Abstract
We explore the physical phenomenon of acoustic waves induced at the interface between two different anisotropic rock media. Specifically, one medium is a transversely isotropic medium with a vertical axis of symmetry (VTI medium) and the other one is a transversely isotropic medium with a tilt axis of symmetry (TTI medium). By solving the Kelvin-Christoffel equation, an eighth-order polynomial is established for reflection and refraction angles, which is confirmed from Snell’s law. Three types of analytical expressions of the polarization coefficients of the induced waves are obtained corresponding to different incident angle regions. An effective algorithm has been developed for numerical analysis of the polarization coefficients. Applying characteristic anisotropic parameters reported in the literature, the influencing factors on reflection and refraction coefficients are analyzed, e.g., the anisotropy, the tilt-angle of rock-layer, and the incident-angle. The calculated reflection and refraction coefficients have been rechecked for energy conservation.
Similar content being viewed by others
References
J. P. Castagna and M. M. Backus, Offset-Dependent Reflectivity — Theory and Practice of AVO Analysis, Society of Exploration Geophysicists Press, Tulsa, OK, USA, 1993
M. A. Mustapa, O. B. Yaakob, Y. M. Ahmed, C. K. Rheem, K. K. Koh, and F. A. Adnan, Wave energy device and breakwater integration: A review, Renew. Sustain. Energy Rev. 77, 43 (2017)
R. J. Przybyla, H. Y. Tang, A. Guedes, S. E. Shelton, B. E. Boser, and D. A. Horsley, 3D ultrasonic rangefinder on a chip, IEEE. J. Solid-St. Circ. 50(1), 320 (2015)
A. S. Goh, J. C. Kohn, D. B. Rootman, J. L. Lin, and R. A. Goldberg, Hyaluronic acid gel distribution pattern in periocular area with high-resolution ultrasound imaging, Surg J. 34(4), 510 (2014)
M. A. Breazeale, L. Adler, and G. W. Scott, Interaction of ultrasonic waves incident at the Rayleigh angle onto a liquid-solid interface, J. Appl. Phys. 48(2), 530 (1977)
A. Atalar, C. F. Quate, and H. K. Wickramasinghe, Phase imaging in reflection with an acoustic microscope, Appl. Phys. Lett. 31(12), 791 (1977)
R. Briers, O. Leroy, and G. Shkerdin, Bounded beam interaction with thin inclusions: Characterization by phase differences at Rayleigh angle incidence, J. Acoust. Soc. Am. 108(4), 1622 (2000)
G. L. Zhang, C. T. Hao, and C. Yao, A study of synthetic seismograms for SKS-wave response to crustal fractured-induce anisotropy, Chin. J. Geophys.-Ch. 59(7), 2498 (2016)
W. K. Cao, L. T. Wu, C. Zhang, J. C. Ke, Q. Cheng, T. J. Cui, and Y. Jing, Asymmetric transmission of acoustic waves in a waveguide via gradient index metamaterials, Sci. Bull. (Beijing) 64(12), 808 (2019)
Y. Wang, S. Y. Zhang, J. Xu, Y. C. Xie, and X. D. Lan, Characteristics of surface acoustic waves in (1120) ZnO film/R-sapphire substrate structures, Sci. China Phys. Mech. Astron. 61(2), 024311 (2018)
H. Zhang, S. Y. Yu, F. K. Liu, Z. Wang, M. H. Lu, X. B. Hu, Y. F. Chen, and X. G. Xu, Using coupling slabs to tailor surface-acoustic-wave band structures in photonic crystals consisting of pillars attached to elastic substrates, Sci. China Phys. Mech. Astron. 60(4), 044311 (2017)
Y. Y. Fu, Y. D. Xu, and H. Y. Chen, Negative refraction based on purely imaginary metamaterials, Front. Phys. 13(4), 134206 (2018)
S. G. Woon, Acoustical Imaging Techniques and Applications: Chapter 14, Negative Refraction, Acoustical Metamaterials and Acoustical Cloaking, John Wiley Sons, Ltd, 2012
L. Xu, Q. N. Wu, Y. Y. Zhou, and H. Y. Chen, Transformation devices with optical nihility media and reduced realizations, Front. Phys. 14(4), 42501 (2019)
K. Zoeppritz, Erdbebenwellen VIIIB, on the reflection and propagation of seismic waves, Gottinger. Nachr. 17(1), 66 (1919)
P. G. Richards and C. W. Frasier, Scattering of elastic waves from depth-dependent inhomogeneities, Geophysics 41(3), 441 (1976)
K. Aki and P. G. Richards, Quantitative Seismology: Theory and Methods, W. H. Freeman Press, New York, USA, 1980
R. T. Shuey, A simplification of the Zoeppritz equations, Geophysics 50(4), 609 (1985)
W. J. Ostrander, Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence, Geophysics 49(10), 1637 (1984)
J. Y. Li, X. H. Chen, Z. J. Hao, and Z. H. Rui, A study on multiple time-lapse seismic AVO inversion, Chin. J. Geophys-Ch. 48(4), 902 (2005)
G. Z. Zhu, L. Liu, and D. Y. Fu, Experimental study on the non-specularly reflected ultrasonic beam on a liquidsolid interface, Sci. China Ser. A 44(2), 225 (2001)
L. Fa, L. Xue, Y. X. Fa, Y. L. Han, Y. D. Zhang, and H. S. Cheng, Acoustic Goos-Hanchen effect, Sci. China Phys. Mech. 60(10), 104311 (2017)
L. Thomsen, Weak elastic anisotropy, Geophysics 51(10), 1954 (1986)
Z. Wang, Seismic anisotropy in sedimentary rocks, part 1: A single-plug laboratory method, Geophysics 67(5), 1415 (2002)
Z. Wang, Seismic anisotropy in sedimentary rocks, part 2: Laboratory data, Geophysics 67(5), 1423 (2002)
G. L. Backus, Long-wave elastic anisotropy produced by horizontal layering, J. Geophys. Res. 67(11), 4427 (1962)
P. F. Daley and F. Hron, Reflection and transmission coefficients for transversely isotropic media, Bull. Seismol. Soc. Am. 67(3), 661 (1977)
P. F. Daley and F. Hron, Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media, Geophysics 44(1), 27 (1979)
A. Rüger, P-wave reflection coefficient for transversely isotropic models with vertical and horizontal axes of symmetry, Geophysics. 1997, 62(14), 713 (1997)
I. Tsvankin, Seismic signature and analysis of reflection data in aisotropic media, Society of Exploration Geophysicists Press, Elsevier, Amsterdam, 2005
J. M. Carcione, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous Media, Elsevier Science Press, Amsterdam, 2001
L. Klimeš, Weak-contrast reflection-transmission coefficients in a generally anisotropic background, Geophysics 68(6), 2063 (2003)
L. Fa, R. L. Brown, and J. P. Castagna, Anomalous postcritical refraction behavior for certain transversely isotropic media, J. Acoust. Soc. Am. 120(6), 3479 (2006)
L. Fa, J. P. Castagna, and H. F. Dong, An accurately fast algorithm of calculating reflection/transmission coefficients, Sci. China. Phys. Mech. 51(7), 823 (2008)
B. A. Auld, Acoustic Fields and Waves in Solids, John Wiley & Sons Press, New York, USA, 1972
L. Fa, F. Y. Fa, Y. D. Zhang, P. F. Ding, J. M. Gong, G. H. Li, L. J. Li, S. J. Tang, and M. S. Zhao, Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave, Sci. Rep. 5(1), 12700 (2015)
L. Fa, W. Y. Li, J. Zhao, Y. L. Han, M. Liang, P. F. Ding, and M. S. Zhao, Polarization state of an inhomogeneously refracted compressional-wave induced at interface between two anisotropic rocks, J. Acoust. Soc. Am. 141(1), 1 (2017)
L. Fa, M. S. Zhao, Y. C. Liu, L. Wang, Y. Q. Wang, and J. G. Sun, Polarization of plane wave propagating inside elastic hexagonal system solids, Sci. China Phys. Mech. Astron. 57(2), 251 (2014)
L. Fa, J. Zhao, Y. L. Han, G. H. Li, P. F. Ding, and M. S. Zhao, The influence of rock anisotropy on elliptical-polarization state of inhomogeneously refracted P-wave, Sci. China Phys. Mech. Astron. 59(4), 644301 (2016)
L. Fa, Z. Zeng, C. Deng, and M. S. Zhao, Effects of geometrical-size of cylindrical-shell transducer on acoustic-beam steering efficiency for a slim-hole acoustic-logging tool, Open Acoustics Journal 3(1), 21 (2010)
L. Fa, J. P. Castagna, Z. Zeng, R. L. Brown, and M. S. Zhao, Effects of anisotropy on time-depth relation in transversely isotropic medium with a vertical axis of symmetry, Chin. Sci. Bull. 55(21), 2243 (2010)
Y. Zhao, N. Zhao, L. Fa, and M. S. Zhao, Seismic signal and data analysis of rock media with vertical anisotropy, J. Mod. Phys. 04(01), 11 (2013)
L. Fa, L. Wang, Y. Zhao, L. Liu, Y. Zheng, N. Zhao, M. Zhao, and G. Li, Research progress in acoustical application to petroleum logging and seismic exploration, Open Acoustics Journal 6(1), 1 (2014)
L. Fa and M. S. Zhao, Recent development of an acoustic measurement system, in: Understanding Plane Waves, Nova Science Publishers, 2019
L. Fa and M. S. Zhao, Recent progress in acoustical theory and applications, in: Understanding Plane Waves, Nova Science Publishers, 2019
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (Grant No. 41974130) and by the Physical Sciences Division at The University of Chicago.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fa, L., Tang, J., Zhang, Q. et al. Reflection and refraction of elastic wave at VTI-TTI media interface. Front. Phys. 15, 22601 (2020). https://doi.org/10.1007/s11467-019-0934-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11467-019-0934-z