## 3 The Christoffel Equation and the Use of Slowness Curves
### Interpretation of Slowness Curve Solutions  
From the previous section, it is clear that the solutions to the Christoffel equation do not take into consideration the boundary conditions (BCs) of the problem. That is, the solution only depends on the material properties of the elastic solid in which the plane waves propagate. As a result, in the literature, it is shown that the Christoffel equation can be used to determine the reflection and refraction characteristics of plane waves at oblique incidence to an interface between anisotropic elastic solids [6; 7; 8]. Additionally, the energy velocity vector and the bulk wave's skew angle can be calculated [4].  
Slowness curves are often plotted in terms of the 'inverse phase velocity'. However, since phase velocity is not a vector, the use of the term 'inverse phase velocity' to describe the slowness vector can be confusing. A different notation that may be easier to conceptually understand is plotting slowness curves on the \(k_{z}/\omega\) and \(k_{x}/\omega\) axis instead of the \(c_{z}^{-1}\) versus \(c_{x}^{-1}\) axis. Therefore, if a ray is drawn from the origin to a point on the slowness curve, it is understood that its unit vector describes the wave propagation direction in the \(x\)-\(z\) plane.  
Since the focus of the partial-wave method is on guided waves, for the purposes of this paper, it is assumed that the wave is propagating parallel to the boundary/interface, which we take to be in the \(x\)-direction. The \(z\)-axis is through the thickness of the elastic solid. This coordinate system will be maintained throughout the entirety of the paper, with additional details depending on the problem being considered. Further, continuing to assume that \(k_{y}=0\), and using the frequency-wavenumber (\(\omega\)-\(k_{x}\)) pairs obtained from any one of the many methods for calculating dispersion curves (i.e., enforcing the BC and solving the eigenvalue problem), the Christoffel equation becomes a complex valued quadratic eigenvalue problem in three dimensions. The solution, should it exist, will always be composed of six complex valued eigensolutions. These six solutions are the partial waves of the partial-wave method. The partial waves can be paired up into three pairs, and each pair is composed of an upward and a downward propagating wave (i.e., the magnitudes of the eigenvalues are the same, but the signs are opposite) [1;4].  
In [PERSON] [2], there is a discussion of regions for an increasing \(k_{x}\), in which the partial waves have specific characteristics. Also, in [PERSON] [2], it is recognized that at an interface, \(k_{x}\) is the same for all of the partial waves in order to satisfy [PERSON]'s law, and that a given guided wave must theoretically have a single \(k_{x}\) value for a given \(\omega\). [PERSON] gives an example for an isotropic material, which has two slowness curves and three regions (see Equations (15) and (16)):  
* Region 1: \(0<c_{p}<c_{S}\) All six partial waves are surface waves (imaginary-valued \(k_{2}\)) (see Figure 1a).
* Region 2: \(c_{S}<c_{p}<c_{L}\): Two partial waves are surface waves (imaginary-valued \(k_{z}\)), and four are bulk waves (real-valued \(k_{2}\)) (see Figure 1b).
* Region 3: \(c_{L}<c_{p}\): All six partial waves are bulk waves (real-valued \(k_{z}\)) (see Figure 1c).  
The inverses of the shear and longitudinal group velocities (\(c_{S}^{-1}\) and \(c_{L}^{-1}\)) are also the radii of the two slowness curves, which are circles for an isotropic material. By [PERSON]'s paper [6], it is also understood that outside of these circles, the value for \(k_{Z}\) becomes imaginary, and the magnitude traces a hyperbola.