## 2 Formulations
### Classical DPSM Formulation: ESD is a Single Point Source
If the number of target points is made equal to the number of source points (both equal to \(M\)) and the target points are placed on the transducer surface then Equation (7) can be written in matrix form as:  
\[\mathbf{V}_{S}=\mathbf{M}_{SS}\mathbf{A}_{S} \tag{8}\]  
where \(\mathbf{V}_{S}\) is the (\(M\times 1\)) vector of the velocity components at \(M\) number of target points on the transducer surface, and \(\mathbf{A}_{S}\) is the (\(M\times 1\)) vector containing the strengths of \(M\) number of point sources. \(\mathbf{M}_{SS}\) is the (\(M\times M\)) square matrix relating the two vectors \(\mathbf{V}_{S}\) and \(\mathbf{A}_{S}\):Vector \(\mathbf{A}_{S}\) of the source strengths is given by:  
\[\{A_{S}\}^{T}=\left[ \begin{array}{cccccccc}A_{1}&A_{2}&A_{3}&\ldots&A_{M-2}&A_{ M-1}&A_{M}\end{array} \right] \tag{10}\]  
The matrix \(\mathbf{M}_{SS}\) is obtained from Equation (7):  
\[\mathbf{M}_{SS}=\left[ \begin{array}{cccccccc}m_{11}&m_{12}&\ldots&m_{1M}\\ m_{21}&m_{22}&\ldots&m_{2M}\\ \vdots&\vdots&\ddots&\vdots\\ m_{M1}&m_{M2}&\ldots&m_{MM}\end{array} \right]=\left[ \begin{array}{cccccccc} f(z_{11},r_{11})&f(z_{12},r_{12})&\ldots&f(z_{1M},r_{1M})\\ f(z_{21},r_{21})&f(z_{22},r_{22})&\ldots&f(z_{2M},r_{2M})\\ \vdots&\vdots&\ddots&\vdots\\ f(z_{2M},r_{M1})&f(z_{2M},r_{M2})&\ldots&f(z_{MM},r_{M1})\end{array} \right]_{ M\times M} \tag{11}\]  
where:  
\[m_{nm}=f\left(z_{nm},r_{nm}\right)=\frac{z_{nm}\text{exp}\left(ik_{f}r_{nm} \right)}{i\omega\rho\cdot r_{nm}^{2}}\left(ik_{f}-\frac{1}{r_{nm}}\right) \tag{12}\]  
Equation (8) can be inverted to obtain the point source strengths:  
\[\mathbf{A}_{S}=\left[\mathbf{M}_{SS}\right]^{-1}\mathbf{V}_{S} \tag{13}\]  
Instead of velocity, if pressure is specified on the transducer surface, then the source strength is obtained from the following equation:  
\[\mathbf{A}_{S}=\left[\mathbf{Q}_{SS}\right]^{-1}\mathbf{P}_{S} \tag{14}\]  
where \(\mathbf{P}_{S}\) is the specified pressure field at the transducer surface and \(\mathbf{Q}\) matrix relates the pressure field and the source strength (obtained from Equation (4)). It should be noted here that Equation (13) can also be rewritten as \(\mathbf{V}_{S}^{*}=\mathbf{M}_{SS}^{*}\mathbf{A}_{S}^{*}\), where \(\mathbf{V}_{S}^{*}\) is a (\(3M\times 1\)) vector containing all three components of velocity at \(M\) test points. For inversion of this equation, when needed, we may have used the concept of ESD, which are triplet sources used at every center point so that \(\mathbf{A}_{S}^{*}\) becomes a (\(3M\times 1\)) vector containing all three strengths of the triplets at \(M\) source center points. See [3] for more detailed derivation. After evaluating the source strength vector \(\mathbf{A}_{S}\) from Equation (13) or (14) the pressure \(p(\mathbf{x})\) or velocity vector \(\mathbf{V}(\mathbf{x})\) at any set of target points \(\mathbf{x}\) (on the transducer surface or away) can be obtained from the following equations:  
\[ \begin{array}{l}\mathbf{P}_{T}=\mathbf{Q}_{TS}\mathbf{A}_{S}\\ \mathbf{V}_{T}=\mathbf{M}_{TS}\mathbf{A}_{S}\\ \mathbf{V}_{T}^{*}=\mathbf{M}_{TS}^{*}\mathbf{A}_{S}\end{array}  \tag{15}\]  
For a set of \(N\) target points the vector \(\mathbf{P}_{T}\) contains \(N\) pressure values at \(N\) target points, so its size is (\(N\times 1\)). \(\mathbf{V}_{T}\) is a (\(N\times 1\)) vector containing \(z\)-components of velocity at \(N\) target points, and \(\mathbf{V}_{T}^{*}\) is a (\(3N\times 1\)) vector containing all three components of velocity at \(N\) target points. Therefore, the velocity field on the transducer surface can be computed at \(N\) number of points where \(N\) is much greater than \(M\). It should be noted that only at \(M\) points the boundary conditions are specified. The velocity variation on the transducer face shown in Figure 2 is obtained with \(M=464\) (the number of source points considered for modeling the transducer) and \(N=2500\) (the number of target points where velocity is computed on the plane of the transducer face).