## 4 Discussion
The focus amplitude was improved when adapting the equation with \(n=0.3\), for the Bessel-MOD, which produced a field more congruent with the measurements. The focus locations of these representations were also more in agreement with the measurements than the classical models presented before, probably due to the reduction of the effective radiating areas. In the post-focus zone, the model can also produce the after-lobe observed in the measurements but with a slightly lower amplitude. This after-lobe was only present in the Bessel-SSR and Bessel-MOD simulations with 8% of rim radiation, which was based on the measurements (see rim radiation in Figure 3). The Bessel-CR condition produced a similar profile at the prefocal zone but with less pronounced peaks. In this case, the after-lobe in the post-focal zone was practically unnoticeable, even with the inclusion of the rim radiation, probably because of the total dominance of the acoustic intensities coming from the main radiating surface. Figure 6 shows the radial distributions at different depths of the three more significant cases for our comparison purposes, namely, the SCUR, the modified Bessel-SSR (Bessel-MOD), and the measurements. Before the focus, in Figure 6a-c, the radial profile using the Bessel-MOD follows adequately the measured field with some variations in amplitude in the central peak. In spite of the complexity of adequately simulating this region, the Bessel-MOD approach correctly produces acceptable results. A similar agreement was obtained in other less complicated regions as the focus and after the focus in both amplitude and field width. In the former, shown in Figure 6d, the Bessel-MOD produced almost the same amplitude as the measurements, with a 0.4% relative error; the focus with the SCUR is 40% larger than the measured one. After the focus (Figure 6e,f), our model still has a similar tendency as the measured field, with a good match of the graphs in Figure 6e and little variation in the central peak in Figure 6f. From these graphs, the SCUR does not represent an adequate model for this kind of radiators, i.e., those with nonuniform Bessel-based vibration patterns. The acoustic fields presented in Figure 7 show the differences among each approach in a detailed manner using a linear color scale. The prefocal zones of both Bessel approaches are similar but with a larger focus for the Bessel-CR approach. Focus locations were also improved using our proposed models. The SCUR approach differs at practically any region from the measured field. For this transducer, the Bessel-MOD presents better results than the other nonuniform distributions studied here, with a clear good agreement in most of the graphs shown in this paper. However, this does not mean that the applicability of this model is universal for curved radiators. Some transducers could still behave as ideal pistons, simply supported curved disks, or clamped curved disks that could require the use of any of the other approaches (SCUR, SSR, CR, and Bessel-CR). The use of any of these models should depend on the real radiation profile measured very close to the radiating surface. This paper has presented a new modeling approach to effectively simulate the acoustic field of focused transducers that can be adapted to most of the devices used in different medical applications. Eventually, using the proposed functions as radiation distributions of this kind of transducers could potentially permit to find more analytical solutions of this kind of radiators that could better match the measurements.