## 4 Discussion  
A proposal to effectively model the acoustic field of focused transducers has been presented. This was based on another previously proposed nonuniform distribution for planar radiators named Bessel-based nonuniform radiation distributions (shown in Figure 3). In this work, two other more general proposals are also used as comparison: the simply supported radiator (SSR) and clamped radiator (CR). The three proposals have shown that can be used to generate acoustic fields in good agreement with the field produced by focused radiators. The distributions were set in a curved boundary, and the results were obtained with a FEM commercial software. Only low-power linear behavior was considered in this work, but the main rationale presented here could be applied for high-power nonlinear measurements. The emission was considered to be purely harmonic operating at the transducer's nominal frequency of 2 MHz; the measured transducer emission bandwidth was 100 kHz, which represents about 5% of the central operation frequency. Because of that narrow bandwidth, the effect of this parameter in the acoustic field was considered negligible. The use of an analytical expression in the radiating boundary instead of the raw measurements, for acoustic field processing, would open the possibility not only to reduce computation but also to find a simpler way to calculate analytically the acoustic field in a closed form under certain conditions, for instance using the Hankel transform [42] or the direct integration of the Rayleigh integral [22,24]. However, this process is beyond the scope of this paper. Figure 4 shows the axial acoustic fields for the SCUR approach and the two classical nonuniform distributions SSR and CR already proposed for planar radiators. It can be noticed that the latter produce good results in the overall field amplitude when they are set in this curved radiating boundary but with poor agreement in the prefocal zone. At this region, the diffraction effects are quite different between the measured profiles in both the SSR and CR models. Actually, the diffraction profile of pure SSR and CR fields are similar to the SCUR field but with a reduced amplitude and an \"offset\". This indicates that reducing the emitted field amplitude at the edges of the radiator provokes also a smooth overall acoustic field. The SSR and CR fields are closer in amplitude to the measured field, and this could occur because the piezoelectric plate of the transducer is fixed at the edges, which produces an emitted radiation closer to a SSR/CR distribution than a SCUR, as seen in the measured field in Figure 3. This is also noticeable in the focus locations, which were closer to the measurements in the  
Figure 7: The normalized acoustic field of the focused transducer at a central longitudinal plane: the transducer would be at the bottom of each figure. The linear color scale: white = 1 and black = 0. (**a**) The measured acoustic field; (**b**) the modeled field using SCUR; (**c**) the modeled field using a modified Bessel-SSR (\(m=1\), \(n=0.3\)); and (**d**) the modeled field using Bessel-CR (\(m=1\), \(n=1\)). SSR and CR distributions than the SCUR; this was improved probably because of the reduction of the effective radiating area in the nonuniform distributions [15]  
Although these classical nonuniform distributions are closer in amplitude to the measured field, their results are not fully satisfactory. Then, it is proposed the use of Bessel-based distributions is closer to the real vibration distribution produced in concave transducers. The results of combining a Bessel with SSR and CR edge conditions are shown in Figure 5. From those results, it can be noticed that the prefocal zone of the acoustic field is better modeled after including the Bessel behavior in the radiation distribution. The peak-distribution in the modeled prefocal zone using Bessel-SSR has the same number of peaks as the measured field and almost at the same position.