# American Institute of Mathematical Sciences

## Complex ray in anisotropic solids: Extended Fermat's principle

 CNRS, University of Bordeaux, Institut de Mécanique et d'Ingénierie - Bordeaux, 351, cours de la libération - 33405 Talence, France

* Corresponding author: Marc Deschamps

Received  May 2018 Revised  June 2018 Published  November 2018

In contrast to homogeneous plane waves, solutions of the Chris-toffel equation for anisotropic media, for which a determined number of rays can be observed in a fixed direction of observation, inhomogeneous plane waves provide a continuum of "rays" that propagate in this direction. From this continuum, some complex plane waves can be extracted for verifying a definition of quasi-arrivals, based on the condition that the time of flight would vary the less in extension to the Fermat's principle that stipulates a stationary time of flight for wave arrivals. The dynamic response in some angular zones contain prominent, although not singular, features whose arrivals cannot be described by the classical ray theory. These wave packet's arrivals can be described by quasi-fronts associated to specific inhomogeneous plane waves. The extent of the phenomena depends on the degree of anisotropy. For weak anisotropy, such quasi-fronts can be observed. For strong anisotropy, the use of inhomogeneous plane waves, due to their complex slowness vector, permits a simple description of quasi-arrivals that refer to the internal diffraction phenomenon. Some examples are given for different wave surfaces, showing how the wave fronts can be extended beyond the cuspidal edges for forming closed wave surfaces.

Citation: Marc Deschamps, Olivier Poncelet. Complex ray in anisotropic solids: Extended Fermat's principle. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019110
##### References:
 [1] B. Audoin, C. Bescond and M. Deschamps, Recovering of stiffness coefficients of anisotropic materials from point-like generation and detection of acoustic waves, J. Appl. Phys., 80 (1996), 3760-3771. [2] B. A. Auld, Acoustic Fields and Waves in Solids, New York, 1973. [3] Ph. Boulanger and M. Hayes, Bivectors and Waves in Mechanics and Optics, Chapman and Hall, London, 1993. doi: 10.1007/978-1-4899-4531-0. [4] C. Corbel, F. Guillois, D. Royer, M. A. Fink and R. De Mol, Laser generated elastic waves in carbon-epoxy composite, IEEE Trans. Ultr. Fer. Freq. Contr., 40 (1993), 710-716. [5] M. Deschamps and O. Poncelet, Complex rays and internal diffraction at the cusp edge, NATO Science series, Mathematics, Physics and Chemistry, 163 (2004), 131-142. doi: 10.1007/1-4020-2387-1_6. [6] M. Deschamps, Reflection and refraction of the inhomogeneous plane wave, Acoustic Interaction with Submerged Elastic Structures, Part I, World Scientific Publishing Company, Edited by A. Guran, J. Ripoche and F. Ziegler, 5 (1996), 164–206. doi: 10.1142/9789812830593_0006. [7] A. G. Every, Ballistic phonons and the shape of the ray surface in cubic crystals, Phys. Rev. B, 24 (1981), 3456-3467. [8] M. R. Hauser, R. L. Weaver and J. P. Wolfe, Internal diffraction of ultrasound in crystal: Phonon focusing at long wavelength, Phys. Rev. Lett., 68 (1992), 2604-2607. [9] M. Hayes, Inhomogeneous plane waves, Arch. Rational Mech. Anal., 85 (1984), 41-79. doi: 10.1007/BF00250866. [10] M. Hayes, Energy flux for trains of inhomogeneous plane waves, Proc. R. Soc. Lond. A, 370 (1980), 417-429. [11] K. Y. Kim, K. C. Bretz, A. G. Every and W. Sachse, Ultrasonic imaging of the group velocity surface about the cubic axis in silicon, J. Appl. Phys., 79 (1996), 1857-1863. [12] A. Mourad, M. Deschamps and B. Castagnède, Acoustic waves generated by a line impact in an anisotropic medium, Acustica - Acta Acustica, 82 (1996), 839-851. [13] B. Poirée, Les ondes planes évanescentes dans les fluides parfaits et les solides élastiques, J. Acoustique, 2 (1989), 205-216. [14] O. Poncelet, M. Deschamps, A.G. Every and B. Audoin, Extension to cuspidal edges of wave surfaces of anisotropic solids: Treatment of near cusp behavior, Review in Quantitative Nondestructive Evaluation, ed. by D.O. Thompson and D.E. Chiment, 20A (2000), 51-58. [15] D. Royer and E. Dieulesaint, Ondes élastiques dans les solides, volume 1, Masson, Paris, 1996. [16] J. H. M. T. Van Der Hijden, Propagation of Transient Elastic Waves in Stratified Anisotropic Media, North-Holland series in Applied Mathematics and Mechanics, 32. North-Holland Publishing Co., Amsterdam, 1987. [17] R. L. Weaver, M. R. Hauser and J. P. Wolfe, Acoustic flux imaging in anisotropic media, Z Phys. B, 90 (1993), 27-46.

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##### References:
 [1] B. Audoin, C. Bescond and M. Deschamps, Recovering of stiffness coefficients of anisotropic materials from point-like generation and detection of acoustic waves, J. Appl. Phys., 80 (1996), 3760-3771. [2] B. A. Auld, Acoustic Fields and Waves in Solids, New York, 1973. [3] Ph. Boulanger and M. Hayes, Bivectors and Waves in Mechanics and Optics, Chapman and Hall, London, 1993. doi: 10.1007/978-1-4899-4531-0. [4] C. Corbel, F. Guillois, D. Royer, M. A. Fink and R. De Mol, Laser generated elastic waves in carbon-epoxy composite, IEEE Trans. Ultr. Fer. Freq. Contr., 40 (1993), 710-716. [5] M. Deschamps and O. Poncelet, Complex rays and internal diffraction at the cusp edge, NATO Science series, Mathematics, Physics and Chemistry, 163 (2004), 131-142. doi: 10.1007/1-4020-2387-1_6. [6] M. Deschamps, Reflection and refraction of the inhomogeneous plane wave, Acoustic Interaction with Submerged Elastic Structures, Part I, World Scientific Publishing Company, Edited by A. Guran, J. Ripoche and F. Ziegler, 5 (1996), 164–206. doi: 10.1142/9789812830593_0006. [7] A. G. Every, Ballistic phonons and the shape of the ray surface in cubic crystals, Phys. Rev. B, 24 (1981), 3456-3467. [8] M. R. Hauser, R. L. Weaver and J. P. Wolfe, Internal diffraction of ultrasound in crystal: Phonon focusing at long wavelength, Phys. Rev. Lett., 68 (1992), 2604-2607. [9] M. Hayes, Inhomogeneous plane waves, Arch. Rational Mech. Anal., 85 (1984), 41-79. doi: 10.1007/BF00250866. [10] M. Hayes, Energy flux for trains of inhomogeneous plane waves, Proc. R. Soc. Lond. A, 370 (1980), 417-429. [11] K. Y. Kim, K. C. Bretz, A. G. Every and W. Sachse, Ultrasonic imaging of the group velocity surface about the cubic axis in silicon, J. Appl. Phys., 79 (1996), 1857-1863. [12] A. Mourad, M. Deschamps and B. Castagnède, Acoustic waves generated by a line impact in an anisotropic medium, Acustica - Acta Acustica, 82 (1996), 839-851. [13] B. Poirée, Les ondes planes évanescentes dans les fluides parfaits et les solides élastiques, J. Acoustique, 2 (1989), 205-216. [14] O. Poncelet, M. Deschamps, A.G. Every and B. Audoin, Extension to cuspidal edges of wave surfaces of anisotropic solids: Treatment of near cusp behavior, Review in Quantitative Nondestructive Evaluation, ed. by D.O. Thompson and D.E. Chiment, 20A (2000), 51-58. [15] D. Royer and E. Dieulesaint, Ondes élastiques dans les solides, volume 1, Masson, Paris, 1996. [16] J. H. M. T. Van Der Hijden, Propagation of Transient Elastic Waves in Stratified Anisotropic Media, North-Holland series in Applied Mathematics and Mechanics, 32. North-Holland Publishing Co., Amsterdam, 1987. [17] R. L. Weaver, M. R. Hauser and J. P. Wolfe, Acoustic flux imaging in anisotropic media, Z Phys. B, 90 (1993), 27-46.
Absolute value of the derivative of the real slowness component on the $\mathbf{X}_{2}$-axis with respect the slowness component in the $\mathbf{n}$-direction versus the phase velocity along the $\mathbf{n}_{e}$-direction (in $\mathrm{mm}/\mathrm{\mu s}$)
Group velocity $c_{e}$ (in $\mathrm{mm}/\mathrm{\mu s}$) in polar coordinates for various directions $\mathbf{n}_{e}$ and various anisotropy parameters $\epsilon$. $L$ and $T$ real waves (solid line), $L_{e}$ and $T_{e}$ complex waves (dashed line)
3D polar waveforms (radius: $r/t$, angle: $\theta$) overlaid with the real and complex wavefronts (grey line). Black: zero amplitude; White: highest amplitude
Group velocity $c_{e}$ in polar coordinates for various directions $\mathbf{n}_{e}$ for various anisotropy parameters $\epsilon$. Real wavefronts (solid line) and complex wavefronts (dashed line). Velocity associated with the position of the maximum of amplitude in the waveform (circle)
Radial displacement $u_{r}$ on an unbounded crystalline space submitted to a delta-function pulse line force versus $1/s_{e}^{\prime}$. Total contribution (solid line), longitudinal contribution (- - -) and transverse contribution (...)
Wavefronts and 3D polar waveforms for a long fibers composite material
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