## 2 Formulations
### Classical DPSM Formulation: ESD is a Single Point Source  
To generate the acoustic field in the fluid medium in front of an ultrasonic transducer or the electric field in front of an emitter by the DPSM technique, a number of point sources are placed slightly behind the transducer (or emitter) face as shown in Figure 1. This figure shows \(M\) number of source points placed at the centers of \(M\) spheres of radius \(r_{s}\). Spheres touch the front face of the transducer. Thus, the point sources are located at a distance of \(r_{s}\) behind the transducer face to avoid the singularity. The DPSM formulation for ultrasonic wave field modeling is presented below. However, for electric field modeling the derivations are identical; the parameter names are simply changed from pressure, velocity, etc., to electric field, magnetic field, etc. From [3] the pressure and velocity fields in a fluid at point \(\mathbf{x}\) at a distance \(r_{m}\) from the \(m\)-th point source of strength \(Am\) can be written as:  
The pressure field:  
\[p_{m}(\mathbf{x})=p_{m}(r_{m})=A_{m}\frac{\exp(ik_{f}r_{m})}{r_{m}} \tag{1}\]  
The velocity in the radial direction, at a distance \(r\) from the \(m\)-th point source:  
\[v_{m}(r)=\frac{A_{m}}{i\omega\rho}\frac{\exp(ik_{f}r)}{r}\left(ik_{f}-\frac{1} {r}\right) \tag{2}\]  
Figure 1: \(M\) point sources are placed slightly behind the transducer face and the ultrasonic field is computed at \(N\) target points in front of the transducer. Therefore, the \(z\)-component of the velocity is:  
\[v_{zm}(r)=\frac{A_{m}}{i\omega\rho}\frac{2\exp(ik_{f}r)}{r^{2}}\left(ik_{f}-\frac {1}{r}\right) \tag{3}\]  
If there are \(M\) point sources distributed over the transducer surface, as shown in Figure 1, then the total pressure and the \(z\)-direction velocity values at point \(\mathbf{x}\) are given by:  
\[p(\mathbf{x})=\sum_{m=1}^{M}p_{m}(r_{m})=\sum_{m=1}^{M}A_{m}\frac{\exp(ik_{f}r _{m})}{r_{m}} \tag{4}\]  
\[v_{z}=\sum_{m=1}^{M}v_{zm}\left(r_{m}\right)=\sum_{m=1}^{M}\frac{A_{m}}{i\omega \rho}\frac{z_{m}\exp\left(ik_{f}r_{m}\right)}{r_{m}^{2}}\left(ik_{f}-\frac{1}{ r_{m}}\right) \tag{5}\]  
where \(z_{m}\) is the distance of point \(\mathbf{x}\) in the \(z\)-direction (perpendicular to the transducer face) measured from the \(m\)-th point source. If the transducer surface velocity in the \(z\) direction is given by \(v_{0}\) then for all \(\mathbf{x}\) values on the transducer surface the velocity should be equal to \(v_{0}\). Therefore:  
\[v_{z}\left(x_{\text{transducer}}\right)=\sum_{m=1}^{M}\frac{A_{m}}{i\omega \rho}\frac{z_{m}\exp\left(ik_{f}r_{m}\right)}{r_{m}^{2}}\left(ik_{f}-\frac{1} {r_{m}}\right)=v_{0} \tag{6}\]  
By satisfying Equation (6) at \(M\) points on the transducer surface, where the spheres touch the transducer face it is possible to obtain a system of \(M\) linear equations with \(M\) unknowns (\(A_{1}\), \(A_{2}\), \(A_{3}\),..., \(A_{m}\)). If the point sources are placed slightly behind the transducer face as shown in Figure 1 then \(r_{m}\) of Equation (6) can never be equal to zero; thus, singular points are avoided. If point \(\mathbf{x}\) in Figure 1 coincides with the \(n\)-th target point and the velocity at the \(n\)-th target point or the observation point is to be computed then from Equation (6) one obtains:  
\[v_{z}\left(x_{n}\right)=\sum_{m=1}^{M}\frac{A_{m}}{i\omega\rho}\frac{z_{nm} \exp\left(ik_{f}r_{nm}\right)}{r_{nm}^{2}}\left(ik_{f}-\frac{1}{r_{nm}}\right) \tag{7}\]  
In Equation (7) the first subscript \(n\) (of \(z\) and \(r\)) corresponds to the \(n\)-th target point and the second subscript m represents the m-th source point; therefore, \(r_{nm}\) is the distance of the \(n\)-th target point from the \(m\)-th source point while \(z_{nm}\) is the distance along the \(z\)-direction between the \(n\)-th target point and \(m\)-th source point.