## 2 Materials and Methods
### Acoustic Field Modeling Using FEM
#### 2.2.3 Radiator with Bessel Acceleration  
The vibration distribution of a piezoelectric plate is closer to a combination of Bessels than a uniform distribution [38; 39; 40]. This behavior depends on the material's piezoelectric properties, the constraint conditions, the coupling layers, and the shape. When the plate is concave, the vibration of the concave-shape piezoelectric plates can also be composed of Bessel-like vibration distributions due to the Lamb waves [17]. The components affecting the vibration of an ultrasound transducer are vast and difficult to quantify. Then, for the proposal of this paper, the vibration was determined by measuring the acoustic pressure very close to the radiating surface [27]. If we suppose the ultrasound behaves as a plane wave in the region very close to the radiating surface (when \(r\to 0\)), the acoustic pressure is related with the acceleration as shown in Equation (2). Thus, the proposal of this paper is to use, as the nonuniform radiation distribution, a Bessel-based function composed by two sub-functions [27]. For this, the acceleration on the curved radiating surface will be expressed by  
\[a_{0}(r)=A_{0}f(r)\cdot g(r), \tag{4}\]  
where \(r\) is the variable radius related to each point in the curved boundary and \(A_{0}\) is the amplitude of the emitted acoustic pressure, provided that the amplitude of the product of \(f(r)\) and \(g(r)\) is 1 at \(r=0\). The function \(f(r)\) is given by the acceleration of Equation (3), and it represents the effect of attaching the piezoelectric plate to the transducer case (edge clamping), then \(f(r)=\sum_{mn}A_{mm}\Big{[}1-(r/R)^{2m}\Big{]}^{n+1}\). This function can be determined by adjusting the parameters to get a correct representation of the measured profile, which can be evaluated by correlation. The function \(g(r)\) represents the radial measured peak distribution (MPD) shown in Figure 3. This was approximated using the Bessel-based function  
\[g(r)=C_{1}J_{0}\Big{(}\beta_{2N}\frac{r}{R}\Big{)}+C_{2}. \tag{5}\]  
Equation (5) was proposed after considering the reports and simulations about Bessel patterns in the vibration distribution on some transducer surfaces [17; 28; 29]. For the profile measured for the transducer used for this paper, we used a pure negative Bessel \(J_{0}\) mounted over either an SSR or CR function to approximate the experimental MPD. Then, the Bessel-SSR and Bessel-CR distributions were obtained by combining Equations (5) and (3) into Equation (4), with adequate constants. Here, \(C_{1}\) and \(C_{2}\) are the coefficients that control the amplitude and the offset of Equation (5), respectively, and \(\beta_{2N}\) is the \(2N\) zero of \(J_{0}\), where \(N\) is the number of peaks in the MPD. When \(C_{1}\) is close to zero, the acoustic field is close to the ideal uniform radiation (SCUR) for \(C_{2}\
eq 0\); for practical concerns, the average amplitude into the effective radiating radius of the function \(a_{0}(r)\), after selecting adequate functions \(f(r)\) and \(g(r)\), should be 1, which can be adjusting with an adequate value of \(A_{mn}\). The effective radius was determined by calculating the effective area containing 95% of the total radiation. Then, the amplitude of the emitted pressure can be controlled with \(A_{0}\). In Figure 3, \(C_{1}=-0.3\), \(C_{2}=0.9\), \(R=1\) cm, \(m=1\), \(n=0.3\), \(N=5\), \(A_{mn}=11.7\), and \(\beta_{2N}=30.67\), which is the tenth zero of Bessel \(J_{0}\). The values of \(m\) and \(n\) of Equation (3) were determined by the values producing the maximum correlation with the measured data. The Bessel-SSR of Figure 3 requires \(n=0\); Bessel-MOD (MOD, modified Bessel-SSR) have the same constants as the Bessel-SSR, but \(n=0.3\). These values are highly dependent on the transducer construction, not only on the piezoelectric constants and disk geometry; they are dependent on the way the disk was attached to the case, the thickness and properties of extra layers (electrodes, glue, etc. ), and the mechanical properties of the case.