A 2D fast near-field method for calculating near-field pressures generated by apodized rectangular pistons

Analytical two-dimensional (2D) integral expressions are derived for fast calculations of time-harmonic and transient near-field pressures generated by apodized rectangular pistons. These 2D expressions represent an extension of the fast near-field method (FNM) for uniformly excited pistons. After subdividing the rectangular piston into smaller rectangles, the pressure produced by each of the smaller rectangles is calculated using the uniformly excited FNM expression for a rectangular piston, and the total pressure generated by an apodized rectangular piston is the superposition of the pressures produced by all of the subdivided rectangles. By exchanging summation variables and performing integration by parts, a 2D apodized FNM expression is obtained, and the resulting expression eliminates the numerical singularities that are otherwise present in numerical models of pressure fields generated by apodized rectangular pistons. A simplified time space decomposition method is also described, and this method further reduces the computation time for transient pressure fields. The results are compared with the Rayleigh–Sommerfeld integral and the FIELD II program for a rectangular source with each side equal to four wavelengths. For time-harmonic calculations with a 0.1 normalized root mean square error (NRMSE), the apodized FNM is 4.14 times faster than the Rayleigh–Sommerfeld integral and 59.43 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 12.50 times faster than the Rayleigh–Sommerfeld integral and 155.06 times faster than the FIELD II program. For transient calculations with a 0.1 NRMSE, the apodized FNM is 2.31 times faster than the Rayleigh–Sommerfeld integral and 4.66 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 11.90 times faster than the Rayleigh–Sommerfeld integral and 24.04 times faster than the FIELD II program. Thus, the 2D apodized FNM is ideal for fast pressure calculations and for accurate reference calculations in the near-field region.