2. Formulation of the problem and basic relations

Consider a semi-infinite elastic medium of unrestricted anisotropy with a free boundary. It will be characterized by the tensor of moduli of elasticity cijkl and the density ρ. The sagittal plane is specified by two unit vectors: the propagation direction m along the surface and the normal n to the surface. Reflection shown schematically in Figure 1b is the superposition of four partial waves: the incident (α =4= i) and reflected (α =1= r1) waves from the outer sheet of the slowness surface, the reflected wave (α =2= r2) from the middle sheet, and the localized wave (α =3= l) from the internal sheet:

$$
\mathcal{C}\begin{pmatrix} \mathbf{u}(\mathbf{r},t) \\ (i/k)\hat{\sigma}(\mathbf{r},t) \end{pmatrix} = \sum\_{a=1}^{4} \mathcal{C}\_{a} \begin{pmatrix} \mathbf{A}\_{a} \\ \mathbf{L}\_{a} \end{pmatrix} \exp\left\{i(\mathbf{k}\_{a} \cdot \mathbf{r} - at)\right\}.\tag{1}
$$

Here, u is the wave field displacement vector, σ^ is the stress tensor, C<sup>α</sup> are the partial waves amplitudes, <sup>ω</sup> is the frequency, and <sup>k</sup><sup>α</sup> <sup>¼</sup> <sup>k</sup> <sup>m</sup> <sup>þ</sup> <sup>p</sup>α<sup>n</sup> � � are the wave vectors of partial components with a common projection k onto the direction of propagation m (Figure 1b). The k value determines the tracing speed v = ω/k of stationary motion of the wave field (1) along the boundary. Vectors A<sup>α</sup> and L<sup>α</sup> characterize the partial field polarizations. Being not independent (as well as u and σ^), these vectors are normalized by the condition: A<sup>2</sup> <sup>α</sup> ¼ 1.

In terms of Eq. (1), the boundary condition of free surface, σijnj � � <sup>y</sup>¼<sup>0</sup> <sup>¼</sup> 0, takes the form:

$$\mathbf{C}\_{r1}\mathbf{L}\_1 + \mathbf{C}\_{r2}\mathbf{L}\_2 + \mathbf{C}\_l\mathbf{L}\_3 + \mathbf{C}\_i\mathbf{L}\_4 = \mathbf{0}.\tag{2}$$

The unknown vectors A<sup>α</sup> and L<sup>α</sup> are found from the so-called Stroh's formalism based on the fact that the combined six-vectors ξ<sup>α</sup> = {Aα, Lα} <sup>T</sup> (the superscript T means transposition) together with parameters p<sup>α</sup> (α = 1,…,6) are eigenvectors and eigenvalues of the 6 � 6 Stroh matrix <sup>N</sup>^ [12],

$$
\hat{N} = -\begin{pmatrix} (mn)^{-1}(nm) & (nm)^{-1} \\ (mn)(nm)^{-1}(nm) - (mm) - \rho v^2 \hat{I} & (mn)(nn)^{-1} \end{pmatrix} . \tag{3}
$$

Here ^<sup>I</sup> is the unit 3 � 3 matrix and the matrices (ab) are defined by the convolutions (ab)jk = aicijklbl of the moduli tensor cijkl with the vectors a and b. The six eigenvectors ξ<sup>α</sup> are complete and orthogonal to each other everywhere apart from points of degeneracy. The orthogonality property may be expressed in the form:

$$\mathbf{A}\_a \cdot \mathbf{L}\_\beta + \mathbf{A}\_\beta \cdot \mathbf{L}\_a = \mathbf{0}, \qquad a \neq \beta. \tag{4}$$

Depending on v, the vectors ξ<sup>α</sup> and the parameters p<sup>α</sup> may be real or form complex conjugated pairs. The reflection considered in this chapter (Figure 1b) belongs to the second supersonic region of the slowness surface. In the above terms, here the wave superposition formally may include four bulk partial waves with real parameters pα, two incident and two reflected, from the external and middle sheets. In addition, at our disposal, there are two inhomogeneous partial waves with complex conjugated parameters pα, one localized and the other nonphysical (increasing into the depth of the medium), related to the internal sheet. The second incident wave and the nonphysical inhomogeneous component were naturally excluded (C<sup>5</sup> = C<sup>6</sup> = 0) from the sums in Eqs. (1) and (2).

The amplitude Ci of the incident wave is assumed to be known, while the remaining amplitudes may be expressed in terms of Ci through scalar multiplication of Eq. (2) by the vector products L<sup>2</sup> � L3, L<sup>1</sup> � L3, or L<sup>1</sup> � L2. As a result, we arrive at the following reflection coefficients in the form of the ratios of mixed products:

$$R\_1 = \frac{\mathbf{C}\_{r1}}{\mathbf{C}\_i} = -\frac{[\mathbf{L}\_4 \mathbf{L}\_2 \mathbf{L}\_3]}{[\mathbf{L}\_1 \mathbf{L}\_2 \mathbf{L}\_3]}, \qquad R\_2 = \frac{\mathbf{C}\_{r2}}{\mathbf{C}\_i} = \frac{[\mathbf{L}\_4 \mathbf{L}\_1 \mathbf{L}\_3]}{[\mathbf{L}\_1 \mathbf{L}\_2 \mathbf{L}\_3]}.\tag{5}$$
