Some Aspects of One-Dimensional Finite Amplitude Elastic Wave Propagation

  • J. B. Haddow
  • R. J. Tait


This chapter is concerned with certain aspects of the propagation of one dimensional waves in solids. The governing equations for the problems considered are partial differential equations in two independent variables, a spatial variable, denoted by x, and a temporal variable t. We concentrate on techniques pertinent to wave propagation and refer the reader to texts such as [1], [2] and [3] for other aspects of the theory of partial differential equations. Additional information relevant to wave propagation may also be found in the books by Whitham [4] and Smoller [5].

We consider first order partial differential equations and systems of first order equations. The application of these systems to problems of one dimensional wave propagation in strings, membranes and unbounded solids is discussed.


Weak Solution Riemann Problem Contact Discontinuity Simple Wave Order Partial Differential Equation 
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  1. 1.
    R. Courant and D. Hilbert Methods of Mathematical Physics, Vol. II, Partial Differential Equations, Wiley-Interscience, New York, 1962.Google Scholar
  2. 2.
    P. Garabedian, Partial Differential Equations, Wiley, New York, 1964.MATHGoogle Scholar
  3. 3.
    I. N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, New York, 1957.MATHGoogle Scholar
  4. 4.
    G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.MATHGoogle Scholar
  5. 5.
    J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, 1983.Google Scholar
  6. 6.
    P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Conf. Board Math. Sci., 11, SIAM, 1973.Google Scholar
  7. 7.
    B. Riemann, Gesammeltte Werke, Teubner, 1896.Google Scholar
  8. 8.
    R. Courant and K. Friederichs, Supersonic Flow and Shock Waves, Wiley-Interscience, New York, 1948.MATHGoogle Scholar
  9. 9.
    G. A. Sod, Numerical Methods in Fluid Dynamics, Cambridge University Press, Cambridge, 1987.Google Scholar
  10. 10.
    R. W. Ogden, Non-Linear Elastic Deformations, Ellis-Horwood, John Wiley and Sons, 1983.Google Scholar
  11. 11.
    A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, 1978.Google Scholar
  12. 12.
    G. C. Ciarlet, Mathematical Elasticity, Vol. 1, Three Dimensional Elasticity, North-Holland, 1978.Google Scholar
  13. 13.
    J. Englebrecht, Nonlinear Wave Processes of Deformation in Solids, Pitman Publishing, 1983.Google Scholar
  14. 14.
    M. Shearer, The Riemann problem for the planar motion of an elastic string, J. of Differential Equations, 61 (1986), 149-163.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    B. Wendroff, The Riemann problem for materials with nonconvex equations of state, J. Math. Anal. AppL, 38 (1972), 454-466.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    J. L. Wegner, Some problems of hyperbolic wave propagation, PhD. Thesis, University of Alberta, 1988.Google Scholar
  17. 17.
    J. L. Zhong, Impact problems for strings and membranes, PhD. Thesis, University of Alberta, 1994.Google Scholar
  18. 18.
    K. Abdella, Propagation of waves in nonlinear elasticity, MSc. Thesis, University of Alberta, 1989.Google Scholar
  19. 19.
    T. P. Liu, The Riemann problem for general 2x2 conservation laws, Trans. Amer. Math. Soc, 199 (1974), 89-112.MathSciNetMATHGoogle Scholar
  20. 20.
    J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure App. Math., 18 (1965), 95-105.MathSciNetCrossRefGoogle Scholar
  21. 21.
    J. L. Wegner, J. B. Haddow and R. J. Tait, Unloading waves in a plucked hyperelastic string, J. Appl. Mech., 56 (1989), 459-465.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    R. J. Tait and J. L. Zhong, Perturbation methods for the impact problem of a nonlinear elastic string, Int. J. Non-Linear Mech., 28 (1993), 713-730.MATHCrossRefGoogle Scholar
  23. 23.
    R. J. Tait and D. B. Duncan, Motion of a mass on a nonlinear elastic string, Int. J. Non-Linear Mech., 27 (1992), 139-148.MATHCrossRefGoogle Scholar
  24. 24.
    R. J. Tai, K. Abdella and D. B. Duncan, Approximate Riemann solvers and waves in a nonlinear elastic string, Comp. Math. Applic, 21 (1991), 77-89.Google Scholar
  25. 25.
    R. J. Tait and J. B. Haddow, On a simple catapult problem, Int. J. Non-Linear Mech., 26 (1991), 741-752.MATHCrossRefGoogle Scholar
  26. 26.
    A. E. Green, P. M. Naghdi and W. L. Wainwright, A general theory of a Cosserat surface, Archiv for Rat. Mech. and Analysis, 2 (1965), 287-308.MathSciNetGoogle Scholar
  27. 27.
    P. M. Naghdi, Finite deformation of elastic rods and shells, in: Proc. IUTAM Symposium on Finite Elasticity, D. E. Carlson and R. T. Shield, eds., Martinus Nijhoff, The Hague, 1982, 47-102.Google Scholar
  28. 28.
    P. M. Naghdi and P. Y. Tang, Large deformations possible in every isotropic elastic membrane, Phil. Trans. Roy. Soc. London, Ser. A, 287 (1977), 145-187.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    J. B. Haddow, J. L. Wegner and L. Jiang, The dynamic response of a stretched circular hyperelastic membrane subject to normal impact, Wave Motion, 16(1992), 137-150.MATHCrossRefGoogle Scholar
  30. 30.
    R. J. Tait and J. L. Zhong, An impact problem for a nonlinear elastic membrane with non-constant boundary conditions, ZAMM, 75 (1995), 605-613.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    C. L. Farrar, Impact response of a circular membrane, Experimental Mechanics, 24 (1984), 144-150.CrossRefGoogle Scholar
  32. 32.
    R. J. Tait and J. L. Zhong, Wave propagation in a nonlinear elastic tube, Bull. Tech. Univ. Istanbul, 47 (1994), 127-150.MathSciNetMATHGoogle Scholar
  33. 33.
    R. J. Tait, D. J. Steigmann and J. L. Zhong, Finite twist and extension of a cylindrical elastic membrane, Acta Mechanica, 117, (1996), 129-143.MATHCrossRefGoogle Scholar
  34. 34.
    R. J. Tait and J. L. Zhong, Dynamic extension and twist of a nonlinear elastic tube, Int. J. Non-Linear Mech., 30 (1995), 887-898.MATHCrossRefGoogle Scholar
  35. 35.
    R. J. Tait, J. B. Haddow and T. B. Moodie, A note on infinitesimal shear waves in a finitely deformed elastic solid, Int. J. Eng. Sc, 22:7 (1984), 823-827.MATHCrossRefGoogle Scholar
  36. 36.
    W. D. Collins, The propagation and interaction of one dimensional nonlinear waves, Q. J. Mech. and Appl. Math., 20 (1967), 429-452.MATHCrossRefGoogle Scholar
  37. 37.
    J. B. Haddow, S. A. Lorimer and R. J. Tait, Nonlinear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid, Int. J. Non-Linear Mech., 22 (1987), 297-306.MATHCrossRefGoogle Scholar
  38. 38.
    R. J. Tait, S. A. Lorimer and J. B. Haddow, Finite amplitude elastic shear wave propagation, Wave Motion, 11 (1989), 251-260.MATHCrossRefGoogle Scholar
  39. 39.
    J. B. Haddow, Some thermodynamic aspects of finite amplitude unloading waves in hyperelastic strings, ZAMM, 71 (1993), 47-54.MathSciNetGoogle Scholar
  40. 40.
    J. B. Haddow, Nonlinear hyperbolic waves in hyperelastic solids, in: IUTAM Symposium, Nonlinear waves in solids, Victoria, August 1993, Applied Mechanics Reviews, 46 (1993), Part 1, 527-539.Google Scholar

Copyright information

© Springer Science+Business Media New York  2002

Authors and Affiliations

  • J. B. Haddow
    • 1
  • R. J. Tait
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsUniversity of AlbertaEdmontonCanada

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