## 3 FEA Theory and Methodology
### Transducer Array Arrangement  
In elastic waves, the superposition of two or more waves results in interference, which could either be constructive or destructive. The superposition of in-phase waves leads to constructive interference, whereas the superposition of out-of-phase waves leads to destructive interference, as illustrated in Figure 6.  
The transducer array arrangement modelled was determined using the concept of linear superposition analysis (LSA) developed by [PERSON] [34]. Several researchers have implemented LSA to achieve unidirectional wave excitation with significant mode purity [34; 35; 36]. The LSA principle is based on the summation of the sound field vectors from each distinct GW source determining the resultant sound field from an array at a given point. The resultant sound field from an array of two UGW sources can be obtained using Equation (11) below:  
\[\mathcal{S}(x,\ell)=Asin(\
ot{\kappa}r_{a}-w\ell)+Asin(\
ot{\kappa}r_{b}-w\ell) \tag{11}\]  
where  
\[\
ot{\kappa}=\frac{2\pi\
ot{\kappa}}{\sigma_{PH}}=\frac{2\pi}{\lambda} \tag{12}\]  
where \(\mathcal{S}\) is a receiver point on a plane, \(r_{a}\) and \(r_{b}\) are the distances between the receiving point and the two corresponding guided wave sources and \(v_{PH}\) is the phase velocity. Using the LSA, [PERSON] developed a 12 \(\times\) 2 configuration shear array design that offered sound wave excitation in a single direction with significant \(\mathcal{SH}0\) mode purity [35]. This dual array design was implemented to achieve desired UGW interference aimed at \(\mathcal{SH}0\) mode beam directionality control. In order to achieve \(\mathcal{SH}0\) mode purity and directionality control, appropriate column/row spacing, out-of-phase excitation and time delay were applied. The result was a trade-off between \(\mathcal{SH}0\) mode destructive interference in one direction with constructive interference in the other, and simultaneous destructive interference of the unwanted Lamb modes in all directions. In order to optimise the \(\mathcal{SH}0\) mode purity and directionality, apodisation of the amplitude of excitation signal was applied on transducers across each row. The apodisation function, also known as the tapering function, was achieved by exciting UGW sources across the array with different voltage amplitudes [37]. This resulted in gradually increasing signal amplitude from the outside to the inside of the array, reducing the contribution of outward UGW sources. The rate of apodisation, \(\
ot{\kappa}_{d}\), was obtained using the following relationship [34]:  
\[\mathbb{A}_{d}=\frac{1-\ \cos\left(\frac{2\pi n}{\mathcal{N}+1}\right)}{2} \tag{13}\]  
where \(\mathcal{N}\) is the number of transducers in the array and \(n\) is the index number of each transducer. The 12 \(\times\) 2 configuration shear transducer array was further improved for the sheet pile inspection by optimising the row and column spacing according to the material properties and web width. Assuming a fixed 8 mm row spacing, the \(\mathcal{SH}0\) mode directionality and \(\mathcal{SH}0\) mode purity were evaluated against column spacing ranging from 10 mm to 30 mm, over the frequency range of 20 kHz to 100 kHz. The LSA results obtained for \(\mathcal{SH}0\)  
Figure 6: Superposition of waves showing (**a**) constructive interference and (**b**) destructive interference (adapted from [34]). mode directionality and purity, shown in Figure 7, indicated that the 12 \(\times\) 2 configuration array design with 26 mm column spacing and 8 mm row spacing could excite \(\mathcal{SH}0\) with relatively high mode purity and directionality at 70 kHz of excitation frequency. The 12 \(\times\) 2 configuration shear transducer array was excessively large to inspect the flange of the pile, and therefore a 7 \(\times\) 2 configuration shear transducer array was also designed for flange inspection.