Properties of acoustic axes in triclinic media
Keywords:triclinic medium, acoustic axis, singular direction, slowness surface
We developed a method of obtaining relationships that describe the position of the acoustic axes in a triclinic medium and the dependencies between them. It is proved that these relations are linearly independent in real number system. However, any relationship algebraically depends on other two relationships. The relation between derived relationships and those obtained in earlier papers is also investigated. The formulae defining the change of these relationships when rotating around the axes of the coordinate system are derived. It is proved that the fulfillment of five relations is necessary and sufficient for the definition of all acoustic axes in a given coordinate system. It is shown that the acoustic axis in a given phase direction exists if and only if the specified two vectors of dimension five are collinear. For an orthorhombic medium, these relations are represented in an explicit form by homogeneous polynomials of the sixth degree in the components of the phase direction vector and the third degree in the stiffness coefficients. It is shown that in symmetry planes, only two of these relations are not identically equal to zero. The theory is illustrated in two numerical examples of anisotropic media. In the first example, for a triclinic medium, the positions of the sixteen acoustic axes are shown as the intersection points of the graphs of three relationships on the plane (phase polar and azimuth angles). In this case, six points corres-pond to the intersections of P and S1 phase velocities sheets, and ten points correspond to the intersections of S1 and S2 phase velocities sheets. The second example demonstrates the definition of all acoustic axes in an orthorhombic medium based on the derived relationships. To illustrate this example, we consider only one quadrant due to symmetry with respect to symmetry planes.
Al’shits, V. I., & Lothe, J. (1979). Elastic waves in triclinic crystals. I, II, III. Sov. Phys. Crystallogr., 24, 387—392, 393—398, 644—648.
Boulanger, Ph., Hayes, M. (1998). Acoustic axes for elastic waves in crystals: Theory and applications. Proc. R. Soc. London, Ser. A., 454, 2323—2346. doi: 10.1098/rspa.1998.0261.
Darinskii, B. M. (1994). Acoustic axes in crystals. Sov. Phys. Crystallogr, 39(5), 773—780.
Fedorov, F. I. (1968). Theory of Elastic Waves in Crystals. New York: Plenum Press, 375 p.
Grechka, V. (2017). Algebraic degree of a general group-velocity surface. Geophysics, 82(4), WA45—WA53. doi: https://doi.org/10.1190/geo2016-0523.1.
Holm, P. (1992). Generic elastic media. Physica Scripta, 44, 122—127. https://doi.org/10.1088/0031-8949/1992/T44/019.
Khatkevich, A. G. (1962). The acoustic axis in crystals. Sov. Phys. Crystallogr., 7, 601—604.
Kim, K. Y., Sachse, W., & Every, A. G. (1993). Focusing of acoustic energy at the conical point in zinc. Physical Review Letters, 70, 3443—3446. doi: http://dx.doi.org/10.1103/PhysRevLett.70.3443.
Musgrave, M. J. P. (1985). Acoustic axes in orthorhombic media. Proc. R. Soc. London, Ser. A, 401, 131—143. doi: http://dx.doi.org/10.1098/rspa.1985.0091.
Norris, A. N. (2004). Acoustic axes in elasticity. Wave Motion, 40(4), 315—328. doi: 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 fractured earth. Geophysics, 62(6), 1954—1974. doi: https://doi.org/10.1190/1.1444297.
Shafarevich, I. R. (2010). Basic Algebraic Geometry 1. Berlin, Heidelberg: Springer-Verlag, 310 p. doi: 10.1007/978-3-642-37956-7.
Shuvalov, A. L. (1998). Topological features of the polarization fields of plane acoustic waves in anisotropic media. Proc. R. Soc. London, Ser. A., 454, 2911—2947. doi: http://dx.doi.org/10.1098/rspa.1998.0286.
Vavryčuk, V. (2005). Acoustic axes in triclinic anisotropy. The Journal of the Acoustical Society of America, 118, 647—653. doi: http://dx.doi.org/10.1121/1.1954587.
How to Cite
Copyright (c) 2020 Geofizicheskiy Zhurnal
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).