This study's primary goal is to improve the accuracy and efficiency of acoustic wave propagation simulations using circular probe models in ultrasonography. This research aims to develop a detailed understanding of how variations in boundary conditions and wave characteristics influence the fidelity of ultrasound imaging. A range of simulation techniques were employed, focusing on non-dispersive and dispersive scenarios, to model acoustic wave behavior comprehensively. The study utilized Gaussian beamforming techniques and improved kernel functions to refine the resolution and decrease computational overhead. Various scenarios were simulated to analyze the impact of wave scattering and dispersion on imaging outcomes. The simulations demonstrated significant improvements in image resolution and accuracy. The refined methods allowed for more apparent distinctions in wave behavior under different boundary conditions, providing deeper insights into wave propagation dynamics. The results confirmed that controlling dispersion and scattering is critical for enhancing imaging quality. This research contributes to the field of ultrasonic imaging by presenting advanced simulation methods that offer more accurate and efficient imaging solutions. The study provides valuable insights into optimizing ultrasonic probes and imaging techniques by focusing on the impact of wave characteristics and boundary conditions. The findings have significant implications for medical diagnostics and material characterization, suggesting potential improvements in ultrasound technology for better patient outcomes and more precise material assessments.