## 5 Rayleigh Waves and Generalized Rayleigh Waves
### Stoneley Waves  
The same approach can be applied to generalized Rayleigh waves. Generalized Rayleigh waves refer to guided waves that demonstrate Rayleigh wave-like characteristics, such as for example, Stoneley waves, Scholte waves, and Love waves [11; 12; 13]. For this example, Stoneley waves will be the focus, but a similar approach can be applied to the others as well. Stoneley waves, which propagate along the interface between two half-spaces, will be analyzed as an example. Stoneley waves typically have a non-dispersive phase velocity that is slightly less than the denser material's bulk shear wave velocity, which means that it is in Region 1 (Figure 1a). A Rayleigh wave-like wave structure should exist in the denser half-space [2]. A tungsten half-space rigidly connected to an aluminum half-space will be used for the example. Using [PERSON]'s example (i.e., on pages 104-105 in [2]), the six eigensolutions are calculated for a frequency of \(3.188\times 10^{6}\) rad/s and a wavenumber of 1172.1 rad/m. The frequency and wavenumber are chosen to meet the non-dispersive phase velocity solution to the Stoneley wave problem [2]. The eigensolutions at this frequency and wavenumber are shown in Tables 3 and 4 for tungsten and aluminum, respectively. Figure 3: Conceptual depiction of the partial waves, which are necessary for the Rayleigh-wave problem, overlaid on a traction-free boundary of an aluminum half-space. For both tungsten and aluminum, the Stoneley wave is in Region 1 (Figure 1a). To construct the Stoneley wave displacement solution, the following partial waves are needed: solutions \((-L)\) and \((-SV)\) from aluminum's Christoffel equation, and solutions \((+L)\) and \((+SV)\) from the tungsten's Christoffel equation. Solutions \((-L)\) and \((-SV)\) from the aluminum are considered to be exponentially decaying solutions for \(+z\) values, while solutions \((+L)\) and \((+SV)\) from the tungsten are considered to be exponentially decaying solutions for \(-z\) values. These are shown in Figure 4 with respect to the interface. The superposition of partial waves for each material, when \(u_{i}(x,z,t)=u_{i}(z)e^{i(-k_{x}x+\omega t)}\), is shown in Equations (32) and (33). For tungsten (using values from Table 3):  
\[u_{i}^{W}(z)=B^{(+L)}\bigg{(}u_{i}^{(+L)}e^{-\hat{h}_{z}^{(+L)}z}\bigg{)}+B^{( +SV)}\bigg{(}u_{i}^{(+SV)}e^{-\hat{h}_{z}^{(+SV)}z}\bigg{)} \tag{32}\]  
while for aluminum (using values from Table 4):  
\[u_{i}^{AI}(z)=B^{(-L)}\bigg{(}u_{i}^{(-L)}e^{-\hat{h}_{z}^{(-L)}z}\bigg{)}+B^{( -SV)}\bigg{(}u_{i}^{(-SV)}e^{-\hat{h}_{z}^{(-SV)}z}\bigg{)} \tag{33}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Partial Waves** & \(k_{x}\)**rad/m** & \(u_{x}\) & \(u_{y}\) & \(u_{z}\) \\ \hline \((+L)\) & \(1058.1i\) & \(-0.742i\) & \(0\) & \(0.671\) \\ \((-L)\) & \(-1059.1i\) & \(0.742i\) & \(0\) & \(0.671\) \\ \((+SV)\) & \(562.5i\) & \(-0.433i\) & \(0\) & \(0.902\) \\ \((-SV)\) & \(-562.5i\) & \(0.433i\) & \(0\) & \(0.902\) \\ \((+SH)\) & \(562.5i\) & \(0\) & \(1\) & \(0\) \\ \((-SH)\) & \(-562.5i\) & \(0\) & \(-1\) & \(0\) \\ \hline \end{tabular}
\end{table} 
Table 4: Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at \(\omega=3.188\times 10^{6}\) rad/s and \(k_{x}=1172.1\) rad/m. Figure 4: Conceptual depiction of the partial waves from tungsten and aluminum, which are necessary for the Stoneley wave problem, overlaid on the two half-spaces.