Granular Matter

, 20:42 | Cite as

Possibility of useful mechanical energy from noise: the solitary wave train problem in the granular chain revisited

  • Sourish ChakravartyEmail author
  • Surajit Sen
Original Paper


A momentary velocity perturbation at an edge of a granular chain with the grains barely touching one another and held between fixed walls propagates as a solitary wave whereas a long lived perturbation, even if it is noisy, ends up as a solitary wave train. Here, we extend our earlier work but with a force instead of a velocity perturbation. Such a perturbation can propagate an extended compression front into the system. We find that a snapshot of the distribution of grain compressions in the solitary wave train shows parabolic as opposed to an approximate exponential decay with the leading edge at the front of the traveling pulse and the trailing edge following it. The system’s time evolution depends on three independent parameters-the material properties, duration of perturbation and the characteristic amplitude of the perturbation. Hence, the coefficients used to describe the parabolic decay of the grain compressions in the solitary wave train depend on these three parameters. When a random finite duration force perturbation is applied we find that the randomness is smoothed out by the system, which in turn suggests that long granular chains (or equivalent systems, such as circuits) can be potentially useful in converting random noisy signals to organized solitary wave trains and hence to potentially usable energy.


Granular chains Nonlinear dynamics Solitary wave trains 



We are grateful to the US Army Research Office for partial support of the work reported here. We would like to thank Robert W. Newcomb and Gary F. Dargush for their interest in the work. SC thanks Abhishek Venketeswaran and Kundan Goswami for useful discussions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Biot, M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4), 1254–1264 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biot, M.A.: Mechanics of deformation and acoustic propagation in porous dissipative media. J. Appl. Phys. 34(9), 1482–1498 (1962)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Hardin, B.O., Richart, F.E.: Elastic wave velocities in granular media. J. Soil. Mech. Found. Div. 89, Proc. Paper 3407 (1963)Google Scholar
  4. 4.
    Karpman, V.I.: Nonlinear Waves in Dispersive Media. Pergamon Press Ltd., Oxford (1974)Google Scholar
  5. 5.
    Liu, C.-H., Nagel, S.R.: Sound in a granular material: disorder and nonlinearity. Phys. Rev. B 48(21), 15646 (1993)ADSCrossRefGoogle Scholar
  6. 6.
    Jia, X., Caroli, C., Velicky, B.: Ultrasound propagation in externally stressed granular media. Phys. Rev. Lett. 82(9), 1863–1869 (1999)ADSCrossRefGoogle Scholar
  7. 7.
    Makse, H.A., Gland, N., Johnson, D.L., Schwartz, L.: Granular packings: nonlinear elasticity, sound propagation, and collective relaxation dynamics. Phys. Rev. E 70(6), 061302 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    Sen, S., Krishna Mohan, T.R., Visco, D.R., Swaminathan, S., Sokolow, A., Avalos, E., Nakagawa, M.: Using mechanical energy as a probe for the detection and imaging of shallow buried inclusions in dry granular beds. Int. J. Mod. Phys. B 19(18), 2951–2973 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    O’Donovan, J., Ibraim, E., O’Sullivan, C., Hamlin, S., Muir Wood, D., Marketos, G.: Micromechanics of seismic wave propagation in granular materials. Granul. Matter 18(03), 1–18 (2016)Google Scholar
  10. 10.
    Gilcrist, L.E., Baker, G.S., Sen, S.: Preferred frequencies for three unconsolidated earth materials. Appl. Phys. Lett. 91(25), 254103 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    Shrivastava, R.K., Luding, S.: Wave propagation of spectral energy content in a granular chain. EPJ Web Conf. EDP Sci. 140, 02023 (2017)CrossRefGoogle Scholar
  12. 12.
    Shrivastava, R.K., Luding, S.: Effect of disorder on bulk sound wave speed: a multiscale spectral analysis. Nonlinear Process Geophys. 24(3), 435 (2017)ADSCrossRefGoogle Scholar
  13. 13.
    Nesterenko, V.F.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24(5), 733–743 (1983)ADSCrossRefGoogle Scholar
  14. 14.
    Lazaridi, A., Nesterenko, V.F.: Observation of a new type of solitary waves in a one-dimensional granular medium. J. Appl. Mech. Tech. Phys. 26(3), 405 (1985)ADSCrossRefGoogle Scholar
  15. 15.
    Nesterenko, V.F.: Dynamics of Heterogenous Materials. Springer, New York (2001)CrossRefGoogle Scholar
  16. 16.
    Sinkovits, R.S., Sen, S.: Nonlinear dynamics in granular columns. Phys. Rev. Lett. 74(14), 2686–2689 (1995)ADSCrossRefGoogle Scholar
  17. 17.
    Sen, S., Sinkovits, R.S.: Sound propagation in impure granular columns. Phys. Rev. E 54(6), 6857–6865 (1996)ADSCrossRefGoogle Scholar
  18. 18.
    Coste, C., Falcon, E., Fauve, S.: Solitary waves in a chain of beads under Hertz contact. Phys. Rev. E 56(5), 6104–6117 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    Chatterjee, A.: Asymptotic solution for solitary waves in a chain of elastic spheres. Phys. Rev. E 59, 5912–5919 (1999)ADSCrossRefGoogle Scholar
  20. 20.
    Mouraille, O., Mulder, W.A., Luding, S.: Sound wave acceleration in granular materials. J. Stat. Mech. Theory Exp. 07, 07023 (2006)CrossRefGoogle Scholar
  21. 21.
    Sen, S., Hong, J., Bang, J., Avalos, E., Doney, R.: Solitary waves in the granular chain. Phys. Rep. 462(2), 21 (2008)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Sokolow, A., Bittle, E.G., Sen, S.: Formation of solitary wave trains in granular alignments. Europhys. Lett. 77, 24002 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    Job, S., Melo, F., Sokolow, A., Sen, S.: Solitary wave trains in granular chains: experiments, theory and simulations. Granul. Matter 10, 13–20 (2007)CrossRefzbMATHGoogle Scholar
  24. 24.
    Hertz, H.: Über den kontakt elastischer körper. J. reine angew. Math. 92, 156–171 (1881)Google Scholar
  25. 25.
    Sun, D., Daraio, C., Sen, S.: Nonlinear repulsive force between two solids with axial symmetry. Phys. Rev. E 83, 066605 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    Goldsmith, W.: Impact—The Theory and Physical Behavior of Colliding Solids. Edwards Arnold, London (1960)zbMATHGoogle Scholar
  27. 27.
    Sellami, L., Newcomb, R.W., Sen, S.: Simulink modeling for circuit representation of granular chains. Mod. Phys. Lett. B 27, 1350093 (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Bolotin, V.V.: Random Vibrations of Elastic Systems—Mechanics of Elastic Stability 8. Martinus Nijhoff, The Hague (1984)CrossRefGoogle Scholar
  29. 29.
    Manciu, M., Sen, S., Hurd, A.J.: Crossing of identical solitary waves in a chain of elastic beads. Phys. Rev. E 63, 016614 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    Manciu, F.S., Sen, S.: Secondary solitary wave formation in systems with generalized Hertz interactions. Phys. Rev. E 66, 016616 (2002)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Job, S., Melo, S., Sokolow, A., Sen, S.: How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys. Rev. Lett. 94, 178002 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    Santibanez, F., Munoz, R., Caussarieu, A., Job, S., Melo, F.: Experimental evidence of solitary wave interaction in Hertzian chains. Phys. Rev. E 84(2), 026604 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22, 431 (1967)ADSCrossRefGoogle Scholar
  34. 34.
    Toda, M.: Waves in nonlinear lattice. Prog. Theor. Phys. Suppl. 45, 174 (1970)ADSCrossRefGoogle Scholar
  35. 35.
    Rosas, A., Lindenberg, K.: Pulse velocity in a granular chain. Phys. Rev. E 69, 037601 (2004)ADSCrossRefGoogle Scholar
  36. 36.
    Przedborski, M., Sen, S., Harroun, T.A.: Fluctuations in Hertz chains at equilibrium. Phys. Rev. E 95, 032903 (2017)ADSCrossRefGoogle Scholar
  37. 37.
    Takato, Y., Sen, S., Lechman, J.: Strong plastic deformation and softening of fast colliding nanoparticles. Phys. Rev. E 89, 033308 (2014)ADSCrossRefGoogle Scholar
  38. 38.
    Takato, Y., Benson, M.E., Sen, S.: Granular chains with soft boundaries: slowing the transition to quasi-equilibrium. Phys. Rev. E 91, 042207 (2015)CrossRefGoogle Scholar
  39. 39.
    Hasan, M.A., Nemat-Nasser, S.: Universal relations for solitary waves in granular crystals under shocks with finite rise and decay times. Phys. Rev. E 93, 042905 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringThe State University of New York at Buffalo BuffaloUSA
  2. 2.The Picower Institute for Learning and MemoryMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of PhysicsThe State University of New York at BuffaloBuffaloUSA

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