# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 2 Modelling Method
### The Distributed Point Source Method  
The analytical method is fast but can only be used for simple cases. The numerical method is widely used but is computationally intensive. To improve the computational speed while ensuring computational accuracy, the semi-analytic method has been proposed. The semi-analytic method is a numerical method that uses a partially analytic solution or analytic function in the calculation process. At present, the semi-analytical method has become the mainstream method for ultrasonic testing simulation and calculation. DPSM is often used to calculate the sound field emitted by the probe using the semi-analytical method. The discrete point source method is usually used to discretize the probe surface into point sources when calculating the sound field emitted by the probe, such as in Rayleigh integration and DPSM [40]. This method uses distributed point sources located slightly below the surface of the transducer to simulate the ultrasonic field. The sound intensity of each point source is obtained by matrix inversion, which requires that the number of target points on the surface of the transducer is equal to the number of point sources. For a rigid baffle-type transducer in a homogeneous medium (as shown in Figure 5), the solution of the linear lossless wave equation can be expressed as a Rayleigh integral [41; 42]:  
\[p\left(\overset{\rightarrow}{x},\omega_{0}\right)=\frac{-j\omega_{0}p_{0}u_{0 }}{2\pi}\iint\limits_{s}\frac{e^{ikR}}{R}ds \tag{7}\]  
where \(p\left(\overset{\rightarrow}{x},\omega_{0}\right)\) is the pressure at the field point; \(\omega_{0}\) is the angular frequency; \(p_{0}\) is the density of the medium; \(V_{0}\) is the amplitude of the particle velocity that is normal to the transducer surface and distributed uniformly over the entire transducer aperture as \(V_{z}=V_{0}e^{-j\omega_{0}t}\); \(t\) is the time; \(k=\omega_{0}/c\) is the wave number and the velocity of sound in a medium is denoted by \(c\); and \(R\) is the distance between the field point and surface element \(ds\). The transducer is located on the \((x,y,0)\) plane; \(ds\) is the surface element of the transducer; \((x,y,z)\) is the coordinate of the field point \(\overset{\rightarrow}{x};R\) is the distance between the surface element and field point; and the shaded area is the rigid baffle. The purpose is to use \(M\) distributed point sources (as shown in Figure 6) to replace the energy device itself to model the ultrasonic field. These point sources are placed on a plane parallel to the surface of the transducer and are slightly offset from the plane \(r_{s}\). The order of the point source is marked as \(1-M\), the point source and the target point are indexed by \(m\) and \(n\), respectively, and the distance between the point source \(m\) and the target point \(n\) is \(r_{nm}\). From this we can simplify the integral summation in Equation (7) and obtain the following equation:  
\[p_{m}\left(\overset{\rightarrow}{x},\omega_{0}\right)=A_{m}\frac{e^{ikr_{m}}}{r_ {m}} \tag{8}\]  
where \(m\) is the index of a point source; \(r_{m}\) is the distance between point source m and field point \(\overset{\rightarrow}{x}\); and \(p_{m}\left(\overset{\rightarrow}{x},\omega_{0}\right)\) is the pressure that is contributed by a point source \(m\). \(A_{m}=-j\omega_{0}p_{0}\tau_{0}ds/2\pi\) is defined as the acoustic intensity of the point source \(m\). [PERSON] found in 1998 that the effect of element size on steering characteristics was small and the simple solution obtained by the discrete point source method was very close to that of the phased array. This gives an idea of using the discrete point source method in the ultrasonic arrays [42]. In 2005, [PERSON] et al. [43] of the University of Arizona used the DPSM to model the ultrasonic field generated by ultrasonic array transducers and studied the interaction of two-phased array transducers placed in a homogeneous fluid. In 2011, [PERSON] et al.