Location of singular points in orthorhombic media
Keywords:singular point, phase velocity, polarization vector, Christoffel matrix, orthorhombic medium
The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types of these singular points and their location on the unit circle are described. It is shown that, by selecting parameters , any singular point in the symmetry plane 13 can be combined with the limiting position of the singularity point in-between the symmetry planes, or this point can be included in the singular curve of the degenerate ORT medium. We derive equations for the semi-axes of an ellipse of conical refraction, which is the image of a singular point from plane 13 in the group domain. Conditions for degeneration of the ellipse of conical refraction into a segment or a point are defined. It is shown that there exists only one ORT medium with a fixed phase velocity of S1- and S2-waves in a given singular direction n. If we present the ORT media as projections of these vectors n onto the plane 12 and mark the value of the Poincaré index of the S1- or S2-wave at the point n, we get 2 regions with indices 1/2 and –1/2 separated by projection of the singular curve in the form of an ellipse or hyperbola. We compute equations for of a degenerate ORT medium in terms of the values, , and velocity of S1-, S2-waves on a singular curve. The singular curve is defined by the intersection of a unit sphere with an elliptic cone. It is proved that a degenerate ORT medium for or is, respectively, a transversally isotropic medium with a vertical or horizontal axis of symmetry. The results are illustrated in several examples.
Roganov, Yu., Stovas, A., & Roganov, V. (2019). Properties of acoustic axes in triclinic media. Geophysical Journal, 41(3), 3—17. https://doi.org//10.24028/gzh.0203-3100.v41i3.2019.172417 (in Russian).
Alshits, V.I., & Lothe, J. (1979). Elastic waves in triclinic crystals 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. Zh. Eksp. Teor. Fiz., 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.tb0 1413.x.
Darinskii, B.M. (1994). Acoustic axes in crystals. Soviet physics, crystallography, 39, 697—703.
Fedorov, F.I. (1968). Theory of Elastic Wavesin Crystals. New York: Plenum Press, 375 p.
Holm, P. (1992). Generic elastic media. Physica Scripta, 44, 122—127. http://dx.doi.org/10.1088/0031-8949/1992/T44/019.
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.
Norris, A.N. (2004). Acoustic axes in elasticity. Wave Motion, 40, 315—328. http://dx.doi.org/10.1016 /j.wavemoti.2004.02.005.
Schoenberg, M., & Helbig, K. (1997). Orthorhombic media: Modeling elastic wave behavior in a vertically fracture dearth. 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. Journal of the Acoustical Society of America, 101, 2381—2382. https://doi.org/10.1121/1.418251.
Stovas, A., Roganov, Yu., & Roganov V. (2021). Wave characteristics in elliptical orthorhombic medium. Geophysics, 86, C89—C99. https://doi.org/10.1190/GEO2020-0509.1.
Vavryčuk, V. (2005). Acoustic axes in triclinic anisotropy. Journal of the Acoustical Society of America, 118, 647—653. http://dx.doi.org/10.1121/1.1954587.
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