## 2 Method
### Frequency Weighting Matrix Beamforming Algorithm  
The traditional beamforming algorithm and the dispersion phenomenon in the plate are introduced above. From the analysis, it can be understood that the traditional beamforming method has the following two difficulties for the leakage location of thin plates such as spacecraft shells:  
1. Lamb wave velocity is difficult to determine, and there is an error in the delay matrix.
2. For practical applications, it is difficult to determine the characteristics of frequency bands by different leaks.  
The FWMB algorithm that is proposed in this paper solves the above problems through the following two angles:  
1. Narrowband filtering is used to delay different overlaid signal speeds with different frequency bands via the dispersion curve.
2. A weighting method is used to compensate the frequency band response of different leak holes in real applications. The weighting matrix is determined by the different energy ratios of different frequency bands to adapt to the frequency band changes.  
A flowchart comparison between the traditional beamforming [18] and the FWMB algorithm is shown in Figure 2. In the figure, the main differences between the two algorithms are marked by a red border. The following describes the FWMB algorithm based on the traditional beamforming algorithm with some improvements in detail.  
First, the frequency band of the signal needs to be determined. The signal from sensor #0 is processed by the fast Fourier transform and meshed at 10 kHz intervals (for example, 100-110 kHz). Since the number of points in each frequency band is consistent, all the values in each frequency band are summed and normalized as the energy value of the frequency band. The frequency bands that reach more than 50% of the peak value are reserved, and the bands are defined by the internal frequency as \(f_{1}\),\(f_{2}\),...,\(f_{k}\) (for example, 100-110 kHz is 105 kHz). The total number of the reserved frequency bands is \(k\). The weight of the peak frequency band is 1, and the corresponding weights determine the frequency weighting matrix \(mf1\), \(mf2\),..., \(mfk\) of the different frequency bands.  
The signals of sensors #0-7 are narrow-band filtered on the above frequency bands to obtain a processed signal matrix. The processed signal is represented as \(f_{k}\)(\(t\),\(n\)). For leakage excitation in the thin plate, a large out-of-plane displacement is generated [19]. For the out-of-plane motion, compared to the \(\mathrm{A_{0}}\) mode, the energy of the \(\mathrm{S_{0}}\) mode in the Lamb wave is relatively small [17]. Because the beamforming method mainly focuses on energy, not the arrival time in the TDOA, the wave velocity of the \(\mathrm{A_{0}}\) mode is selected as the wave velocity for each bands. The wave velocity matrixes \(sf1\), \(sf2\),..., \(sfk\) at the above frequencies are determined by the dispersion curve. Therefore, the signals of each sensor are delayed and superimposed to obtain the following delayed sum signal \(h\)(\(t\),\(\theta\)) (corresponds to \(g\)(\(t\),\(\theta\)) in Section 2.1.):  
\[h(t,\theta^{\prime})=\sum_{n=1}^{7}\sum_{k=1}^{K}mfk\times\left[f_{k}(t-\frac{ d_{n}}{sfk},n)+f_{k}(t,0)\right], \tag{7}\]Through the above changes, the FWMB algorithm can improve the traditional beamforming algorithm's ability to locate thin plate leaks and its adaptability to different leaks.