## 2 Theory
### The Lamb Wave Characteristics of Impact Signal in Stiffened Panel  
In the plate, ultrasonic waves are constantly reflected at the boundary and propagate in the form of Lamb waves. According to the movement type of the particle relative to the plate, it can be divided into a symmetrical (S) mode and an antisymmetrical (A) mode Lamb wave. Since the Lamb waves have dispersion characteristics, each mode of Lamb waves is further divided into multiple modes having different phase velocities and group velocities as the frequency changes. The equation of the dispersion is shown in Equation (1) [46].  
\[\frac{\tan{qd/2}}{\tan{pd/2}}=-\left(\frac{4k^{2}pq}{\left(q^{2}-k^{2}\right)^{ 2}}\right)^{\pm 1}, \tag{1}\]  
where \(p=\sqrt{{k_{l}}^{2}-k^{2}}\), \(q=\sqrt{{k_{l}}^{2}-k^{2}}\), \(k_{l}=\omega/c_{l}\), \(k_{t}=\omega/c_{t}\), \(k=\omega/c_{p}\), the indices \(1\) and \(-1\) represent the symmetrical and antisymmetrical modes, \(d\) represents the plate thickness, \(\omega\) is the angular frequency, \(c_{l}\), \(c_{t}\) and \(c_{p}\) are the longitudinal wave velocity, transverse wave velocity and phase velocity, respectively, \(k_{l}\) and \(k_{t}\) are the wave numbers of the longitudinal wave and transverse wave, respectively, \(k\) represents the wave number along the horizontal direction of the panel, and \(f\) is the frequency. The group velocity \(c_{g}\) is shown in Equation (2).  
\[c_{g}=c_{p}\cdot\left[1-\frac{1}{1-\frac{c_{p}}{\left(fd\right)\frac{\mathrm{ d}p}{\mathrm{d}(fd)}}}\right] \tag{2}\]  
Figure 2 shows the phase and group velocity dispersion curves for a \(3\) mm thick \(5\)A06 aluminum plate obtained by numerical calculation.  
Figure 1: Flow chart of the adaptive energy compensation threshold filtering (AECTF) method.  
The sound waves generated by impact propagate from the impact point in the form of Lamb waves in the plate, and are received by the sensors as the impact signals. When an impact occurs, there are mainly three modes of Lamb waves of S0, A0 and S2 [5], and the wave velocity for impact locating is the group velocity. The S2 mode is relatively high in the frequency band. As the propagation distance increases, the signal energy is attenuated severely, so it is not suitable for large-scale impact source locating. For the S0 and A0 modes, the A0 mode has higher energy and larger amplitude, which is widely used in the leak location field. However, for the impact source location, the A0 mode wave velocity is slower than the S0 mode, and it overlaps with the S0 mode frequency in the frequency domain. When it passes through the stiffeners, reflection and modal transformation occur, it is difficult to obtain the accurate arrival time of the A0 mode in the time domain. The S0 mode is in a lower frequency band and has a faster wave velocity, which is less susceptible to interference by other modes. However, the S0 mode's energy is much smaller than the A0 mode's, the energy is further attenuated after passing through a plurality of stiffeners. In addition, the effects of factors such as frequency response, coupling, and environmental noise cannot be ignored. Taking into account the above factors, this paper takes the S0 mode Lamb wave in the frequency band of 50 kHz-500 kHz as the research object.