TY - JOUR
T1 - Amplitude equation and pattern selection in Faraday waves
AU - Chen, Peilong
AU - Viñals, Jorge
PY - 1999
Y1 - 1999
N2 - A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter [Formula Presented]. For [Formula Presented], the selected wave pattern comprises a single standing wave (stripe pattern). For [Formula Presented], patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, [Formula Presented] patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.
AB - A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter [Formula Presented]. For [Formula Presented], the selected wave pattern comprises a single standing wave (stripe pattern). For [Formula Presented], patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, [Formula Presented] patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.
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U2 - 10.1103/PhysRevE.60.559
DO - 10.1103/PhysRevE.60.559
M3 - Article
C2 - 11969795
AN - SCOPUS:18144379623
SN - 1063-651X
VL - 60
SP - 559
EP - 570
JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
IS - 1
ER -