PARAMETRIC REPRESENTATION OF THE ELASTIC WAVE IN ANISOTROPIC MEDIA

Abstract

Shortly after his appointment to the first geophysical professorship (1895 at the Jagiellonian University of Cracow), Rudzki had published two papers in which he made a strong case for anisotropy of crustal rocks [Beitr. Geophys. 2 (1898); Bull. Acad. Sci. Crac. (1899)]. He had solved the Christoffel equation for transversely isotropic media in terms of what we today would call the ''slowness surface''. Rudzki regarded this as the representation of the wave surface in line coordinates. The conversion to point coordinates lead to an equation of 12th degree. Rudzki had determined a few points, but this was not sufficient to obtain an impression of the wavefront. Costanzi [Boll. Soc. Sismol. Ital. 7 (1901)] had suggested to simplify the coordinate conversion by expressing the solution of the Christoffel equation in a parameter form. The first part of the current paper describes the implementation of this idea. For the first time, the cusps in the wave surface became visible. The results of this first part have been discussed and expanded by Helbig [Beitr. Geophys. 67 (1958) 177; Bull. Seismol. Soc. Am. 56 (1966) 527; Helbig, K., 1994. Foundations of Anisotropy for Exploration Geophysics. Pergamon] and Khatkevich [Isv. Akad. Nauk. SSSR, Ser. Geofiz. 9 (1964) 788]. In a second part, Rudzki applied the ideas to orthorhombic media. The process is straightforward: the elements of the characteristic determinant are of order 2 in the three line coordinates (the three slowness components), with squares of coordinates in the diagonal elements and products of two coordinates in the off-diagonal elements. The elements are easily manipulated so that they are expressed in terms of squares only. Next, the determinant is expanded in terms of rows. This leads to three (equivalent) expressions. The vanishing of any of the three expressions means that the characteristic determinant vanishes, i.e., it corresponds to a solution of the Christoffel equation. Each of the equations can be used to determine one of the sheets of the line coordinates of the wave surface (point coordinates of the slowness surface). To this end, it is expressed in terms of two parameters, which have been chosen strictly for mathematical convenience. After conversion of the line coordinates to point coordinates (formation of the envelopes), one obtains a parameter expression for the wave surface. Until today, the second part of the Rudzki's paper has not been closely studied. However, a blind test of the equations showed that they indeed describe the wave surface of orthorhombic media. The final sections discuss a few interesting aspects, among them the stability conditions for orthorhombic media and the condition under which a transversely isotropic medium transmits pure P- and S-waves.