## 4 Reinterpreting the Dispersion Curves
That is, the quasi-Rayleigh wave and the Rayleigh wave are both composed of partial waves (with slight differences in amplitude and exponential decay). Many higher-order Lamb modes have wave structures that are sinusoidal through the thickness, but have slightly larger or smaller amplitudes near the traction-free boundaries. In Region 2 (Figure 1b), this could be explained by surface waves affecting the wave structure near the traction-free boundaries, while the reflecting bulk waves create the sinusoidal appearance. This reinterpretation of guided waves provides for the relatively simple calculation of wave structures based on the superposition of the partial-wave eigensolutions. Based on the preceding concepts and with wave propagation in the \(x\)-direction, \(c_{p}=\omega/k_{x}=c_{x}\). Solving Equations (15) and (16) for \(c_{z}^{-1}\) results in:  
\[c_{z}^{-1}=\pm\sqrt{c_{L}^{-2}-c_{x}^{-2}} \tag{17}\]  
and:  
\[c_{z}^{-1}=\pm\sqrt{c_{S}^{-2}-c_{x}^{-2}} \tag{18}\]  
respectively. Equations (9), (10) and (17) can then be substituted into Equation (11) to get matrix \(A\) as a function of \(c_{x}^{-1}\) and \(c_{L}^{-1}\):  
\[A^{(L)}:=\left[ \begin{array}{ccc}(\lambda+\mu)\Big{(}c_{L}^{-2}-c_{x}^{-2} \Big{)}&0&\pm\sqrt{c_{L}^{-2}-c_{x}^{-2}}c_{x}^{-1}(\lambda+\mu)\\ 0&-(\lambda+\mu)c_{L}^{-2}&0\\ \pm\sqrt{c_{L}^{-2}-c_{x}^{-2}}c_{x}^{-1}(\lambda+\mu)&0&(\lambda+\mu)\big{(} -c_{x}^{-2}\big{)}\end{array} \right] \tag{19}\]  
And Equations (9), (10) and (18) can be substituted into Equation (11) to get matrix \(A\) as a function of and \(c_{x}^{-1}\) and \(c_{S}^{-1}\):  
\[A^{(S)}:=\left[ \begin{array}{ccc}(\lambda+2\mu)c_{x}^{-2}+\mu\Big{(}c_{S}^{ -2}-c_{x}^{-2}\Big{)}-\mu c_{S}^{-2}&0&\pm\sqrt{c_{S}^{-2}-c_{x}^{-2}}c_{x}^{ -1}(\lambda+\mu)\\ 0&0&0\\ \pm\sqrt{c_{S}^{-2}-c_{x}^{-2}}c_{x}^{-1}(\lambda+\mu)&0&\mu c_{x}^{-2}+( \lambda+2\mu)\Big{(}c_{S}^{-2}-c_{x}^{-2}\Big{)}-\mu c_{S}^{-2}\end{array}  \right] \tag{20}\]  
Figure 2: Lamb wave dispersion curves for an aluminum plate divided according to the three regions shown in Figure 1.  
Finding the reduced row echelon form of \(A^{(L)}\) and \(A^{(S)}\) gives the solution to the eigenvectors, which contain, at most, two free variables (i.e., \(m\) and \(p\)). Considering \(A^{(L)}\) and \(A^{(S)}\), and also the \"\(\pm\)\" terms in them, which correspond with the sign of \(k_{z}\), results in four possible eigenvectors (not including the multiplicity). For the positive \(k_{z}\) eigenvalue of the longitudinal wave, the eigenvector is:  
\[u_{i}^{(+L)}=\left[ \begin{array}{c}\frac{c_{x}^{-1}}{\sqrt{c_{L}^{-2}-c_{x} ^{-2}}}m\\ 0\\ m\end{array} \right] \tag{21}\]  
While for the negative \(k_{z}\) eigenvalue of the longitudinal wave, the eigenvector is:  
\[u_{i}^{(-L)}=\left[ \begin{array}{c}-\frac{c_{x}^{-1}}{\sqrt{c_{L}^{-2}-c_{x} ^{-2}}}m\\ 0\\ m\end{array} \right] \tag{22}\]  
For the positive \(k_{z}\) eigenvalue of the shear wave, the eigenvector is:  
\[u_{i}^{(+S)}=\left[ \begin{array}{c}\frac{-\sqrt{c_{S}^{-2}-c_{x}^{-2}}}{c_{ x}^{-1}}m\\ p\\ m\end{array} \right] \tag{23}\]  
While for the shear wave's negative \(k_{z}\) eigenvalue, the eigenvector is:  
\[u_{i}^{(-S)}=\left[ \begin{array}{c}\frac{\sqrt{c_{S}^{-2}-c_{x}^{-2}}}{c_{ x}^{-1}}m\\ p\\ m\end{array} \right] \tag{24}\]  
The superscripts \(L\) and \(S\) correspond to the longitudinal and the shear waves, respectively. The \(S\) can be made more specific by using \(SV\) and \(SH\) to denote the shear vertical and shear horizontal polarizations. Examining Equations (21) and (22), it is clear that for \(c_{x}^{-1}>c_{L}^{-1}\), the square root becomes imaginary-valued, and for \(c_{x}^{-1}<c_{L}^{-1}\), the square-root becomes real-valued. These two scenarios correspond with the surface wave and bulk wave solutions, respectively. The same holds true for Equations (23) and (24) with \(c_{S}\) instead of \(c_{L}\), with \(p=0\). For \(c_{x}^{-1}\gg c_{L}^{-1}\) or \(c_{x}^{-1}\gg c_{S}^{-1}\) and positive \(k_{z}\), the corresponding eigenvector is:  
\[u_{i}^{(+)}=\left[