Behavior of S waves in vicinity of singularity point in elliptic orthorhombic media

Abstract

In low-symmetry anisotropic models, the S waves are generally coupled and have very complex behavior due to triplications and singularity points. We focus on the effect of singularity points on S waves phase and group velocities in a simple elliptical orthorhombic model with no triplications and only one fundamental singularity point located in one of the symmetry planes. We show that the slowness ellipse (defined by the zero Gaussian curvature) is the image of a singularity point on the group velocity surface, and the group velocity ellipse (also known as the internal refraction cone) is the image of a singularity point on the slowness surface. The equations for the slowness and group velocity ellipses are explicitly defined. The square of S1 and S2 waves lacunae is also computed.

The S waves geometrical spreading in elliptical orthorhombic media

“…From Figure 2, one can see that the geometrical spreading factor for S2 waves in the vicinity of the fundamental singularity point is undefined (tends to infinity). Stovas et al (2021b) show that for an elliptical ORT model M1 with f = 0, equation det N = 0 shapes the ellipse in (p 1 , p 2 ) plane, 1). The singularity point is shown by a black dot.…”

“…In Figure 3(b), at offset Y (1) and Y (2) , 1/ |det N| = 0 (for both S1 and S2 waves), and at singularity point Y (R1) , 1/ |det N| → ∞ (for S2 waves). Stovas et al (2021b) show that the image of the singularity point in the group domain is given by the ellipse that (for…”

“…Stovas et al . (2021b) show that for an elliptical ORT model M1 with $$f = 0$$, equation $${\det {\bf N}} = 0$$ shapes the ellipse in $$( {{p_1},{p_2}} )$$ plane, $$\begin{eqnarray}{E_1} &=& \frac{{4{c_{44}}{c_{55}}{c_{66}}}}{{\left( {{c_{44}} - {c_{55}}} \right)\left( {{c_{66}} - {c_{44}}} \right)}}p_1^2\nonumber\\ && +\; \frac{{4c_{44}^2{c_{66}}\left( {{c_{66}} - {c_{55}}} \right)}}{{{{\left( {{c_{66}} - {c_{44}}} \right)}^2}\left( {{c_{44}} - {c_{55}}} \right)}}{\left( {{p_2} - p_{2c}^{\left( {R1} \right)}} \right)^2} - 1 = 0,\end{eqnarray}$$where the coordinate of the centre of the ellipse …”

“…Stovas et al . (2021b) show that the image of the singularity point in the group domain is given by the ellipse that (for model 1) is defined by two equations: $$\begin{eqnarray} && \frac{{4{c_{44}}}}{{\left( {{c_{44}} - {c_{55}}} \right)\left( {{c_{66}} - {c_{44}}} \right)}}V_1^2 + \frac{{4{c_{44}}\left( {{c_{66}} - {c_{55}}} \right)}}{{\left( {{c_{44}} - {c_{55}}} \right){{\left( {{c_{66}} - {c_{44}}} \right)}^2}}}\nonumber\\ &&\quad{\left( {{V_2} - \frac{{\left( {{c_{44}} + {c_{66}}} \right)}}{2}\sqrt {\frac{{\left( {{c_{44}} - {c_{55}}} \right)}}{{{c_{44}}\left( {{c_{66}} - {c_{55}}} \right)}}} } \right)^2} - 1 = 0,\end{eqnarray}$$and …”

On pathological orthorhombic models

“…and the symmetry plane 2-3 is the essential one (Musgrave, 1981(Musgrave, ,1985Stovas et al, 2022aStovas et al, , 2023a, which implies,…”