## 2 Nonlinear Guided Waves
### Emat  
EMAT is composed of a generating coil for generating an eddy current of the electromagnet and the transmission and a receiving detecting coil. When the probe is approached to the metal specimen, the inside of the specimen is affected by the electromagnet. As Shown in Figure 3, the eddy current is formed on the surface of the specimen by an alternating current flowing in the generating coil inside the probe. Lorentz force is generated between the eddy current and the low power line, and mechanical displacement occurs on the surface of the specimen. This mechanical displacement is called ultrasound.  
The purpose of this study was to investigate the nonlinearity of materials by inducing an ultrasonic wave in the longitudinal mode in steel wire rod. The permanent magnet was magnetically biased in the axial direction on the specimen. Then, when the coil was wound in the circumferential direction and the current was applied, the longitudinal mode of the guided ultrasonic wave was generated in the cylindrical structure such as the pipe or the wire rod.  
EMAT was generated by Lorentz force effect in meander coil. The alternative current induce in the coil and the magnetic field is built in coil by high-power permanent magnet. Here **J** is eddy currents field and B is the magnetic field. The Lorentz force is generated by the eddy currents and magnetic field  
\[\overset{\rightarrow}{F}=\overset{\rightarrow}{\int}\times\overset{ \rightarrow}{B} \tag{3}\]  
As shown in Figure 4, the geometry used for describing the reception of EMAT is considering various factors. Equation (3) explain the relationship between magnetic field and eddy current. The influence of magnetic field for inducing eddy current is indicated by Equations (4) and (5). Produce a voltage pick up by receiving coil is Equation (6). EMAT does not require a contact medium in flaw detection. As a noncontact method using electromagnetic, it shows higher performance than  
Figure 3: Principle of electromagnetic acoustic transducer (EMAT) with magnetic flux.  
Figure 2: Nonlinear parameters according to distance.  
PZT for measuring nonlinearity of materials. However, it should be considered that the receiving sensitivity is lower than PZT.  
\[\begin{array}{c}E_{ed}=\
u\times B\\ J_{ed}=\sigma\times E_{ed}\end{array}\overset{\rightarrow}{F}=\overset{ \rightarrow}{J}\times\overset{\rightarrow}{B} \tag{4}\]  
\[\begin{array}{c}\
abla^{2}\mathrm{A}_{0}=j\omega\sigma_{0}\mu_{0}\mathrm{A}_{ 0}\\ \
abla^{2}\mathrm{A}_{1}=-\alpha^{2}\sigma_{1}\mu_{1}\mathrm{A}_{1}\overset{ \rightarrow}{F}=\overset{\rightarrow}{J}\times\overset{\rightarrow}{B}\\ \
abla^{2}\mathrm{A}_{2}=j\omega\sigma_{2}\mu_{2}\mathrm{A}_{2}\\ \
abla^{2}\mathrm{A}_{3}=j\omega\sigma_{3}\mu_{3}\mathrm{A}_{3}\end{array} \tag{5}\]  
\[\begin{array}{c}\mathrm{E}_{r}=-\frac{\partial\Lambda}{\partial t}\\ V_{r}=N\times l\times\mathrm{E}_{r}\end{array}\overset{\rightarrow}{F}= \overset{\rightarrow}{J}\times\overset{\rightarrow}{B} \tag{6}\]  
where \(\
u\) is the velocity of particle vibration, B is the static magnetic field, \(\sigma\) is the conductivity of the test piece, \(\mathrm{E}_{ed}\) and \(\mathrm{J}_{ed}\) are the induced electric field and the induced eddy current sheet due to the particle vibration at the depth \(z_{ed}\) respectively.