## 2 Ultrasonic Guided Wave Arc Sparse Array in a Plate and Receiving Aperture  
The ultrasonic guided wave transducer, functioning as an array element, can be conceptualized as a continuum array. Within this continuum, there are various configurationsof array element distributions. By adjusting the positions of certain array elements in a regular array, either radially or laterally, the arc array can be achieved, exhibiting a sparse distribution pattern. The arc array can be considered as an evolution from a linear array, while the arc sparse array can be viewed as an evolution from the arc array. Therefore, the arc array serves as the reference array for the arc sparse array. In the detection of arc sparse arrays, the transmitting aperture consists of different array elements, with the number of array elements used to describe the specific transmitting aperture configuration. In the case of 'one send and multiple receive' arc sparse arrays, a specific array element forms the transmitting aperture, while the receiving aperture is composed of multiple array elements. Extensive research has demonstrated that the receiving matrix contains valuable imaging information, and the data received by the receiving aperture can be effectively represented by the receiving matrix. The receiving matrix comprises both the amplitude receiving matrix and the transit time receiving matrix. Regarding the arc sparse array, considering the perspective of the receiving aperture, the amplitude matrix and the transit time matrix correspond to three typical receiving apertures: the 'left array receiving aperture', the 'right array receiving aperture', and the 'inner array receiving aperture', respectively. As depicted in Figure 1, the arc sparse array, comprised of \(N\) array elements, exhibits a partial left array receiving aperture configuration. The arrangement of the arc sparse array is laid out from left to right, using 14 ultrasonic wave transducers suitable for multi-frame imaging detection with an arc sparse array. The production steps and materials for the transducers come from the same batch. Additionally, the left array's receiving aperture is arbitrary, with the number of receiving aperture elements being \(s\) (\(1\leq s\leq N\)). In the context of ultrasonic guided waves in arc sparse arrays within a plate, the term 'left array receiving aperture' refers to the scenario where the acoustic wave signal is transmitted through the entire medium. After encountering scattering from a scatterer within the medium, the acoustic wave signal is then received by the array element positioned on the left side. This specific receiving aperture is referred to as the left array receiving aperture. Figure 1 illustrates the diagram of the receiving aperture for the left array within the arc sparse array. Figure 1: Chart of receiving aperture of arc sparse array and left array. (The curve connecting the transducers in the figure is the trajectory of the array). In the context of imaging with ultrasonic guided wave arc sparse arrays, a similar principle applies. Following the transmission of an acoustic wave signal, the wave propagates throughout the entire medium. Upon encountering scattering from a scatterer within the medium, the acoustic wave signal is received by all of the array elements positioned internally, forming the inner array receiving aperture. On the other hand, the right array receiving aperture is defined as the scenario where, during ultrasonic guided wave arc sparse array imaging detection, the acoustic wave signal propagates through the entire medium after transmission. Subsequently, after encountering scattering from a scatterer within the medium, the acoustic signal is received by the array element located on the right side, constituting the posterior array receiving aperture. The left array receiving aperture, inner array receiving aperture, and right array receiving aperture are three common types of receiving apertures.