## 4 Reinterpreting the Dispersion Curves  
As mentioned previously, the Christoffel equation is only dependent upon the material properties of the elastic solid in which the plane waves propagate. The BCs are not used until the partial-wave method is applied. Not only that, but the boundary conditions are not limited to the conventional traction-free BCs (e.g., Lamb waves). As a result, the partial-wave method is not limited to any specific type of waveguide. In fact, most types of guided waves should be representable using slowness curves. For some, such as the shear horizontal guided waves, this approach is more cumbersome than the standard derivation. However, its value lies in how it provides a fundamental foundation for all of the guided waves, and a common method for interpreting and comparing each one. When considering reflection/transmission, mode conversion, or multi-mode propagation, this perspective is quite beneficial. In modern literature, perhaps the most commonly used dispersion curves are of Lamb waves in an aluminum plate in terms of phase velocity versus the frequency-thickness product. Using the three regions depicted in Figure 1, we can divide the phase velocity dispersion curves into three regions, as shown in Figure 2. By doing this, it is possible to view how the dispersion curve solutions match up with the partial waves' characteristics. For example, this is a good way to dispel the potential misconception that guided waves are composed of multiple reflecting bulk waves. That is, the original assumption of plane wave propagation (Equation (4)), which is fundamental to the Christoffel equation, does not exclude the surface wave solution (i.e., imaginary-valued \(k_{z}\)) as \(k_{x}\) increases. It would be more accurate to say that guided waves are a superposition of reflecting bulk waves and surface waves with elliptical particle motion (this will be shown in the later calculations and examples). In fact, guided waves composed solely of reflecting bulk waves exist only in Region 3 (c), where they are highly dispersive. However, there is one exception to this rule: there exists a range of phase velocities in Region 2 (Figure 0(b)), where only bulk waves propagate, which will be discussed in a later section. The A0 mode and Rayleigh waves exist in Region 1 (Figure 0(a)), where each guided wave can be represented in terms of six surface waves. The low-frequency S0 mode is in Region 2 (b), and is composed of two surface waves and two bulk waves (as will be shown later in the paper), excluding the two shear horizontal partial waves. The line separating Region 1 (Figure 0(a)) and Region 2 (Figure 0(b)) has great significance, since many Lamb wave modes appear nearly non-dispersive near it. That is, the higher-order Lamb waves approach the shear wave speed, and the A0 and S0 modes approach the Rayleigh wave speed for large \(\omega\) values. The shear horizontal modes also converge to this line for large \(\omega\) values. The line separating Region 2 (Figure 0(b)) and Region 3 (Figure 0(c)) has many higher-order Lamb modes, which appear nearly non-dispersive close to it as well. Figure 1: Depiction of the characteristics of (**a**) region 1, (**b**) region 2, and (**c**) region 3 with respect to the slowness curves of an isotropic solid. Slowness curves with solid lines denote real-valued \(k_{z}\) solutions, and dotted lines denote imaginary-valued \(k_{z}\) solutions. This figure is a compilation of elements from the figures in Auld [2] and Crandall [6]. The A0 and S0 Lamb wave modes' tendency to approach the Rayleigh wave phase velocity can also be explained using this reinterpretation, since increasing \(k_{x}\) will increase the magnitude of the imaginary-valued \(k_{z}\) (see Figure 1a). Once \(k_{z}\) increases sufficiently, the surface waves decay with the depth from the surface before interacting with the opposite traction-free boundary, thereby imitating a Rayleigh wave. This also explains why quasi-Rayleigh waves [9] have similar wave structures to Rayleigh waves.