## 2 Method
### The Sound Field of the Transducer  
Piezoelectric and magnetostrictive transducers are commonly used in guided wave generation. Because of the differences in the principles of energy conversion and transducer structures, piezoelectric transducers are good at out-of-plane displacement generation, while magnetostrictive transducers have advantages in in-plane displacement generation. The magnetostrictive effect of the ferromagnetic materials is made use of in the wave generation, and the inverse magnetostrictive effect works in receiving waves. Details of the working principles of this kind of transducer can be found in [39]. The transducer used in this study is named the magnetostrictive sandwich transducer (MST), with a structure shown in Figure 4. The magnetostrictive strip is the main component for energy conversion in both wave generation and receiving, with the residual magnetism along the direction \(u_{2}\) by pre-magnetization. The soft magnet is used to reduce the magnetic resistance of the dynamic magnetic fields, improving energy conversion efficiency. There are two coils in the MST to generate and sense the dynamic magnetic fields. Coil 1 is \(\lambda/4\) ahead of Coil 2 in the direction \(u_{1}\). \(\lambda\) is the wavelength of the guided wave. The burst signals in Coil 1 and Coil 2 have a phase difference of \(\pi/2\). Wires in the working region of the MST are distributed with equal intervals and numbered with \(wn\) from 1 to 40. The current direction \(C_{d}\) and initial phase \(C_{ip}\) of all the wires are listed in Table 1.  
Each wire in the working region can be regarded as a line sound source, and the sound field \(S_{wn}\) for the \(wn\)th wire can be derived as [40]  
\[S_{wn}\left(r^{\prime},\theta^{\prime},t\right)\approx S_{m}L\sqrt{\frac{2}{\pi k r ^{\prime}}}\left[\frac{\sin\left(\frac{1}{2}kL\sin(\theta^{\prime})\right)}{ \frac{1}{2}kL\sin(\theta^{\prime})}\right]e^{i(k^{\prime}-\omega t+\frac{\pi }{4})} \tag{6}\]  
where the wave number \(k=\frac{2\pi}{\lambda}\), \(S_{m}\) is a constant coefficient, and \(r^{\prime}\) and \(\theta^{\prime}\) denote the relative position between the \(wn\)th wire and the receiving point as in Figure 4. \(L\) is half the length of the working region. Equation (6) is valid when \(r^{\prime}\gg L\). The sound field of the MST can be calculated as the sound field combinations of all wires:  
\[S\approx\sum_{wn=1}^{40}C_{d}(wn)S_{wn}e^{iC_{ip}(wn)}. \tag{7}\]  
As mentioned in Section 2.1, the thickness of the iron plate used in this study is 4 mm. The group velocity \(c_{g}\) and phase velocity \(c_{p}\) of the SH0 mode are both 3208 m/s, as in Figure 2. The wave length \(\lambda\) of the SH0 mode is 25.1 mm as the width of the magnetostrictive strip available is 50.2 mm. Thus, the main frequency \(f\) of the designed MST is 128 kHz, which is below the cutoff frequency of the SH1 mode (401 kHz). It should be noted that, as mentioned above, the main frequency of the transducer should be determined with the consideration of both detection sensitivities and testing ranges in field tests. The normalized sound fields of the MSTs with different lengths are shown in Figure 5. The waves are controlled to propagate in one direction, and the amplitudes of the main lobes in the sound fields are much larger than those of the side lobes. It can also be seen that the widths of the main lobes decrease with the increase in the lengths of the MSTs. The divergence angle \(\theta_{d}\), defined as the largest angle at which the amplitudes are no less than half of the largest amplitude, is used to denote the width of the main lobes. The relationship between the divergence angles and the lengths of the MSTs from 2 mm to 150 mm are shown in Figure 6.