## 1 Introduction
There is no static bias magnetic field in the high-energy pulse EMAT. Lorentz force is generated by the interaction of dynamic magnetic field \(\mathbf{B}\) and pulse eddy \(\mathbf{J_{e}}\). The frequency of \(\mathbf{F_{L}}\) is twice the frequency of the current in the excitation coil, and the sensitivity of detection defects can be improved by the high-frequency acoustic signal [28]. The discharge circuit of the high-energy pulse electromagnetic acoustic excitation source can be simplified as a series connection of a discharge capacitor \(C\), an excitation coil \(L\), and an equivalent resistance \(R\). The discharge process can be simplified to a \(LC\) second-order circuit zero input response. The capacitor's initial voltage \(u_{c}\) and inductor current \(i\) are associated with reference direction. The characteristic equation of the circuit of high-energy pulse excitation source can be expressed as,  
\[LC\frac{\mathrm{d}^{2}u_{c}}{\mathrm{d}l^{2}}+RC\frac{\mathrm{d}u_{c}}{ \mathrm{d}t}+u_{c}=0 \tag{5}\]  
\[LCs^{2}+RCs+1=0 \tag{6}\]  
\[i=C\frac{\mathrm{d}u_{c}}{\mathrm{d}t} \tag{7}\]  
where \(L\) is the value of coil inductance, \(C\) is the value of capacitor capacitance, \(R\) is the circuit equivalent resistance, \(u_{c}\) is the initial capacitor voltage, and \(s\) is the Laplace transform. The requirement about the discharge circuit parameters of the high-energy pulse excitation source should be met, that \(R\) is much smaller than \(2\sqrt{(L/C)}\). In this way, the circuit operates in the zero-input underdamped state of the second-order circuit. The variation of current with time is a gradual decayingsinusoidal function relationship, in the discharge circuit. The Equations (5)-(7) combined to solve the current function relationship is Equation (8):  
\[i=\frac{u_{c}}{\sqrt{1-\alpha^{2}}}\sin(\sqrt{1-\alpha^{2}}\frac{t}{2\pi\sqrt{LC }})\sqrt{\frac{C}{L}}\mathrm{e}^{[-\alpha(t/\sqrt{LC})]}. \tag{8}\]  
The \(\alpha=\frac{R}{2}\sqrt{\frac{C}{L}}\) represents the damping coefficient, and its value is proportional to the equivalent resistance in the circuit, which affects the rate of decay of current in the circuit. Due to the lack of a static bias magnetic field, hysteresis loops on the magnetization curve do not be formed in the magnetization process, but a cluster of hysteresis loops that decay to zero. The magnetization process is shown in Figure 1. The alternating magnetic field reaches the maximum M\({}_{0}\) from zero, and the high-energy pulse electromagnetic acoustic magnetization changes along the magnetization curve O-m\({}_{0}\). The magnetic flux density \(B\) increases with the magnetic field strength \(H\) to the maximum. The alternating magnetic field varies from a maximum value M\({}_{0}\) to N\({}_{0}\) and then to M\({}_{1}\) in a decay cycle. The magnetic flux density \(B\) also varies in a period along m\({}_{0}\)-n\({}_{0}\)-m\({}_{1}\) due to the hysteresis effect in ferromagnetic materials. In ferromagnetic materials, the magnetic flux density \(B\) decays periodically with the magnetic field strength \(H\), and finally decays to zero. Due to the action of the alternating magnetic field, acoustic waves are excited by the mechanical vibration which is caused by periodic stretching and compression of the material. The magnetizing force \(F_{M}\) is generated by the alternating magnetic field, is shown in Equation (9),  
\[F_{M}=\mu_{0}M\cdot\
abla H \tag{9}\]  
where \(\mu_{0}\) represents magnetic permeability in the vacuum, and \(M\) represents the magnetization vector. In addition to the magnetic field affecting the magnetization of ferromagnetic materials, the EMAT conversion mechanism of ferromagnetic materials also adds a magnetostrictive conversion mechanism. For ferromagnetic materials, the magnetostrictive constitutive relationship can be expressed as,  
\[\left\{