## 3 Results  
In this paper, we propose a Bessel-based nonuniform vibration distribution for the surface of a spherically curved radiator to get a modeled acoustic field closer to a real measured field. Three classical approaches based on uniform (SCUR) and nonuniform distributions are included to contrast our main results. Figure 3 shows the measured pressure profile at 1 mm from the piezoelectric curved plate that is composed of local peaks distributed along the radius. This distribution can be represented with Equation (4), combining an adequate Bessel function (Equation (5)) and a function representing the radiator clamping using the traditional nonuniform approaches (SSR or CR with Equation (3)). Using a radiating function instead of the measured data for determining the acoustic field could permit the generalization of the radiation coming from this kind of transducer and eventually simplify the calculations by proposing analytic solutions for certain conditions. Although SSR and CR nonuniform distributions have been suggested since the 1960s decade for planar radiators, these have not been formally proposed for curved radiators [26; 37]; their inclusion in this paper is not just for comparison purposes but as two valid alternatives for producing certain types of acoustic fields. Then, the use of SSR/CR nonuniform distribution for curved radiators is not discarded by this work, so this can be an effective approach if the operation conditions of the transducer produce these kinds of vibration profiles. In Figure 4, it is shown the relative pressure profile along the propagation axis (z-direction) of classical SCUR and two \"traditional\" nonuniform vibration distributions compared with the measured field. The improvement in the fields using nonuniform distributions on the radiator can be noticed. When the vibration of a curved radiator is uniform, the field has very sharp local peaks along the propagation axis that are smoothed when the emitted radiation is gradually reduced at the plate rim, as in the proposed nonuniform approaches for \(r\to R\). The concordances of these fields and the measurements are much better, except in the prefocus zone in which the diffraction patterns differ. In both nonuniform approaches, the post-focus zone is smoother, following the same tendency as the measured profile. There is an after-lobe at about 3 cm, which is well-represented by both nonuniform approaches but only when those included the rim radiation. Without this component, this lobe disappears [41], keeping the most other components of the field. This rim radiation was also included in the main results of this paper. Figure 4: The amplitude of the relative acoustic pressure along the propagation axis of the spherically curved radiator with uniform distribution (SCUR), simply supported radiator (SSR), and clamped radiator (CR) conditions vs. measured field. The pressure is relative to the average pressure on the radiator surface of each condition. The results of using the Bessel-based radiation distributions are shown in Figure 5. For these proposals, Equations (3) to (5) were used to set the acceleration of the curved radiating boundary; also, the radiation from a 3.5 mm rim was included (boundary 6). For the Bessel-SSR approach, it can be noticed that the acoustic field at the prefocal zone has a peak distribution very close to the measured field, and the post-focal zone has the after-lobe produced by the rim radiation with an amplitude equivalent to that of the measurements. The Bessel-CR approach did not provide a good result in the prefocal zone, preserving only one peak of the three located at the measured field. The after-lobe in the far field was not present for this approximation, even when it included the rim radiation, which could indicate this approach is useless for this particular transducer. However, controlling the value of \(n\) in Equation (3) can reduce the amplitude of the focus of the pure SSR condition to make the model fit with the measurements if correlation does not provide an acceptable result.