Controlling Faraday waves with multi-frequency forcing

1. Controlling Faraday waves with multi-frequency forcing Mary SilberEngineering Sciences & Applied Mathematics Northwestern University http://www.esam.northwestern.edu/~silber Work with Jeff Porter (Univ. Comp. Madrid) & Chad Topaz (UCLA), Cristián Huepe, Yu Ding & Paul Umbanhowar (Northwestern), and Anne Catllá (Duke)

2. FARADAY CRISPATIONS – M. Faraday, Phil. Trans. R. Soc. Lond. (1831)

3. FARADAY CRISPATIONS – M. Faraday, Phil. Trans. R. Soc. Lond. (1831)

4. Edwards and Fauve, JFM (1994) 12-fold quasipattern Bordeaux to Geneva: 5cm, depth: 3mm

5. Kudrolli, Pier and Gollub, Physica D (1998) Superlattice pattern Birfurcation theoretic investigations of superlattice patterns: Dionne and Golubitsky, ZAMP (1992) Dionne, Silber and Skeldon, Nonlinearity (1997) Silber and Proctor, PRL (1998)

6. Arbell & Fineberg, PRE 2002

7. FARADAY CRISPATIONS

8. LINEARSTABILITY ANALYSIS Benjamin and Ursell, Proc. Roy. Soc. Lond. A (1954) Considered inviscid potential flow: in modulated gravity with free surface given by: Find satisfy the Mathieu equation: gravity-capillary wave dispersion relation with

9. MATHIEU EQUATION Subharmonic resonance From: Jordan & Smith

10. Unique capabilities of the Faraday system • Huge, easily accessible control parameter space • Multiple length scales compete (or cooperate) (Naïve) Schematic of Neutral Stability Curve: overall forcing strength m/2 n/2 p/2 q/2 wave number k cf. Huepe, Ding, Umbanhowar, Silber (2005)

11. Unique capabilities of the Faraday system • Huge, easily accessible control parameter space • Multiple length scales compete (or cooperate) Goal: Determine how forcing function parameters enhance (or inhibit) weakly nonlinear wave interactions. Benefits: Helps interpret existing experimental results. Leads to design strategy: how to choose a forcing function that favors particular patterns in lab experiments. Approach: equivariant bifurcation theory. Exploit spatio-temporal symmetries (and remnants of Hamiltonian structure) present in the weak-damping/weak-driving limit. Focus on (weakly nonlinear) three-wave interactions as building blocks of spatially-extended patterns.

12. Resonant triads • Lowest order nonlinear interactions • Building blocks of more complex patterns k1 + k2 = k3 k2 k3 qres k1 Resonant triads & Faraday waves: Müller, Edwards & Fauve, Zhang & Viñals,…

13. Resonant triads • Role in pattern selection: a simple example k2 k3 critical modes damped mode qres k1 spatial translation, reflection, rotation by p

14. Resonant triads • Role in pattern selection: a simple example k2 k3 critical modes damped mode (eliminate) qres k1 center manifold reduction

15. Resonant triads • Role in pattern selection: a simple example k2 “suppressing”, “competitive” “enhancing”, “cooperative” qres k1 rhombic equations:

17. consider free energy: nonlinear coupling coefficient:

18. Organizing Center forcing Expanded TW eqns. Hamiltonian structure SW eqns. time translation, time reversal symmetries damping Porter & Silber, PRL (2002); Physica D (2004)

19. Travelling Wave eqns. • Parameter (broken temporal) symmetries u=m denotes dominant driving frequency time translation symmetry:

20. Travelling Wave eqns. • Parameter (broken temporal) symmetries time reversal symmetry: Hamiltonian structure (for ): (See, e.g., Miles,JFM (1984))

21. Travelling Wave eqns. • Enforce symmetries  Travelling wave amplitude equations damping parametric forcing damping

22. Travelling Wave eqns. Time translation invariants: • Enforce symmetries  Travelling wave amplitude equations Example: (m,n) forcing, =m-n

23. Travelling wave eqns. Focus on Possible only for At most 5 relevant forcing frequencies for fixed W Perform center manifold reduction to SW eqns. • Enforce symmetries  Travelling wave amplitude equations Porter, Topaz and Silber, PRL & PRE 2004

24. Key results • Strongest interaction is for W = m • Parametrically forcing damped mode can strengthen interaction • Phases fu may tune interaction strength • Only W = n – m is always enhancing (Hamiltonian argument) ex. (m,n, p = 2n – 2m) forcing, W = n – m>0 > 0 Pp(F) > 0 bres > 0 for this case(can get signs for some other cases)

25. Zhang & Viñals, J. Fluid Mech 1997

26. Direct Reduction to Standing Wave eqns k2 q k1 Solvability condition at :

27. Demonstration of key results • Strongest interaction is for W = m • Parametrically forcing damped mode can strengthen interaction • Phases fu may tune interaction strength • Only W = n – m is always enhancing (bres > 0) ex. (6,7,2) forcing, b()computed from Zhang-Viñals equations: W = n – m = 1 W = m = 6

28. Example: Experimental superlattice pattern Kudrolli, Pier and Gollub, Physica D (1998) Topaz & Silber, Physica D (2002)

29. Example: Experimental superlattice pattern 6/7/2 forcing frequencies: Epstein and Fineberg, 2005 preprint. 3:2 5:3

30. Example: Experimental superlattice pattern Epstein and Fineberg, 2005 preprint. 3:2 4:3 5:3

31. Example: Experimental quasipattern q = 45 Arbell & Fineberg, PRE, 2002 { (3,2,4)forcing } (3,2)forcing

32. Example: Impulsive-Forcing (See J. Bechhoefer & B. Johnson, Am. J. Phys. 1996)

33. Example: Impulsive-Forcing (Catllá, Porter and Silber, PRE, in press) One-dim. waves Weakly nonlinear analysis from Z-V equations. C sinusoidal Capillarity parameter Prediction based on 2-term truncated Fourier series:

34. Linear Theory: Shallow and Viscous Case Huepe, Ding, Umbanhowar, Silber, 2005 preprint Forcing function Neutral Curve

35. Linear Theory: Shallow and Viscous Case Linear analysis, aimed at finding envelope of neutral curves ( following Cerda & Tirapegui, JFM 1998): Lubrication approximation: shallow, viscous layer, low-frequency forcing Transform to time-independent Schrödinger eqn.,1-d periodic potential WKB approximation: Matching across regions gives transition matrices Periodicity requirement determines stability boundary

36. Linear Theory: Shallow and Viscous Case Exact numerical: WKB approximation:

37. Conclusions • Determined how & which parameters in periodic forcing function influence weakly nonlinear 3-wave interactions. • Weak-damping/weak-forcing limit leads to scaling laws and phase dependence of coefficients in bifurcation equations. • Hamiltonian structure can force certain interactions to be “cooperative”, while others are “competitive”. • Results suggest how to control pattern selection by choice of forcing function frequency content. ( cf. experiments by Fineberg’s group). • Symmetry-based approach yields model-independent results; arbitrary number of (commensurate) frequency components. (even infinite -- impulsive forcing) • Shallow, viscous layers present new challenges…