The classical wave equation is a cornerstone in mathematical physics and mechanics. Its modifications are widely used in order to describe wave phenomena. In mechanics deformation waves are related to impact problems, acoustic waves are used in Nondestructive Evaluation, seismic waves may cause a lot of damage, etc. In this paper it is shown how the classical wave equation can be modified in order to model better the physics of processes. The examples cover microstructured and inhomogeneous materials together with linear and nonlinear models. Beside usual two-wave models, the evolution equations are described which govern the distortion of a single wave.
1. Stewart, I. 17 Equations that Changed the World. Profile, London, 2012.
2. Wheeler, G. F. and Crummelt, W. P. The vibrating string contraversy. Am. J. Physics, 1987, 55, 1, 33–37.
http://dx.doi.org/10.1119/1.15311
3. Jost, J. Partial Differential Equations. Springer, New York et al., 2002.
4. Truesdell, C. and Noll, W. The non-linear field theories of mechanics. In Enc. of Physics (Flügge, S., ed.), III/3, Springer, Berlin et al., 1965.
5. Engelbrecht, J. Nonlinear Wave Dynamics. Complexity and Simplicity. Kluwer, Dordrecht, 1997.
6. Kolsky, H. Stress Waves in Solids. 2nd ed. Dover, New York, 1963.
7. Achenbach, J. D. Wave Propagation in Elastic Solids. North Holland, Amsterdam, 1973.
8. Bland, D. R. Wave Theory and Applications. Clarendon Press, Oxford, 1988.
9. Graff, K. F. Wave Motion in Elastic Solids. Dover, New York, 1991 (1st ed. 1975).
10. Taniuti, T. and Nishihara, K. Nonlinear Waves. Pitman, London et al., 1983 (in Japanese 1977).
11. Engelbrecht, J. Nonlinear Wave Processes of Deformation in Solids. Pitman, London, 1983.
12. Cantrell, J. H. Acoustic nonlinearity parameters and higher-order elastic constants for crystals. Proc. IOA, 1989, 11, Part 5, 445–452.
13. Naugolnykh, K. and Ostrovsky, L. Nonlinear Wave Processes in Acoustics. Cambridge University Press, Cambridge, 1998.
14. Mindlin, R. D. Micro-structure in linear elasticity. Arch. Rat. Mech. Anal., 1964, 16, 51–78.
http://dx.doi.org/10.1007/BF00248490
15. Eringen, A. C. Microcontinuum Field Theories I: Foundations and Solids. Springer, New York et al., 1999.
16. Engelbrecht, J., Berezovski, A., Pastrone, F. and Braun, M. Waves in microstructured materials and dispersion. Phil. Mag., 2005, 85, 33–35, 4127–4141.
http://dx.doi.org/10.1080/14786430500362769
17. Berezovski, A., Engelbrecht, J., Salupere, A., Tamm, K., Peets, T. and Berezovski, M. Dispersive waves in microstructured solids. Int. J. Solids Structures, 2013, 50, 1981–1990.
http://dx.doi.org/10.1016/j.ijsolstr.2013.02.018
18. Christov, C., Maugin, G. and Porubov, A. V. On Boussinesq’s paradigm in nonlinear wave propagation. C.R. Mécanique, 2007, 335, 521–535.
http://dx.doi.org/10.1016/j.crme.2007.08.006
19. Engelbrecht, J., Salupere, A. and Tamm, K. Waves in microstructured solids and the Boussinesq’s paradigm. Wave Motion, 2011, 48, 717–726.
http://dx.doi.org/10.1016/j.wavemoti.2011.04.001
20. Ravasoo, A. Nonlinear waves in characterization of inhomogeneous elastic material. Mech. Mater., 1999, 31, 205–213.
http://dx.doi.org/10.1016/S0167-6636(98)00061-1
21. Ravasoo, A. Interaction of bursts as a detector of material inhomogeneity. Acta Acustica united with Acustica, 2012, 98, 864–869.
22. Chen, P. J. Growth and decay of waves in solids. In Enc. of Physics (Flügge, S., ed.), VI/3, Springer, Berlin, 1973.
23. Engelbrecht, J. and Braun, M. Nonlinear waves in nonlocal media. Appl Mech. Rev., 1998, 51, 475–488.
http://dx.doi.org/10.1115/1.3099016
24. Whitham, G. Linear and Nonlinear Waves. Wiley-Interscience, New York, 1974.
25. Zabusky, N. and Kruskal, M. D. Interactions of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 1965, 15, 240–243.
http://dx.doi.org/10.1103/PhysRevLett.15.240
26. Newell, A. C. Solitons in Mathematics and Physics. SIAM, Philadelphia, 1985.
http://dx.doi.org/10.1137/1.9781611970227
27. Randrüüt, M. and Braun, M. On one-dimensional solitary waves in microstructured solids. Wave Motion, 2010, 47, 4, 217–230.
http://dx.doi.org/10.1016/j.wavemoti.2009.11.002
28. Ablowitz, M. J. Nonlinear Dispersive Waves. Cambridge University Press, Cambridge, 2011.
29. Amstead, D. C. and Karls, M. A. Does the wave equation really work? Primus, 2006, XVI, No. 2, 162–176.http://dx.doi.org/10.1080/10511970608984144
