*2.1. Theory of Lamb Waves*

The existence of Lamb waves was described by Horace Lamb in 1917 [20]. Lamb waves are guided between two parallel free surfaces and can thus be sustained in thin plates. Due to the dispersive nature of Lamb waves, their velocity and number of wave modes depend on excitation frequency. In general, two different mode families can be distinguished: Symmetric and antisymmetric wave modes. When the wave motion is associated with symmetrical displacement and stresses with respect to the middle plane, the symmetric wave mode propagates (Figure 1a). Antisymmetric displacements and stresses are associated with antisymmetric modes propagation (Figure 1b). The relationship between wavenumber and frequency for symmetric modes and antisymmetric modes can be obtained by solving the following dispersion equations, respectively: 2 2 2 2 tan tan 4 *ω* 2 2 2 2 2 2 2 2

$$\frac{\tan(qd)}{\tan(pd)} = -\frac{4k^2pq}{\left(k^2 - q^2\right)^2} \tag{1}$$

$$\frac{\tan(qd)}{\tan(pd)} = -\frac{\left(k^2 - q^2\right)^2}{4k^2pq} \tag{2}$$

where *k* is the wavenumber, *d* is specimen thickness, and *p* and *q* depend on angular frequency ω:

$$p^2 = \frac{\omega^2}{c\_L^2} - k^2 \tag{3}$$

$$q^2 = \frac{\omega^2}{c\_T^2} - k^2 \tag{4}$$

**Figure 1.** Lamb wave modes: (**a**) Symmetric mode; (**b**) antisymmetric mode.

The velocities of shear and longitudinal waves in an infinite medium denoted as *c<sup>T</sup>* and *c<sup>L</sup>* can be calculated based on known material parameters of the medium:

$$c\_L = \sqrt{\frac{\lambda + 2\mu}{\rho}}\tag{5}$$

$$c\_T = \sqrt{\frac{\mu}{\rho}}\tag{6}$$

where µ and λ are Lame's constants. The group velocity is the derivative of angular frequency over the wavenumber:

$$
\omega\_{\mathcal{S}} = \frac{d\omega}{d\mathbf{k}}\tag{7}
$$

For the given angular frequency, there is an infinite number of possible solutions that fulfill Equations (1) and (2). The wavenumber *k* can be real, imaginary, or complex; however, if the plate is considered unloaded, it is sufficient to consider real values only. The number of possible solutions also increases with frequency range. For higher frequencies, the number of possible wave modes increases.
