Nonlinear Strain Waves in Elastic Waveguides

  • A. M. Samsonov
Part of the CISM Courses and Lectures book series (CISM, volume 341)


Strain wave propagation in nonlinearly elastic wave guides is considered. The general idea is how, starting from the first principles, to reduce the initial highly nonlinear elastic wave problem governed by coupled p.d.e. to the only one “double dispersion” equation, describing longitudinal strain waves in a one-dimensional wave guide, e.g., in a rod. This study is aimed to arrange real physical experiments including generation, detection and observation of strain solitary waves in solids. The porblem of wave motion in a rod, having slowly varying cross-section or elastic moduli is solved. Special attention is paid to nonlinear dissipative waves, corresponding to the wave motion in a rod embedded into an active or a dissipative medium.

The general approach to find travelling wave solutions for various nonlinear problems with dispersion and dissipation is based essentially on the theorem proved recently, that allows to reduce many autonomous nonlinear dissipative 2nd order o.d.e.’s to the Abel equation of the 1st order. Using the approach, new exact solutions are obtained for some nonlinear wave problems. Finally, some physical experiments concerning strain soliton generation in nonlinearly elastic materials are discussed, in particular, the propagation of a soliton induced by a laser generated weak shock wave and detected by the optical holography method.


Solitary Wave Travel Wave Solution Solitary Wave Solution Wave Guide Weak Shock Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    AMIRANOFF, F. et al., (1985). Laser-driven shock wave studies using optical shadowgraphy, Phys. Rev., A32, 6, pp 3535–3546.Google Scholar
  2. 2.
    BLAND, D.R., (1969). Nonlinear dynamic elasticity, Blaisdell Publ. Co., Oxford.Google Scholar
  3. 3.
    BHARATHA, S., LEVINSON, M., (1977). On physically nonlinear elasticity, J. Elasticity, Vol 7. No. 3.Google Scholar
  4. 4.
    BLATZ, P.J., (1969). Application of large deformation theory to the thermo-mechanical behaviour of rubber-like polymers porous, unfilled and filled. In: Rheology-Theory and Applications, Vol 5, ed. F.R. Eirich, Academic Press, pp 1–55.Google Scholar
  5. 5.
    CLARKSON, P.A., LEVEQUE, R.J., SAXTON, R., (1986). Solitary wave interactions in elastic rods, Stud. Appl. Math., Vol. 75, pp 95–122.Google Scholar
  6. DREIDEN, G.V. et al.,(1988). Formation and propagation of strain solitons in non-linearly elastic solids, Soy. Phys. - Techn. Phys.,Vol 33, No. 10, pp 1237–1241.Google Scholar
  7. 7.
    ENGELBRECHT, J. (1983). Nonlinear wave processes of deformation in solids, Pitman, London.Google Scholar
  8. 8.
    HUNTER, J.K., SCHEURLE, J. (1988). Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32, pp 253–268.CrossRefGoogle Scholar
  9. 9.
    INCE, E., (1964). Ordinary differential equations, Dover, New York.Google Scholar
  10. 10.
    JEFFREY, A., KAWAHARA, T., (1982). Asymptotic methods in nonlinear wave theory, Pitman, London, pp 256.Google Scholar
  11. 11.
    JOHNSON, R.S., (1970). A nonlinear equation incorporating damping and dispersion, J. Fl. Mech., Vol 42, pp 49-.Google Scholar
  12. KARABUTOV, A.A. et al.,(1984). Direct observation of shock acoustic wave formation in solids, Vestnik Mosk. Univ., Ser. 3, Fizika, Astronomiya,Vol. 25, 3, pp 89–91 (in Russian).Google Scholar
  13. 13.
    KERR, A.D., (1964). Elastic and viscoelastic foundation models, J. Appl. Mech., Vol. E31, pp 491–498.CrossRefGoogle Scholar
  14. 14.
    KIVSHAR, Y.S., MALOMED, B.A. (1989). Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys., Vol 61, 4, pp 763–915.CrossRefGoogle Scholar
  15. 15.
    KODAMA, J., ABLOWITZ, M., (1981). Perturbation of solitons and solitary waves, Stud. Appl. Math., Vol. 64, pp 225–245.Google Scholar
  16. 16.
    LOVE, A.E.H., (1952). A treatise on the mathematical theory of elasticity (4th ed), Cambridge University Press.Google Scholar
  17. 17.
    LURIE, A.I., (1980). Nonlinear theory of elasticity, Moscow, Nauka, pp 512, (English edition, 1990, Elsevier, Amsterdam, North Holland ).Google Scholar
  18. 18.
    MOONEY, M., (1940). A theory of large elastic deformation, J. Appl. Phys., Vol. 11, pp. 582–592.CrossRefGoogle Scholar
  19. 19.
    MOLOTKOV, I.A., VAKULENKO, S.A., (1980). Nonlinear elastic waves in inhomogeneous rods, Zapiski nauchn., Semin. LOMI., Vol. 99, pp 64–73 (in Russian).Google Scholar
  20. 20.
    NARIBOLI, G.A., SEDOV, A., (1970). Burgers’-KdV equation for viscoelastic rods and plates, J. Math. Anal. Appl., Vol. 32, 3, pp 661–677.CrossRefGoogle Scholar
  21. 21.
    OSTROVSKY, L.A., SUTIN, A.M., (1977). Nonlinear elastic waves in rods, Prildad, Matem, i Mekhan., Vol. 41, 3, pp 531–537 (in Russian).Google Scholar
  22. 22.
    PARKER, D.F., (1984). On the derivation of nonlinear rod theories from three-dimensional elasticity, J. Appl. Math. and Phys., (ZAMP), Vol 35, pp 833–847.CrossRefGoogle Scholar
  23. 23.
    PASTERNAK, N.L., (1954). New method for calculation of foundation on the elastic basement, Moscow, Gosstroiizdat, pp 56, (in Russian).Google Scholar
  24. 24.
    PLEUS, P., SAYIR, M., (1983). A second order theory for large deflections of slender beams, J. Appl. Math. and Phys., (ZAMP), Vol 34, pp 192–217.CrossRefGoogle Scholar
  25. 25.
    PORUBOV, A.V., SAMSONOV, A.M., (1993). Refinement of longitudinal strain wave propagation in a non-linearly elastic rod, J. Tech. Phys. Lett., Vol. 19, No. 12, pp 26–29.Google Scholar
  26. 26.
    POTAPOV, A.I., SOLDATOV, I.N., (1984). Quasiplane nonliner longitudinal wave beam in a plate, Akusticheski zhurnal, Vol. 30, 6, pp 819–822 (in Russian).Google Scholar
  27. 27.
    SHIELD, R.T., (1983). Equilibrium solutions for finite elasticity, Trans ASME, J. Appl. Math., Vol. 50, pp 1171–1180.Google Scholar
  28. 28.
    SAMSONOV, A.M., (1993). On exact travelling wave solutions for nonlinear acoustical problems with damping. Advances in Nonlinear Acoustics, World Science ed. H. Hobak, pp. 57–62.Google Scholar
  29. 29.
    SAMSONOV, A.M., (1993). Some exact wave solutions in terms of the Weierstrass functions for nonlinear hyperbolic equations. In: Future Directions in Nonlinear Dynamics, Plenum, New York, eds. Christiansen, Eilbeck, Parmentier, pp. 125–128.CrossRefGoogle Scholar
  30. 30.
    SAMSONOV, A.M., (1991). On some exact travelling wave solutions for nonlinear hyperbolic equations, In: Nonlinear waves and dissipative effects, eds. D. Fusco, A. Jeffrey, Longman, London, 1991, pp. 123–132.Google Scholar
  31. 31.
    SAMSONOV, A.M., (1990a). Deformation waves in a nonlinearly elastic inhomogeneous wave guide, Atti. Accad. Peloritana dei Pericolanti, Messina, Italy, Vol 68, Sec. 1, pp. 535–551.Google Scholar
  32. 32.
    SAMSONOV, A.M., (1990b). Nonlinear acoustic strain waves in elastic waveguides, In: “Frontiers on Nonlinear Acoustics”, eds. M.F. Hamilton & D.E. Blackstock, Elsevier, pp. 588–593.Google Scholar
  33. 33.
    SAMSONOV, A.M., (1988a). Existence and amplification of solitary strain waves in non-linearly elastic waveguides, Leningrad, A.F. Ioffe, Physical Technical Institute, 1988, pp 23, preprint no. 1259.Google Scholar
  34. 34.
    SAMSONOV, A.M., (1988b). On existence of longitudinal strain solitons in an infinite nonlinearly elastic rod, Soy. Phys.–Doklady, Vol. 33, pp. 298–300.Google Scholar
  35. 35.
    SAMSONOV, A.M., (1987). Transonic and subsonic localized waves in nonlinearly elastic waveguides, In: Proc. Intern. Conf. on Plasma Phys, Kiev, Naukova dumka, Vol. 4, pp. 88–90.Google Scholar
  36. 36.
    SAMSONOV, A.M., (1984). Soliton evolution in a rod with variable cross section, Soy. Physics–Doklady, Vol. 29, No. 7, pp. 586–587.Google Scholar
  37. 37.
    SAMSONOV, A.M., SOKURINSKAYA, E.V., (1989). Energy exchange between nonlinear waves in elastic wave guides and external media, In: Nonlinear Waves in Active Media, ed. J. Engelbrecht, Springer, Berling et al., pp. 99–104.Google Scholar
  38. 38.
    SAMSONOV, A.M., SOKURINSKAYA, E.V., (1988). On the excitation of a longitudinal deformation soliton in a nonlinear elastic rod, Soy. Phys. — Techn. Phys., Vol. 33, No. 8, pp. 989–991.Google Scholar
  39. 39.
    SAMSONOV, A.M., SOKURINSKAYA, E.V., (1987). Longitudinal solitary waves in inhomogeneous nonlinearly elastic rod, J. Appl. Math. and Mech. — PMM, Vol. 51, 3, pp. 483–488.Google Scholar
  40. 40.
    SOERENSEN, M.P., CHRISTIANSEN, P.L., LOMDAHL, P.S., (1984). Solitary waves in nonlinear elastic rods, I. J. Acoust. Soc. Amer., Vol. 76, pp. 871–879.CrossRefGoogle Scholar
  41. 41.
    SOERENSEN, M.P., CHRISTIANSEN, P.L., LOMDAHL, P.S., SKOVGAARD, O., (1987). Solitary waves in nonlinear elastic rods II, J. Acoust. Soc. Amer., Vol. 81, 6, pp. 871–879.CrossRefGoogle Scholar
  42. 42.
    TRELOAR, L.R.G., (1958). Physics of rubber and elasticity,Oxford University Press.Google Scholar
  43. 43.
    WRIGHT, T., (1984). Weak shocks and steady waves, Int. J. Solids and Struct., Vol. 20, pp. 911–919.CrossRefGoogle Scholar
  44. 44.
    WU, J. et al.,(1987). Observation of envelope solitons in solids, Phys. Rev. Lett.,Vol. 59, 24, pp. 2744–2747.Google Scholar
  45. 45.
    ZAREMBO, L.K., KRASILNIKOV, V.A., (1970). Nonlinear phenomena connected with elastic wave propagation in solids, Soviet Physics–Uspekhi, Vol. 102, 4, pp. 549–586.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1994

Authors and Affiliations

  • A. M. Samsonov
    • 1
  1. 1.A.F. Ioffe Physical Technical InstituteSt. PetersburgRussia

Personalised recommendations