Properties of singular points in a special case of orthorhombic media

Authors

  • Yu.V. Roganov Tesseral Technologies Inc., Kyiv, Ukraine, Ukraine
  • A. Stovas NTNU, Trondheim, Norway, Norway
  • V.Yu. Roganov Glushkov Institute of Cybernetic of NASU, Kyiv, Ukraine

DOI:

https://doi.org/10.24028/gj.v45i2.278334

Keywords:

singular point, phase velocity, Christoffel matrix, orthorhombic medium

Abstract

The position of singular lines for orthorhombic (ORT) media with fixed diagonal elements of the elasticity matrix cij, i=1…6 is studied under the condition that c11, c22, c33>c66>c44>c55. In this case, the off-diagonal coefficients of the elasticity matrix c12, c13, c23 are chosen so that some of the values of d12=c12+c66, d13=c13+c55, d23=c23+c44 are zero. For orthorhombic medium, where the only one of d12, d13, d23 is zero, contains only singular points in the planes of symmetry. If two or all three dij are zero, then the ORT medium contains singular lines and discrete singular points. We call such media pathological. A degenerate ORT medium with positive d12, d13, d23 has at most two singular lines, which are the intersection of a quadratic cone with a sphere. The pathological media may have up to 6 singular lines on the surface of the slowness. Singular lines for pathological media are described by more complex equations than conventional degenerate ORT models. The article proposes to using squares x, y, z of the components of the slowness vector in the equations. In a new coordinate system, equations defining singular lines for pathological media become linear or quadratic. Intersecting with the plane x+y+z =1, they define the straight lines, ellipses, or hyperbolas. If non-zero values d12, d13, d23 increase, the singular lines pass through four fixed points on the plane x+y+z =1, which makes it possible to describe the evolution of their change. Conditions are derived under which the singular curves of pathological ORT models are limiting the singular curves for degenerate ORT models with positive values of d12, d13, d23. Formulas are derived for transforming surfaces of slowness and singular lines of pathological media into the region of group velocities. The results are demonstrated with examples of pathological models obtained from the standard model of the ORT medium by changing the elasticity coefficients c12, c13, c23 so that some of the values d12, d13, d23 are zero

References

Alshits, V.I., & Lothe, J. (1979). Elastic waves in triclinic crystals. Articles I, II, III. Soviet Physics. Crystallography, 24, 387—392, 393—398, 644—648.

Alshits, V.I., & Shuvalov, A.L. (1984). Polarization fields of elastic waves near the acoustic axes: Soviet Physics. Crystallography, 29, 373—378.

Alshits, V.I., Sarychev, A.V., & Shuvalov, A.L. (1985). Classification of degeneracies and analysis of their stability in the theory of elastic waves. Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki, 89, 922—938.

Boulanger, Ph., & Hayes, M. (1998). Acoustic axes for elastic waves in crystals: Theory and applications. Proc. Roy. Soc. London, Ser. A, 454, 2323—2346. https://doi.org/10.1098/rspa.1998. 0261.

Crampin, S. (1991). Effects of point singularities on shear-wave propagation in sedimentary basins: Geophysical Journal International, 107, 531—543. https://doi.org/10.1111/j.1365-246X.1991.tb01413.x.

Crampin, S., & Yedlin, M. (1981). Shear-wave singularities of wave propagation in anisotropic media. Journal of Geophysics, 49, 43—46.

Darinskii, B.M. (1994). Acoustic axes in crystals. Soviet Physics. Crystallography, 39, 697—703.

Fedorov, F.I. (1968). Theory of Elastic Waves in Crystals. New York: Plenum Press, 375 p.

Khatkevich, A.G. (1962). The acoustic axis in crystals. Soviet Physics. Crystallography, 7, 601— 604.

Musgrave, M.J.P. (1985). Acoustic axes in orthorhombic media. Proc. Roy. Soc. London, Ser. A, 401, 131—143. http://dx.doi.org/10.1098/rspa. 1985.0091.

Musgrave, M.J.P. (1981). On an elastodinamic classification of orthorhombic media. Proc. Roy. Soc. London, Ser. A, 374, 401—429. https://doi.org/10.1098/rspa.1981.0028.

Norris, A.N. (2004). Acoustic axes in elasticity. Wave Motion, 40, 315—328. http://dx.doi.org/10.1016/j.wavemoti.2004.02.005.

Holm, P. (1992). Generic elastic media. Physica Scripta, 44, 122—127. http://dx.doi.org/10.1088/ 0031-8949/1992/T44/019.

Roganov, Yu., & Roganov, V. (2011). Modeling and use of converted wavefields to determine fractu¬re azimuths. Geofizicheskiy Zhurnal, 33(2), 64—79. https://doi.org/10.24028/gzh. 0203-3100.v33i2.2011.117298 (in Russian).

Roganov, Y.V., Stovas, A., & Roganov, V.Y. (2019). Properties of acoustic axes in triclinic media. Geofizicheskiy Zhurnal, 41(3), 3—17. https://doi.org/10.24028/gzh.0203-3100.v41i3.2019.172417 (in Russian).

Roganov, Yu., Stovas, A., & Roganov, V. (2022). Location of singular points in orthorhombic media. Geofizicheskiy Zhurnal, 44(3), 3—20. https://doi.org/10.24028/gj.v44i3.261965 (in Ukrainian).

Schoenberg, M., & Helbig, K. (1997). Orthorhombic media: Modeling elastic wave behavior in a vertically fractured earth. Geophysics, 62, 1954—1974. https://doi.org/10.1190/1.1444297.

Shuvalov, A.L. (1998). Topological features of the polarization fields of plane acoustic waves in anisotropic media. Proc. Roy. Soc. London, Ser. A, 454, 2911—2947. http://dx.doi.org/10.1098/rspa.1998.0286.

Shuvalov, A.L., & Every, A.G. (1997). Shape of the acoustic slowness surface of anisotropic solids near points of conical degeneracy. The Journal of the Acoustical Society of America, 101(4), 2381—2382. https://doi.org/10.1121/1.418251.

Stovas, A., Roganov, Yu., & Roganov, V. (2022). Behavior of S waves in vicinity of singularity point in elliptic orthorhombic media. Geophysics, 87, C77—C97. https://doi.org/10.1190/geo2021-0522.1.

Stovas, A., Roganov, Yu., & Roganov, V. (2023), On singularity points in elastic orthorhombic media. Geophysics, 88, 1—22. https://doi.org/10.1190/GEO2022-0009.1.

Vavryčuk, V. (2005). Acoustic axes in triclinic ani¬sotropy. The Journal of the Acoustical Socie¬ty of America, 118, 647—653. http://dx.doi.org/10. 1121/1.1954587.

Zeng, X., & MacBeth, C. (1993). Algebraic processing techniques for estimating shear-wave splitting in near-offset VSP data. Geophysical Prospecting, 41, 1033—1066. https://doi.org/10.1111/j.1365-2478.1993.tb00897.x.

Downloads

Published

2023-05-14

How to Cite

Roganov, Y., Stovas, A., & Roganov, V. (2023). Properties of singular points in a special case of orthorhombic media. Geofizicheskiy Zhurnal, 45(2). https://doi.org/10.24028/gj.v45i2.278334

Issue

Section

Articles