Abstract
The evolution of a solitary wave propagating through a microstructural material (composite) is studied on the basis of wavelet analysis. A specific feature of the solution technique proposed is the use of Mexican hat (MH) wavelets, which are elastic wavelets, i.e., they are solutions of the basic system of wave equations for an elastic material with a microstructure. The initial wave profile is also chosen in the form of the MH-wavelet. Primary attention is given to the relationship among the profile behavior, wave bottom length, and characteristic microstructure length. A computer analysis conducted demonstrates that the approach proposed allows us to detect the basic wave effects: splitting of the wave into two modes with different phase velocities, simultaneous propagation of both modes in the components of the composite, and strong dependence of the evolution rate on the characteristic lengths of the wave and microstructure
Similar content being viewed by others
REFERENCES
V. O. Geranin, L. D. Pisarenko, and J. J. Rushchitsky, Wavelet Theory with Elements of Fractal Analysis: A Handbook (32 Lectures) [in Ukrainian], VPF Ukr INTEI, Kiev (2002).
V. P. D'yakonov, Wavelets: From Theory to Practice [in Russian], SOLON-R, Moscow (2002).
J. J. Rushchitsky, Elements of Mixture Theory [in Russian], Naukova Dumka, Kiev (1991).
J. J. Rushchitsky and S. I. Tsurpal, Waves in Microstructural Materials [in Ukrainian], Inst. Mekh. NAN Ukrainy, Kiev (1997).
C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms, Prentice-Hall, New Jersey (1998).
C. Cattani and L. Toscano, “Hyperbolic equations in wavelet bases,” Acoust. Bullet., 3, No. 3, 3–9 (2000).
C. Cattani and M. Pecorado, “Nonlinear differential equations in wavelet bases,” Acoust. Bullet., 3, No. 4, 4–10 (2000).
C. Cattani, “The wavelet-based technique in dispersive wave propagation,” Int. Appl. Mech., 39, No. 4, 493–501 (2003).
I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania (1982).
G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston-Basel-Berlin (1994).
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego-New York-London (1999).
D. E. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd ed., Prentice Hall, London (1993).
J. J. Rushchitsky, “Nonlinear waves in solid mixtures,” Int. Appl. Mech., 33, No. 1, 1–34 (1997).
J. J. Rushchitsky, “Interaction of waves in solid mixtures,” Appl. Mech. Rev., 52, No. 2, 35–74 (1999).
J. J. Rushchitsky, “Extension of the microstructural theory of two-phase mixtures to composite materials,” Int. Appl. Mech., 36, No. 5, 586–614 (2000).
J. J. Rushchitsky, C. Cattani, and E. V. Terletskaya, “The effect of the characteristic dimension of a microstructural material and the trough length of a solitary wave on its evolution,” Int. Appl. Mech., 39, No. 2, 197–202 (2003).
C. Cattani and J. J. Rushchitsky, “Solitary elastic waves and elastic wavelets,” Int. Appl. Mech., 39, No. 6, 741–752 (2003).
J. J. Rushchitsky and C. Cattani, “On the interdependence of the elastic wavelets and solitary elastic waves,” in: Book of Abstracts of the 2nd Conf. on Wavelets and Splines, St. Petersburg, Russia (2003), p. 37.
C. Cattani and J. J. Rushchitsky, “Cubically nonlinear elastic waves: Wave equations and methods of analysis,” Int. Appl. Mech., 39, No. 10, 1115–1145 (2003).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rushchitsky, J.J., Cattani, C. & Terletskaya, E.V. Wavelet Analysis of the Evolution of a Solitary Wave in a Composite Material. International Applied Mechanics 40, 311–318 (2004). https://doi.org/10.1023/B:INAM.0000031914.84082.d2
Issue Date:
DOI: https://doi.org/10.1023/B:INAM.0000031914.84082.d2