## 3 Results
Figure 6: The normal velocity variation on the transducer surface: dashed-dotted line—true value; solid line—computed results using source clusters, as shown in Figure 3; and dashed line—computed by superimposing individual point source contributions, as shown in Figure 1.  
From the symmetry of the problem it is obvious that the fluid velocity parallel to the transducer surface in the \(x\)-direction should be zero along the \(y\)-axis going through the transducer center. This velocity component on the transducer surface is plotted in Figure 7. It is close to zero for the new formulation. However, for the classical formulation, strong velocity values are observed near the periphery of the transducer. Both formulations give zero velocity near the central axis of the transducer. From Figures 6 and 7 it is clear that the new formulation significantly improves the field computed in the fluid adjacent to the transducer face. To investigate how the computed fields away from the transducer face are affected by these two formulations the normal velocity components are plotted along the \(y\)-axis for four different distances (\(z=0\), \(r_{s}/3\), \(2r_{s}/3\) and \(r_{s}\)) from the transducer surface. These figures are shown in Figure 8. It should be noted that \(r_{s}\) is the radius of the source sphere, as shown in Figure 3.0 Therefore, \(r_{s}\) is also the distance of the point source from the transducer surface. Figure 8 clearly shows that as the observation points move away from the transducer face by a distance equal to \(r_{s}\), or when they are located at a distance \(2r_{s}\) from the point sources, then the oscillations in the computed field disappear for the classical DPSM also. However, the curves generated by the classical DPSM and the family ESD DPSM do not match, although their shapes appear to be similar. The classical DPSM results need to be multiplied by a scaling factor to match with the new results. Figure 3.1 The normal velocity variation on the transducer surface (bottom figure) and near it (the other three figures), ’\(z\)’ is the distance from the transducer face; solid line—computed results using source clusters, as shown in Figure 3; and dashed line—computed by superimposing individual point source contributions, as shown in Figure 3.2  
Figure 3.3 The X-component of the velocity variation on the transducer surface: solid line—computed results using source clusters, as shown in Figure 3; and dashed line—computed by superimposing individual point source contributions, as shown in Figure 3.4  
Classical and family ESD DPSM results are then compared with the analytical solution in Figure 3.5 For a circular transducer of radius of the analytical expression of the pressure field along the central axis is given by [23]:  
Figure 3.6  
where \(p\) is the fluid pressure at a distance \(z\) from the transducer face vibrating at frequency \(\omega\) (rad/s) in a perfect fluid of density \(\rho\). The normal velocity component (\(v_{z}\)) at the transducer surface is \(v_{0}\). The wave number \(k_{f}\) is equal to \(\omega/c_{f}\), where \(c_{f}\) is the acoustic wave speed in the fluid. The dashed-dotted line of Figure 9 shows the analytical solution, the continuous line is the family DPSM solution, and the dotted line is the classical DPSM solution. Clearly, the family DPSM solution is closer to the analytical solution (error is 5.57%) than the classical DPSM solution (error is 32.7%), but neither solution matches with the analytical solution. This is because we arbitrarily assumed the Gaussian radiation pattern for each family. This assumption might be better than the spherical radiation pattern generated by a point source in classical DPSM modeling but it is still not necessarily the actual radiation pattern needed for perfect matching. However, both curves can be matched with the analytical solution simply by multiplying them with appropriate scaling factors.