Instability of a horizontal water half-cylinder under vertical vibration


We present the results of an experimental investigation on parametrically driven waves in a water half-cylinder on a rigid horizontal plate, which is sinusoidally vibrated in the vertical direction. As the forcing amplitude is raised above a critical value, stationary waves are excited in the water half-cylinder. Parametrically excited subharmonic waves are non-axisymmetric and qualitatively different from the axisymmetric Savart–Plateau–Rayleigh waves in a vertical liquid cylinder or jet. Depending on the driving frequency, stationary waves of different azimuthal wave numbers are excited. A linear theory is also supplemented, which captures the observed dispersion relations quantitatively.

Graphic abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. Adou AE, Tuckerman LS (2016) Faraday instability on a sphere: floquet analysis. J Fluid Mech 805:591.

    MathSciNet  Article  MATH  Google Scholar 

  2. Binks D, Van de Water W (1997) Nonlinear pattern formation of Faraday waves. Phys Rev Lett 78:4043.

    Article  Google Scholar 

  3. Chandrasekhar SC (1961) Hydrodynamic and hydromagnetic stability, 3rd edn. Clarendon Press, Oxford (reprinted by Dover Publications, New York, 1981)

    Google Scholar 

  4. Ciliberto S, Douady S, Fauve S (1991) Investigating space-time chaos in Faraday instability by means of the fluctuations of the driving acceleration. Europhys. Lett. 15:23

  5. de Jesus VLB (2017) Experiments and video analysis in classical mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  6. Donnelly RJ, Glaberson W (1966) Experiments on the capillary instability of a liquid jet. Proc R Soc Lond A 290:547.

    Article  Google Scholar 

  7. Douady S (1990) Experimental study of the Faraday instability. J Fluid Mech 221:383.

    Article  Google Scholar 

  8. Douglas B (2016) More details are avialable on the webpage of the free software Tracker at the link:

  9. Edwards WS, Fauve S (1994) Patterns and quasi-patterns in the Faraday experiment. J Fluid Mech 278:123.

    MathSciNet  Article  Google Scholar 

  10. Faraday M (1831) On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil Trans R Soc Lond 121:299.

    Article  Google Scholar 

  11. Fauve S, Kumar K, Laroche C, Beysens D, Garrabos Y (1992) Parametric instability of a liquid–vapour interface close to the critical point. Phys Rev Lett 68:3160.

    Article  Google Scholar 

  12. Kudrolli A, Gollub JP (1996) Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97:133.

    Article  Google Scholar 

  13. Kumar K (1996) Linear theory of Faraday instability in viscous fluids. Proc R Soc Lond A 452:1113.

    Article  MATH  Google Scholar 

  14. Kumar K, Bajaj KMS (1995) Competing patterns in the Faraday experiment. Phys Rev E 52:R4606.

    Article  Google Scholar 

  15. Lamb H (1932) Hydrodynamics. Cambridge University Press, Cambridge

    Google Scholar 

  16. McHale G, Elliott SJ, Newton MI, Herbertson DL, Esmer K (2009) Levitation-free vibrated droplets: resonant oscillations of liquid marbles. Langmuir 25:529.

    Article  Google Scholar 

  17. Moseler M, Landman U (2000) Formation, stability, breakup of nanojets. Science 289:1165.

    Article  Google Scholar 

  18. Müller HW (1993) Periodic triangular patterns in the Faraday experiment. Phys Rev Lett 71:3287.

    Article  Google Scholar 

  19. Müller HW, Wittmer H, Wagner C, Albers J, Knorr K (1997) Analytic stability theory for Faraday waves and the observation of the harmonic surface response. Phys Rev Lett 78:2357.

    Article  Google Scholar 

  20. Noblin X, Buguin A, Brochard-Wyart F (2005) Triplon modes of puddles. Phys Rev Lett 94:166102.

    Article  Google Scholar 

  21. Plateau JAF (1843) Acad Sci Brux Mem 16:3

  22. Plateau JAF (1849) Acad Sci Brux Mem 23:5

    Google Scholar 

  23. Plateau JAF (1873) Statique experimentale et théorique des Liquides soumis aux seules forces moléculaires, vol 2. Gauthier Villars, Paris

    Google Scholar 

  24. Rayleigh L (1879) Proc R Soc Lond 10:4

  25. Rayleigh Lord (1892) On the instability of a cylinder of viscous liquid under capillary force. Phil Mag 34:145.

    Article  MATH  Google Scholar 

  26. Rayleigh L (1896) The theory of sound. Macmillan, London (reprinted by Dover Publications, New York, 1945)

    Google Scholar 

  27. Savart F (1833) Wemoire sur la constitution des veines liquids lancees par des orifices circulaires en mince paroi. Ann de Chimie 53:337

    Google Scholar 

  28. Tarasov N, Perego AM, Churkin DV, Staliunas K, Turitsyn SK (2016) Mode-locking via dissipative Faraday instability. Nat Commun 7:12441.

    Article  Google Scholar 

  29. Xu J, Attinger D (2007) Acoustic excitation of superharmonic capillary waves on a meniscus in a planar microgeometry. Phys Fluids 19:108107.

    Article  MATH  Google Scholar 

Download references


Partial support from SERB, India through Project Grant No. EMR/2016/000185 is acknowledged. The authors acknowledge fruitful suggestions from anonymous referees, which improved the manuscript.

Author information



Corresponding author

Correspondence to Krishna Kumar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (MP4 16,906 kb)

Supplementary material 1 (MP4 6,771 kb)

Supplementary material 1 (MP4 4,478 kb)

Supplementary material 1 (CLS 47 kb)

Supplementary material 1 (BST 33 kb)

Supplementary material 1 (BST 30 kb)

Supplementary material 1 (BST 28 kb)

Supplementary material 1 (CLO 4 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maity, D.K., Kumar, K. & Khastgir, S.P. Instability of a horizontal water half-cylinder under vertical vibration. Exp Fluids 61, 25 (2020).

Download citation