Properties of acoustic axes in triclinic media

Abstract

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.