## 3 Results  
Numerical results are generated for a Gaussian radiation pattern, as shown in Figure 4, generated by every family. Quantum source strengths within every family are obtained for this radiation pattern following the classical DPSM approach and ignoring the presence of other families. Then the pressure field in the fluid in front of the uniformly vibrating circular transducer with a radius of 6 mm is computed by considering all families, as shown in Figure 3. For clarity, 50 families are shown in Figure 3. However, during actual computation 464 families were considered. Every family is composed of 30 quantum sources. The transducer vibration frequency is 1 MHz. Fluid properties considered for this simulation are those of water--density (\(\rho\)) = 1000 kg/m\({}^{3}\) and P-wave speed (\(c\)) = 1480 m/s. The normal velocity component is first computed at 2500 points on a square surface that coincides with the transducer surface. The computed results with the new formulation is shown in Figure 5a and with the old formulation (classical DPSM) is shown in Figure 5b. Clearly, the oscillations are much less (almost disappearing) in Figure 5a. In both formulations the final matrix that was inverted had dimensions of 464 \(\times\) 464. In the new formulation 464 clusters gave this matrix dimension, while in the old formulation 464 point sources generated this matrix. To improve the accuracy of the classical DPSM computation, the number of point sources and, hence, the matrix dimension must be increased while, in the new formulation, the number of quantum sources can be increased without increasing the number of clusters and, therefore, the matrix dimension. Figure 4: Normal velocity variation (or radiation pattern) from one source family. Figure 6 shows the variation of the normal velocity along the diameter (\(y\)-axis going through the center of the transducer face). The continuous line shows the field computed by the new formulation while the dotted line is obtained from the classical DPSM formulation. Nine points where the two curves match correspond to the boundary points where the velocity is specified. These nine points out of 464 points appear on the diameter along which the velocity values are computed. Classical DPSM gives correct values only at those nine points and deviates significantly from the correct value of 1 at other points. The results generated by the new formulation are much closer to the true values and match with the true values at several other points where the boundary conditions are not specified. To get a quantitative measure of the difference between these curves the error is computed using the following equation:  
\[\text{Error}=100\times\sqrt{\frac{\sum_{i}\left\{V\left(i\right)_{\text{true} }-V\left(i\right)_{\text{computed}}\right\}^{2}}{\sum_{i}V\left(i\right)_{ \text{true}}^{2}}} \tag{21}\]  
where \(V(i)_{\text{true}}\) is the true value of the velocity amplitude at the \(i\)-th point, which is 1 on the transducer surface and zero outside the transducer, as shown in Figure 6. The true value is denoted as the objective function (dashed-dotted line) in this figure. \(V(i)_{\text{computed}}\) are the computed values (continuous and dotted lines of Figure 6). The computed error is 34.8% for classical DPSM and 3.05% for proposed DPSM with the family ESD technique. Figure 5: The normal velocity on the transducer face computed by the DPSM technique when the transducer face is modeled by (**a**) source clusters, as shown in Figure 3 (new formulation); and (**b**) individual point sources as shown in Figure 1 (old formulation).