critical mode. Associated
with each resonant triad is an angle 0, which separates the two wave vectors of
equal magnitude; two examples are given in Figure 1.
Figure 1: (a) A schematic neutral stability curve 9,(k) showing the forcing
amplitude gz, at which the flat fluid surface first becomes
unstable to perturbations of wavenumber k. The curve represents a bicritical
situation. (b) An associated spatially resonant triad km1, km2
and kn = km1 + km2 with
resonant angle 0, indicated. (c) Another example of a spatially resonant triad
km1, km2, and kn1, = kn1, - kn2.
Our bifurcation analysis of the two-frequency Faraday wave problem shows that
weakly damped harmonic modes strongly influence the pattern selection process
near onset, whereas near critical subharmonic modes do not 18, 91. This work
reveals a surprising mechanism responsible for stabilizing the superlattice
patterns observed in laboratory experiments of Kudrolli, Pier and Gollub [4].
Specifirally, our study indicates that modes associated with a harmonic
resonance tongue at small wavenumber k, are critical to stabilizing the observed
wave patterns with onset critical wavenumber k,, where kc =7ks.
The experimentally observed superlattice pattern exhibits structure on both
these lengthscales. These results are based on a weakly nonlinear analysis of
equations derived by Zhang and Vinals [101, which describe small amplitude
surface waves on a semi-infinite weakly inviscid fluid layer. We have recently
extended parts of our analysis to the full hydrodynamic equations, which apply
to a layer of viscous fluid of finite thickness, and will present these results.
We are currently extending our analysis to the vibrational Rayleigh-Bernard
convection problem, with the immediate goal of determining the role of Boussinesq symmetry in the
pattern formation problem. We are also extending our linear analysis to Faraday
waves on the interface between two immiscible fluids, and to the case of waves
that are parametrically excited by a piecewise constant, periodic acceleration.
Preliminary results will be presented.
Silber, M., Topaz, C.M., Competing Harmonic and Subharmonic Instabilities and Parametrically Excited Surface Wave Patterns, Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 1275-1277, August 9, 2000.