On singularity point in-between the symmetry planes in elastic orthorhombic media

The singularity points are very important for elastic waves propagation in low-symmetry anisotropic media (Stovas et al., 2021a). Being converted into the group velocity domain, they result in internal refraction cone with anomalous amplitudes and very complicated polarization fields. In elastic orthorhombic (ORT) media, there is always one singularity point in the essential symmetry plane, (0,1,2) singularity points in remaining non-essential symmetry planes and (0,1) points in-between the symmetry planes (Stovas et al., 2021b).

I analyze the conditions for existence of a singularity point in-between the symmetry planes.

In order to do that I fix the diagonal elements of the stiffness coefficient matrix, c_jj , j=1,6, and introduce new variables d₁₂ = c₁₂ +c_66, d₁₃ = c₁₃ +c₅₅ and d₂₃ = c₂₃ +c₄₄. I also assume that the symmetry plane 2-3 is the essential one by introducing the inequality c₅₅ < c₄₄ < c₆₆ . If c₆₆ < c₄₄ < c_55, the 2-3 plane is still essential one but the properties of non-essential planes will interchange. In case of other inequalities for “S wave” stiffness coefficients, the corresponding properties of singularity points can be obtained by a cyclic rotation of stiffness coefficients and symmetry planes (Stovas et al., 2023).

For selected essential symmetry plane (2-3), I propose to fix two variables d₁₂ and d_13, and set the variable d₂₃ as a free variable. By changing d₂₃ only, I can define the trajectory of a singularity point in-between the symmetry planes. This trajectory is given by a continuous line connecting the symmetry planes. Then I define the traces of this trajectory on symmetry planes (maximum two points for each plane) by a specific value of the variable d₂₃. These 6 values can be used for intervals of d₂₃ where the singularity point in-between the symmetry planes exists. Analysis shows that there are 7 zones in (d₁₂ , d₁₃) plane with different intervals of d₂₃, which guarantee the existence of singularity point in-between the symmetry planes. There are 3 intervals of d₂₃ in one zone, two intervals in two zones and one interval in three zones. There is no singularity point in-between the symmetry planes for any d₂₃ in remaining zone. These zones are separated by three straight lines that defined by d₁₂ = d_12(critical) , d₁₃ = d_13(critical)  and d₁₃ = α d_12, where α guarantees that trajectory of singularity point meets the essential symmetry plane.

 References

Stovas, A., Roganov, Yu., and V. Roganov, 2021a, Geometrical characteristics of P and S wave phase and group velocity surfaces in anisotropic media, Geophysical Prospecting, 68(1), 53-69.

Stovas, A., Roganov, Yu., and V. Roganov, 2021b, Wave characteristics in elliptical orthorhombic medium, Geophysics, 86(3), C89-C99.

Stovas, A., Roganov, Yu., and V. Roganov, 2023, On singularity points in elastic orthorhombic media, Geophysics, 88(1), C11-C32.