## 3 Results
For this paper, correlation was useful, and the value of \(m\) and \(n\) were easily found by this method, which permitted to find the constants for the modified SSR profile (Bessel-MOD) in Figure 5, with a more accurate result comparable with the measurements. The graphs of Figure 6 show the radial profiles at different depths. Important regions were chosen to plot these graphs: Figure 6a is the graph obtained at the first depth measured for the XZ plane; Figure 6b,c is at two equally space depths before the focus; Figure 6d is at the focus; Figure 6e is at the first minimum after the focus; and Figure 6f is at the after-lobe (after the focus). In these figures, the radial profiles at different \(z\)-distances for the Bessel-MOD are closer to the measurements than the uniform approach, not only in amplitude but also in shape. Other approaches presented in this paper were omitted from this figure for clarity, even if they also would have shown an improvement compared with the uniform approach. For instance, the pure SSR produced in the prefocal zone a profile like the Bessel-MOD without the peaks (a smoother profile), while after the focus, the SSR profiles were the same as those presented for the Bessel-MOD in Figure 6e,f. Figure 5: The amplitude of the relative pressure on the propagation axis using the Bessel-based nonuniform approximations with SSR and CR conditions on the curved radiating surface. Bessel-MOD is the Bessel-SSR with \(n=0.3\), which emulates correctly the measured field distribution. The pressure is relative to the average emitted pressure. The three models include an 8% of rim radiation, equivalent to the radiation measured at that zone. In Figure 7, the full acoustic fields of the more representative models of this work are shown. The Bessel-based approaches with modified SSR (SSR with \(n=0.3\)) and CR components are presented in Figure 7c,d, respectively, in order to be compared with the measured field of Figure 7a and the most used classical SCUR distribution of Figure 7b. This ideal SCUR distribution is composed of a very characteristic diffraction pattern before the focus, which is not present in the measured profile. The Bessel-MOD proposal provides a better result in the prefocal zone, with a similar diffraction pattern to the measured field. The Bessel-CR still has a similar diffraction pattern but with a narrower acoustic field and larger focus size, probably because of the reduced effective radiating area [41]. The focus locations in the Bessel-based models were more in agreement with the measured field than the SCUR; this was probably because of the reduction of the effective radiating areas, which were determined from each data set (models and measurements) to normalize the average emitted radiation that was the reference of the relative pressure used in the field comparisons. Figure 6: The radial relative pressure at different depths of the measured acoustic field compared with the SCUR distribution and the Bessel-MOD distribution, i.e., Bessel-SSR with \(n=0.3\) in Equation (3). Other analyzed distributions did not fully match the measured profile, and they were not included in these graphs for clarity, namely, SSR, CR, and Bessel-CR. The depths are (**a**) 0.2 cm, (**b**) 0.5 cm, (**c**) 1.0 cm, (**d**) 1.7 cm (focus); (**e**) 2.7 cm (down peak); and (**f**) 3.1 cm (lobe after focus). All the graphs have the same type of lines as in (**a**) and the same axis labels. The pressure is relative to the average pressure of Figure 3 of each distribution.