Encyclopedia of Thermal Stresses

2014 Edition
| Editors: Richard B. Hetnarski

Wave Solutions

  • Ivan C. Christov
Reference work entry
DOI: https://doi.org/10.1007/978-94-007-2739-7_33


In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave’s phase speed on its wave number. For systems of nonlinear first-order hyperbolic...

This is a preview of subscription content, log in to check access.


  1. 1.
    Scales JA, Snieder R (1999) What is a wave? Nature 401:739–740Google Scholar
  2. 2.
    LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, New YorkzbMATHGoogle Scholar
  3. 3.
    Guenther RB, Lee JW (1988) Partial differential equations of mathematical physics and integral equations. Prentice Hall, Englewood Hills, NJGoogle Scholar
  4. 4.
    Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73zbMATHMathSciNetGoogle Scholar
  5. 5.
    Joseph DD, Preziosi L (1990) Addendum to the paper “Heat Waves” [Rev. Mod. Phys. 61, 41 (1989)]. Rev Mod Phys 62: 375–391Google Scholar
  6. 6.
    Courant R, Hilbert D (1953) Methods of mathematical physics, vol I. Wiley, New YorkGoogle Scholar
  7. 7.
    Caviglia G, Morro A (1992) Inhomogeneous waves in solids and fluids, vol 7, Series in theoretical and applied mechanics. World Scientific, SingaporezbMATHGoogle Scholar
  8. 8.
    Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkzbMATHGoogle Scholar
  9. 9.
    Noda N, Hetnarski RB, Tanigawa Y (2003) Thermal stresses, 2nd edn. Taylor & Francis, New YorkGoogle Scholar
  10. 10.
    Courant R, Hilbert D (1962) Methods of mathematical physics, vol II. Wiley, New YorkzbMATHGoogle Scholar
  11. 11.
    Lax PD (2006) Hyperbolic partial differential equations, vol 14, Courant lecture notes in mathematics. American Mathematical Society, Providence, RIzbMATHGoogle Scholar
  12. 12.
    Friedrichs KO, Lax PD (1971) Systems of conservation equations with a convex extension. Proc Natl Acad Sci USA 68:1686–1688zbMATHMathSciNetGoogle Scholar
  13. 13.
    Reverberi AP, Bagnerini P, Maga L, Bruzzone AG (2008) On the non-linear Maxwell–Cattaneo equation with non-constant diffusivity: shock and discontinuity waves. Int J Heat Mass Transfer 51:5327–5332zbMATHGoogle Scholar
  14. 14.
    Christov IC, Jordan PM (2010) On the propagation of second-sound in nonlinear media: shock, acceleration and traveling wave results. J Thermal Stresses 33:1109–1135Google Scholar
  15. 15.
    Gilding BH, Kersner R (2004) Travelling waves in nonlinear diffusion-convection reaction. Birkhäuser, BaselzbMATHGoogle Scholar
  16. 16.
    Polyanin AD, Zaitsev VF (2003) Handbook of nonlinear partial differential equations. Chapman & Hall/CRC, Boca Raton, FLGoogle Scholar
  17. 17.
    Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, OxfordGoogle Scholar
  18. 18.
    Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W-function. Adv Comput Math 5:329–359zbMATHMathSciNetGoogle Scholar
  19. 19.
    Destrade M, Gaeta G, Saccomandi G (2007) Weierstrass criterion and compact solitary waves. Phys Rev E 75:047,601MathSciNetGoogle Scholar
  20. 20.
    Wilhelm HE, Choi SH (1975) Nonlinear hyperbolic theory of thermal waves in metals. J Chem Phys 63:2119–2123MathSciNetGoogle Scholar
  21. 21.
    Vázquez JL (2007) The porous medium equation: mathematical theory. Oxford University Press, OxfordGoogle Scholar
  22. 22.
    Zel’dovich YB, Raizer YP (1967) Physics of shock waves and high-temperature hydrodynamic phenomena, vol II. Academic Press, New YorkGoogle Scholar
  23. 23.
    Ablowitz MJ, Zeppetella A (1979) Explicit solutions of Fisher’s equation for a special wave speed. Bull Math Biol 41:835–840zbMATHMathSciNetGoogle Scholar
  24. 24.
    Courant R, Friedrichs KO (1948) Supersonic flow and shock waves. Wiley, New YorkzbMATHGoogle Scholar
  25. 25.
    Morse PH, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New YorkzbMATHGoogle Scholar
  26. 26.
    Bland DR (1988) Wave theory and applications. Oxford University Press, OxfordzbMATHGoogle Scholar
  27. 27.
    Straughan B (2011) Heat waves, applied mathematical sciences, vol 177. Springer, New YorkGoogle Scholar
  28. 28.
    Ignaczak J, Ostoja-Starzewski M (2010) Thermoelasticity with finite wave speeds. Oxford University Press, New YorkzbMATHGoogle Scholar
  29. 29.
    Trangenstein JA (2007) Numerical solution of hyperbolic partial differential equations. Cambridge University Press, New YorkGoogle Scholar
  30. 30.
    Hetnarski RB, Ignaczak J (1995) Soliton-like waves in a low-temperature nonlinear rigid heat conductor. Int J Eng Sci 33:1725–1741zbMATHMathSciNetGoogle Scholar
  31. 31.
    Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiazbMATHGoogle Scholar
  32. 32.
    Dauxois T, Peyrard M (2006) Physics of solitons. Cambridge University Press, CambridgezbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA