# Modes Control of Lamb Wave in Plates Using Meander-Line Electromagnetic Acoustic Transducers
## 3 Mode Control of Lamb Waves
### Modes Control with Paired EMAT Configurations  
According to the characteristic Equations (11)-(18), the displacement forms of symmetric and antisymmetric modes are shown in Figure 3, where the deformations are symmetric or antisymmetric to the middle plane of the plate, respectively. If the distributions of the excitation forces are the same as those of the displacements of the symmetric and antisymmetric motion, the corresponding Lamb wave modes will be obtained from the stress-strain relationship of elastic materials. Therefore, a paired EMAT configuration is proposed based on the mechanism of Lorentz force to generate the forces corresponding to symmetric motion. The paired EMAT consists of two traditional meander-line NB-EMATs which embrace the specimen symmetrically, as shown in Figure 4a, where the normal bias magnetic fields are provided by two vertically polarized magnets. The Lorentz forces excited by this configuration are symmetric from the middle plane of the plate, as shown in Figure 4b. The traction forces on the upper and lower surfaces are expressed as  
\[\tau_{yx}=\left\{ \begin{array}{cc}-F_{x}&y=d\\ -F_{x}&y=-d\end{array} \right.\text{ and }\tau_{yy}=\left\{ \begin{array}{cc}F_{y}&y=d \\ -F_{y}&y=-d\end{array} \right. \tag{27}\]  
Figure 2: Phase velocity dispersion curves for A0, A1, S0, and S1 Lamb wave modes in aluminum plates. Substituting the boundary loading Equation (27) into Equations (9) and (10), respectively, we can obtain  
\[\tau_{yx}\big{|}_{y=d} = \mu[-2 ikpA_{2}\sin(pd)+(k^{2}-q^{2})B_{1}\sin(qd)] \tag{28}\] \[+\mu[2 ikpA_{1}\cos(pd)+(k^{2}-q^{2})B_{2}\cos(qd)]=-F_{x}\] \[\tau_{yx}\big{|}_{y=-d} = \mu[2 ikpA_{2}\sin(pd)-(k^{2}-q^{2})B_{1}\sin(qd)]\] (29) \[+\mu[2 ikpA_{1}\cos(pd)+(k^{2}-q^{2})B_{2}\cos(qd)]=-F_{x}\] \[\tau_{yy}\big{|}_{y=d} = \mu[(k^{2}-q^{2})A_{2}\cos(pd)-2 ikqB_{1}\cos(qd)]\] (30) \[+\mu[(k^{2}-q^{2})A_{1}\sin(pd)+2 ikqB_{2}\sin(qd)]=F_{y}\] \[\tau_{yy}\big{|}_{y=-d} = \mu[(k^{2}-q^{2})A_{2}\cos(pd)-2 ikqB_{1}\cos(qd)]\] (31) \[-\mu[(k^{2}-q^{2})A_{1}\sin(pd)+2 ikqB_{2}\sin(qd)]=-F_{y}\]  
If we subtract Equation (29) from Equation (28) and add Equation (30) to Equation (31), Equations (32) and (33) are obtained:  
\[\mu[-2 ikpA_{2}\sin(pd)+(k^{2}-q^{2})B_{1}\sin(qd)]=0 \tag{32}\]  
\[\tau_{yx}\big{|}_{y=d} = \mu[-2 ikpA_{2}\sin(pd)+(k^{2}-q^{2})B_{1}\sin(qd)] \tag{28}\]0  
Combining Equations (32) and (33), the symmetric mode frequency of Equation (20) is obtained. It follows a similar way to excite antisymmetric wave modes. A paired PB-EMAT is proposed to generate antisymmetric Lamb waves. The paired PB-EMAT consists of two EMATs, whose bias magnetic fields are parallel to the plate surface. As shown in Figure 5a, the parallel bias magnetic fields are provided by two horizontally polarized magnets. In this case, the Lorentz forces are antisymmetrically distributed along the plate section, as shown in Figure 5b. The traction forces on the upper and lower surfaces are expressed as  
\[\tau_{yx}=\left\{ \begin{array}{cc}-F_{x}&y=d\\ F_{x}&y=-d\end{array} \right.\text{ and }\tau_{yy}=\left\{ \begin{array}{cc}F_{y}&y=d \\ F_{y}&y=-d\end{array} \right. \tag{34}\]  
Figure 4: (**a**) The paired normal biased EMAT (NB-EMAT) configuration, and (**b**) the Lorentz forces generated by the paired NB-EMAT. Only the Lorentz forces generated by the wires in the dotted frame are plotted in (b). The symbol \(\bigotimes\) denotes the inward current flows on the paper vertically, and \(\bigcirc\) denotes the outward current flows on the paper vertically. Figure 3: The displacement form of symmetric and antisymmetric Lamb waves: (**a**) symmetric mode, and (**b**) antisymmetric mode. Similarly, substituting the boundary loading Equation (34) into Equations (9) and (10), we can obtain  
\[\mu[2 ikpA_{1}\cos(pd)+(k^{2}-q^{2})B_{2}\cos(qd)]=0 \tag{35}\]  
\[\mu[(k^{2}-q^{2})A_{1}\sin(pd)+2 ikqB_{2}\sin(qd)]=0 \tag{36}\]  
Combining Equations (35) and (36), the antisymmetric mode frequency Equation (21) is obtained.