## 5 Rayleigh Waves and Generalized Rayleigh Waves  
### Rayleigh Waves  
The Rayleigh wave, which travels at the surface of a half-space, is a good starting example, since its solution is well-known, and this different approach will mirror the conventional derivation in the literature. The term 'generalized Rayleigh waves' will be used to refer to a class of wave propagation solutions developed following the original Rayleigh wave derivation, as was done by [PERSON] [11]. This will be considered later in section V.2. Assuming an aluminum half-space with the boundary at \(z=0\), an example Rayleigh wave calculation is completed using the previously mentioned methods. The eigensolutions for the six partial waves at a frequency of \(3.188\times 10^{6}\) rad/s and a wavenumber of \(1103.4\) rad/m are shown in Table 2. The frequency and wavenumber were chosen to match the non-dispersive phase velocity solution to the standard Rayleigh wave problem. There are several notable results in Table 2:  
* All of the \(k_{z}\) values are imaginary, since Rayleigh waves exist in Region 1 (Figure 1a);
* \(k_{z}^{(\pm L)}\) is larger in magnitude than \(k_{z}^{(\pm SV)}\) and \(k_{z}^{(\pm SH)}\) because of the hyperbolic shape of the imaginary-valued slowness curves;
* \(k_{z}^{(+SV)}\), \(k_{z}^{(-SV)}\), \(k_{z}^{(+SH)}\), and \(k_{z}^{(-SH)}\) have a multiplicity due to the isotropic material properties;
* The eigenvectors are complex, which suggests elliptical particle motion [9];
* \(u_{i}^{(+SH)}\) and \(u_{i}^{(-SH)}\) are not complex-valued. As shown in Figure 3, only the exponentially decaying solutions are necessary for the Rayleigh wave solution, because the domain is a half-space. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Aluminum [4]** & **Tungsten [4]** & **Air [10]** \\ \hline \(\rho_{R}\)**, kg/m3** & 2700 & 19.250 & 1.2 \\ \(\lambda_{B}\)**, GPa** & 56.98 & 199.4 & - \\ \(\mu_{B}\)**, GPa** & 25.95 & 158.6 & - \\ \(\alpha_{L}\)**, m/s** & 6350 & 5180 & 343 \\ \(c_{S}\)**, m/s** & 3100 & 2870 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties used for the calculations shown in this paper. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Partial Waves** & \(k_{x}\)**rad/m** & \(u_{x}\) & \(u_{y}\) & \(u_{z}\) \\ \hline \((+L)\) & 982.6\(i\) & \(-0.747i\) & 0 & 0.665 \\ \((-L)\) & \(-982.6i\) & 0.747i & 0 & 0.665 \\ \((+SV)\) & 400.0i & \(-0.341i\) & 0 & 0.940 \\ \((-SV)\) & \(-400.0i\) & 0.341i & 0 & 0.940 \\ \((+SH)\) & \(400.0i\) & 0 & 1 & 0 \\ \((-SH)\) & \(-400.0i\) & 0 & \(-1\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at \(\omega=3.188\times 10^{6}\) rad/s and \(k_{x}=1103.4\) rad/m. Although the derivation method is different, closer inspection of the \(k_{z}\) solutions shows that they match the values of \(q\) and \(s\) given by [PERSON] [9]. In the conventional derivation, the Rayleigh wave solution is a superposition of two exponentially decaying terms that are dependent upon the phase velocity and material properties, so this close agreement is expected. A superposition of the two partial waves, when \(u_{i}(x,z,t)=u_{i}(z)e^{i(-k_{x}x+\omega t)}\), can be represented as:  
\[u_{i}(z)=B^{(-L)}\bigg{(}u_{i}^{(-L)}e^{-ik_{z}^{(-L)}z}\bigg{)}+B^{(-SV)} \bigg{(}u_{i}^{(-SV)}e^{-ik_{z}^{(-SV)}z}\bigg{)} \tag{29}\]  
The amplitudes of each partial wave, \(B^{(-L)}\) and \(B^{(-SV)}\), are solved for by substituting into one of the BCs:  
\[\sigma_{zz}=0\;\text{at}\;z=0, \tag{30}\]  
\[\sigma_{xz}=0\;\text{at}\;z=0, \tag{31}\]  
and assuming \(B^{(-L)}=1\). This approach is taken in lieu of satisfying both BCs (by setting the determinant of the coefficients matrix equal to zero), since \(k_{x}\) is an approximate value, and the determinant is never exactly zero, so most attempts at getting a reduced row echelon form of the matrix result in the trivial solution of the eigenvector. To avoid this dilemma, we set \(B^{(-L)}=1\), and normalized as necessary.