TY - JOUR
T1 - Pattern formation in weakly damped parametric surface waves
AU - Zhang, Wenbin
AU - Viñals, Jorge
PY - 1997/4/10
Y1 - 1997/4/10
N2 - We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary-gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.
AB - We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary-gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.
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U2 - 10.1017/S0022112096004764
DO - 10.1017/S0022112096004764
M3 - Article
AN - SCOPUS:0031126953
SN - 0022-1120
VL - 336
SP - 301
EP - 330
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
ER -