## 2 Theoretical Background on Rayleigh and Lamb Waves  
In this study, we assume that within an appropriate frequency region, Rayleigh (surface) waves are generated in a full-depth pristine concrete region, whereas Lamb (plate) waves are generated in a delaminated region. The behaviors and characteristics of Rayleigh and Lamb waves are summarized in this section.  
Rayleigh waves propagate along the surface of a semi-infinite half-space of an elastic medium. The characteristic equation for the Rayleigh wave velocity is given by [16]:  
\[\left(\frac{c_{R}}{c_{S}}\right)^{6}-\left(\frac{c_{R}}{c_{S}}\right)^{4}+8 \left(\frac{c_{R}}{c_{S}}\right)^{2}\left(3-2\left(\frac{c_{S}}{c_{P}}\right) ^{2}\right)+16\left(\left(\frac{c_{S}}{c_{P}}\right)^{2}-1\right)=0, \tag{1}\]  
where \(c_{P}\), \(c_{S}\), and \(c_{R}\) are the longitudinal (primary or P-), transverse (secondary or S-), and Rayleigh wave velocities, respectively. The particle motion of a Rayleigh wave is elliptic, and its amplitude exponentially decays with depth. Hence, most of its wave energy is limited to a depth of approximately one wavelength. For homogeneous and isotropic media, the Rayleigh wave velocity does not vary with the frequency (i.e., nondispersive).  
Lamb waves are generated and propagated in isotropic, elastic plate-like structures where traction-free boundary conditions are imposed at the top and bottom surfaces. Lamb waves are multimodal and dispersive: each wave mode propagates with a different velocity that varies with the frequency. Depending on the symmetry of the wave motion, Lamb wave modes are divided into symmetric and antisymmetric families. The modal solutions(wavenumbers) of Lamb waves can be obtained by numerically solving the Rayleigh-Lamb equations given by [17]:  
\[\frac{\tan\left(\beta\frac{d}{2}\right)}{\tan\left(\alpha\frac{d}{2}\right)}=- \left[\frac{4\alpha\beta k^{2}}{\left(k^{2}-\beta^{2}\right)^{2}}\right]^{\pm 1}, \tag{2}\]  
where \(d\) is the plate thickness, \(k\) is the wavenumber, and the superscripts \(+1\) and \(-1\) represent symmetric and antisymmetric Lamb wave modes, respectively, and  
\[\alpha^{2}=\frac{\omega^{2}}{c_{p}{}^{2}}-k^{2}\text{ and }\beta^{2}=\frac{ \omega^{2}}{c_{S}{}^{2}}-k^{2}, \tag{3}\]  
where \(\omega\) is the angular frequency. Once the wavenumber solutions are obtained, the phase velocity (\(c_{ph}\)) of each Lamb wave mode at a frequency \(\omega\) is computed by \(\omega\)/\(k\).  
Example wavenumber solutions (dispersion curves) for Lamb waves in a concrete plate are displayed in Figure 1. In the plot, the red and blue curves indicate the symmetric and antisymmetric Lamb wave modes, respectively. In the low-frequency region (below 19 kHz in Figure 1), only two fundamental modes--fundamental symmetric (S0) and antisymmetric (A0) modes--exist. The higher-order modes appear as the frequency increases. At higher frequencies (above 60 kHz in Figure 1), the wavenumbers of the S0 and A0 modes approach those of Rayleigh waves. The frequencies of several Lamb wave-based nondestructive inspection techniques are limited to the low-frequency region where only the S0 and A0 modes exist, to avoid the complexity in the analysis of the higher-order modes. However, the higher-order modes at certain frequencies (S1-ZGV and A2-ZGV in Figure 1) are also employed owing to their unique properties with ZGVs. At these ZGV frequencies, most of the wave energy does not propagate, but it is localized near an excitation source. In particular, the IE method utilizes the S1-ZGV mode to estimate the thickness of a plate \(d\):  
\[d=\frac{\beta c_{p}}{2f_{S1-ZGV}}, \tag{4}\]  
where \(f_{S1-ZGV}\) is the ZGV frequency of the S1 mode and \(\beta\) is an empirical parameter that depends on the Poisson's ratio of the material [18].  
Figure 1: Example dispersion curves for Lamb waves in a linear, isotropic plate. The assumed \(c_{p},c_{S}\), and \(d\) values are 3730 m/s, 2280 m/s, and 0.06 m, respectively.