## 1 Introduction  
Modeling of ultrasonic and sonic fields generated by a planar transducer of finite dimension is a fundamental problem whose solution is available in textbooks in this area [1; 2; 3; 4]. A good review of the earlier developments of the ultrasonic field modeling in front of a planar transducer can be found in [5]. A list of more recent developments is given by [PERSON] et al. [6]. The pressure field in front of a planar transducer in a homogeneous isotropic fluid has been computed both in the time domain [5; 7; 8] and in the frequency domain [9; 10; 11; 12; 13; 14]. In the frequency domain analysis, which is more popular, the transducers are assumed to be vibrating with constant amplitudes at certain frequencies, and the pressure fields in front of the transducers are computed. Most of these models give the steady state response of the transducers. Recently phased array transducers have been modeled as well [15; 16; 17; 18; 19; 20]. In all modeling approaches that are based on the Huygens-Fresnel superposition principle and Rayleigh-Sommerfeld Integral (RSI) the strengths of the point sources modeling the transducer are assumed to be known. For type-I RSI the pressure distribution, and for type II RSI, the velocity distribution, on the transducer surfaces are assumed to be known. The distributed point source method (DPSM) [21; 22; 23] has been developed relatively recently for solving electrostatic, electromagnetic,and ultrasonic wave propagation problems. DPSM was first proposed by [PERSON] and his co-workers for solving electrostatic and magnetostatic problems [21]. Later it was developed further by [PERSON] and [PERSON] [22] and [PERSON] et al. [23] for solving ultrasonic problems. In ultrasonic modeling the source strengths can be treated as unknowns and obtained from the transducer surface boundary conditions as velocity or pressure. Alternately, all point sources can be assumed to have the same strength in order to model the uniform velocity condition at the transducer surface. In recent years DPSM has been extended to solve high frequency fluid flow problems [24] and fluid mechanics problems involving flow around an obstacle [25]. DPSM showed a significant savings in computational time compared to the Finite Elements Method (FEM) FEM technique for modeling ultrasonic problems [23; 24; 26] or electromagnetic problems [27]. A three-dimensional (3D) wave field modeling problem in a homogeneous fluid in front of a square transducer was solved in 2 min by DPSM while COMSOL Multi-Physics FEM code took 35 h to solve the identical problem on the same computer [23]. [PERSON] and [PERSON] [26] used DPSM and FEM for solving ultrasonic wave reflections from a rough surface and concluded, _\"[FEM solution is] two orders of magnitude slower than the equivalent DPSM simulation on the same machine\"_. In the above works on DPSM, point sources were uniformly distributed near the boundaries, interfaces, and front faces of the transducer (or emitter). The source strengths were then calculated by solving a system of linear equations that are obtained by satisfying boundary and continuity conditions for the problem geometry. In the near field, very close to the point sources, the computed field satisfies the boundary conditions only at the discrete points where the boundary conditions are specified. In between those points the boundary conditions are generally not satisfied. To overcome this shortcoming the use of a scaling factor was proposed [23]. In [23] [PERSON] et al. showed that the computed values in the far field (away from the transducer face) could be accurately modeled after incorporating the scaling factor. After considering this scaling factor for both near and far field computations, it was observed that at sections very close to the transducer face the average value of the computed field matched well with the true solution, although for individual points the deviation from the true solution could still be significant.