## 2 Theoretical Background  
### Fundamentals of Guided Waves  
Owing to advances being made in the fields of mathematics and mechanics of wave propagation, GWT has become a mainstream NDT inspection technique in the industry [11]. In order to develop a reliable inspection technique using UGW, it is imperative to understand the elastic wave propagation within the structural boundaries. The Navier-Stokes equation of motion for an isotropic elastic unbounded medium is as follows:  
\[(\lambda+\ \upmu)\
abla(\
abla\cdot\boldsymbol{\omega})+\ \upmu\
abla^{2} \boldsymbol{\omega}=\rho\left(\frac{\partial^{2}\boldsymbol{\omega}}{ \partial t^{2}}\right) \tag{1}\]  
where \(\lambda\) and \(\upmu\) are Lame constants, \(\boldsymbol{\omega}\) is the displacement vector, \(\
abla\) is the Laplace operator and \(\rho\) is the material density. To simplify, Helmholtz decomposition can be utilised to express \(\boldsymbol{\omega}\) in the form of a gradient of a scalar and the curl of the zero divergence vector as shown in Equation (2):  
\[\boldsymbol{\omega}=\
abla\phi+\
abla\times\mathbb{H} \tag{2}\]  
\[\
abla\cdot\mathbb{H}=0 \tag{3}\]  
where \(\phi\) and \(\mathbb{H}\) are scalar and vector potentials of Helmholtz decomposition, respectively. Then, Equation (1) can be decomposed as two simplified wave equations by substituting Equations (2) and (3) of vector potentials:  
\[c_{l}^{2}=\frac{\lambda+2\upmu}{\rho} \tag{4}\]  
\[c_{s}^{2}=\frac{\upmu}{\rho} \tag{5}\]  
where \(c_{l}\) and \(c_{s}\) are the longitudinal and shear wave velocities, respectively. Based on this derivation, longitudinal waves and shear waves are the two elastic wave types that can travel within solids in any direction. However, solid media with multiple boundaries support several types of wave mode, acknowledged as guided waves [14]. Lamb waves are forms of guided waves that exist in a plate and are guided between two parallel tractions, the upper and lower surface of a plate. Lamb waves can be classified into two types, symmetric and asymmetric modes. Symmetric modes are characterised by particle motion being symmetric with respect to the mid-plane of the plate, whereas asymmetric modes are characterised by particle motion being opposite on either side of the mid-plane [15]. Both of these modes are equally used for structural health monitoring (SHM) and are dispersive in nature. Depending on the specimen geometry and material properties, the number of wave modes increases with pulse excitation frequency [12]. In order to identify these wave modes, guided wave nomenclature was specified as:  
\[\mathcal{X}m \tag{6}\]  
where \(\mathcal{X}\) is used to represent the type of vibration mode and \(m\) denotes the order in which modes arrive. Therefore, the fundamental symmetric and asymmetric modes are referred to as \(\mathcal{S}\)0 and \(\mathcal{A}\)0 respectively. Shear modes are also addressed similarly, and the fundamental shear horizontal mode is referred to as \(\mathcal{SH}\)0. The particle vibration of shear modes is polarized parallel to the surface of the plate. Like Lamb modes, shear modes are dispersive in nature, with the exception of the \(\mathcal{SH}\)0 mode [15]. Therefore, \(\mathcal{SH}\)0 has been the preferred mode of excitation in this study. Dispersion curves plot the variation of frequency with UGW modes' velocities, and therefore these curves allow one to evaluate the modes that can be excited at a given frequency [16; 17]. The dispersion curves for the S355 steel grade U-pile are shown in Figure 2. The dimensions and material properties of the S355 steel U-pile are listed in Table 1. Figures 3 and 4 illustrate the group velocity and phase velocity dispersion curves, respectively. The dispersion curves were calculated using a graphical user interface for guided ultrasonic waves (GUIGUW), a MATLAB based software that calculates wave propagation characteristics using the SAFE (semi-analytical finite element) technique.