## 3 The Left Rank of the Arc Sparse Array under the Condition of Weld Scattering
### Left-Rank Arc Sparse Array and Left-Rank Receiving Aperture
When the sound wave emitted by the array element m is reflected or scattered by the scatterer \(P\), the echo energy \(S_{mn}(x_{p},y_{p})\) is received by the array element \(A_{n}\). This energy is equivalent to the integral of the sound wave energy density, denoted as \(\rho_{mn}(x_{p},y_{p})\), along the width of \(dx_{n}\) at the front edge of the receiving array element \(A_{n}\). This integration is taken over the propagation time \(t_{mn}\) of the sound wave. Mathematically, it can be expressed as follows:  
\[S_{mn}(x_{p},y_{p},t_{mn})=\int_{x_{n}}^{x_{n}+dx_{n}}\rho_{mn}(x_{p},y_{p},t_{ mn}) \tag{9}\]  
The array element of the ultrasonic guided wave transducer, denoted as \(t_{mn}\), signifies the process of receiving the ultrasonic echo signal by the guided wave transducer. It represents the integral of the received echo signal at a specific position by each unit on the piezoelectric chip. The echo energy is then converted by the transducer into a magnitude that reflects the sound pressure. Consider \((x_{m},y_{m})\) as the coordinates of the emitting array element \(m\) and \((x_{n},y_{n})\) as the coordinates of the receiving array element \(n\). Additionally, let \(c_{g}\) represent the group velocity of the sound wave in the target area. When the transducer emits a sound wave signal and the emitting and receiving array elements of the transducer are in operation, the time taken for emission and reception can be expressed as follows:  
\[t_{mn}=\frac{\sqrt{\left(x_{m}-x\right)^{2}+\left(y_{m}-y\right)^{2}}+\sqrt{ \left(x_{n}-x\right)^{2}+\left(y_{n}-y\right)^{2}}}{c_{g}} \tag{10}\]  
Through the superimposed processing of the sound field information from each array element, it is possible to obtain the total sound field data matrix. This superposition involves  
Figure 2: The transmitting/receiving schematic diagram of an ultrasonic guided wave left-rank arc sparse array element in the plate. (The curve connecting the transducers in the figure is the trajectory of the array). combining the emitted signals of the array elements, each having an ideal single-frequency component. Mathematically, this superposition can be expressed as follows:  
\[B_{B-SAFT}(d_{m},\theta)=\sum_{n=0}^{N}\mathrm{e}^{\mathrm{j}\omega_{0}(-\frac{2d _{m}}{c_{g}}-t_{mn})} \tag{11}\]  
In the equation, \(\theta\) represents the deflection angle of the emitting array element relative to the scattering point, \(d_{m}\) is the distance from the emitting array element \(m\) to the scatterer, \(t_{mn}\) is the time of sound wave propagation, \(\omega_{1}\) is the angular frequency corresponding to the selected initial frequency, and \(\omega_{2}\) is the angular frequency corresponding to the selected cut-off frequency. Once the array parameters are determined, the size of the left-rank receiving aperture becomes a crucial factor that influences the characteristics of the array's sound field. Consequently, it also impacts the imaging resolution. When the transducer emits signals as array elements, the resulting sound wave comprises multiple frequency components within a specific frequency bandwidth. For the left-rank receiving aperture signal of the guided wave emitted and received by the transducer, with a particular frequency bandwidth serving as the array element, it can be expressed as follows:  
\[B_{B-SAFT}(d_{m},\theta)=2\pi\sum_{n=1}^{N}\left[\int_{f_{1}}^{f_{2}}\mathrm{e} ^{\mathrm{j}\omega(-\frac{2d_{m}}{c_{g}}-t_{mn})}d\omega\right] \tag{12}\]  
In the equation, \(f_{1}\) and \(f_{2}\) represent the central frequencies. To conduct research, a self-developed ultrasonic guided wave transducer is employed for the array elements, forming an arc sparse array. The ultrasound excited by these array elements predominantly adopts the SH0 mode. The emitted and received echo signals exhibit a compact \"wave packet\" structure, with relatively concentrated adjacent peaks. This characteristic greatly benefits imaging detection.