## 4 Discussion  
### Driving Force Analysis of Liquid Solder CF Process  
For the analysis of the LF production mechanism in liquid solder, a 2D theoretical model of the EU wave production was constructed, as shown in Figure 6. The AC flowing \(J\) into the coil can be given by the following:  
\[J=\sqrt{2}J_{A}\cos(\omega t) \tag{5}\]  
where \(J_{A}\) is the effective current density. At present, an alternating MF \(B_{e}\) was induced in the surrounding air, which in turn induced an eddy current \(J_{e}\) of the same magnitude and frequency as the AC but in the opposite direction in the liquid solder. Due to the \(J_{e}\) and the static MF \(B\), the LF \(F_{1}\) was generated inside the liquid solder, which can be expressed by the following formula [31]:  
\[F_{1}=J_{e}\bullet B=\sqrt{2}J_{e}\bullet B\exp\left(-\frac{x}{\delta}\right) \cos\left(-\frac{x}{\delta}+\omega t\right) \tag{6}\]  
\(F_{2}\) was produced by a combination of the \(J_{e}\) and \(B_{e}\) in the air as given by the following:  
\[F_{2}=J_{e}\bullet B_{e}=\sqrt{2}J_{e}\bullet B_{e}\exp\left(-\frac{x}{\delta} \right)\cos\left(-\frac{x}{\delta}+\omega t\right) \tag{7}\]  
Figure 5: CF process of liquid solder at (**a**) 5 ms, (**b**) 11 ms, (**c**) 17 ms, and (**d**) 26 ms.  
Figure 6: Theoretical model of EU wave generation. The distribution of the LF in the liquid solder is shown in Figure 7. The LF as a whole was outwardly directed, with a right to left direction on the left coil and an opposite direction on the right. The presence of an alternating MF resulted in a macroscopic vertical downward repulsion, which was attributed to the vertical component of the LF. The LF on the surface of the liquid solder in all directions is shown in Figure 8, since the LF in Figure 7 had horizontal and vertical components. The liquid solder under the coil was subjected to a high LF, and the maximum horizontal and vertical values were 48.91 kN\(\cdot\)m\({}^{-3}\) and 60.93 kN\(\cdot\)m\({}^{-3}\), respectively. This indicates that the liquid solder under the coil plays a major role in the CF process of the liquid solder. Under the influence of LF \(F_{1}\) and \(F_{2}\), the liquid solder occurred periodic vibration, extrusion, and stretching when the AC frequency was 20 kHz. This resulted in a change in the density state in the liquid solder, ultimately forming the EU wave. According to [PERSON]. [32; 33], ultrasonic vibration could produce pressure in the liquid, that is, sound pressure. In addition, according to [PERSON]'s second law, when the liquid only receives the action of sine or cosine pressure, it will perform a reciprocating motion with the initial position as the center. Therefore, in order to analyze the mechanism of an EU wave on liquid solder, it is necessary to analyze the variation law of sound pressure in the liquid solder. Figure 9 shows the sound pressure distribution in the liquid solder under the influence of the EU wave. In the first half of the cycle, the top and bottom were dominated by negative pressure and positive pressure, respectively. When the time was 60 \(\upmu\)s, the maximum sound pressure value in the liquid solder was 0.309 MPa. Subsequently, the direction of the alternating current changed, and the positive and negative pressures in the liquid solder were converted to each other. The maximum sound pressure value in the second half of the cycle was 0.227 MPa. Figure 8: LF distribution on the surface of liquid solder in (**a**) horizontal direction and (**b**) vertical direction. Figure 7: Distribution of LF inside the liquid solder. To analyze the sound pressure change of the liquid solder entering the substrate gap, micro-points F, G, and H were taken at the bottom of the liquid solder. Figure 10 shows the sound pressure change analysis of three micro-points. The sound pressure variation curve for each micro-point is shown in Figure 11. The curve for each micro-point exhibited a sinusoidal periodic change.