An analytical expression is derived for time-harmonic calculations of the near-field pressure produced by a circular piston. The near-field pressure is described by an efficient integral that eliminates redundant calculations and subtracts the singularity, which in turn reduces the computation time and the peak numerical error. The resulting single integral expression is then combined with an approach that divides the computational grid into sectors that are separated by straight lines. The integral is computed with Gauss quadrature in each sector, and the number of Gauss abscissas in each sector is determined by a linear mapping function that prevents large errors from occurring in the axial region. By dividing the near-field region into 10 sectors, the raw computation time is reduced by nearly a factor of 2 for each expression evaluated in this grid. The grid sectoring approach is most effective when the computation time is reduced without increasing the peak error, and this is consistently accomplished with the efficient integral formulation. Of the four single integral expressions evaluated with grid sectoring, the efficient formulation that eliminates redundant calculations and subtracts the singularity demonstrates the smallest computation time for a specified value of the maximum error.
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