An annular superposition integral for axisymmetric radiators
Coordinate axis used in the derivation. A piston of radius is excited by a radially varying particle velocity specified by an aperture function , where is the radial position on the piston. The radiator is surrounded by an infinite rigid baffle in the plane. The angle and relative distance correspond to the notation used in Eq. (1) .
Normalized pressure fields generated by “smooth” pistons modeled by Eq. (10) . (a) The pressure produced by a piston with radius with transition parameter , which closely resembles the field produced by a uniform piston. (b) The field generated by a piston with the same radius .
Normalized magnitude spectrum of the “smooth piston” pressure fields displayed in Fig. 2 . In each panel, the nearfield pressure is calculated in a transverse plane at via Eq. (10) followed by a two-dimensional Fourier transform. (a) The magnitude spectrum, displayed on a normalized decibel scale, for a smooth piston with . (b) The spectrum for a smooth piston with . Panel (b) contains significantly less spectral information than (a) due to the wider transition band.
Reference pressure fields for polynomial apodization given by Eq. (11) . (a) The effect of quadratic apodization . (b) Quartic apodization .
Pulsed fields generated by smooth pistons with the same parameters used in Fig. 2 . The time evolution for smooth pistons with and are compared. (a) and (b) Normalized pressure fields corresponding to at and , respectively. (c) and (d) Normalized pressure fields corresponding to at and , respectively.
Number of Gauss abscissas vs specified peak error (a) and computation time vs specified peak error (b) for the annular superposition method, the Rayleigh-Sommerfeld integral, and the generalized King integral applied to a parabolic radiator. The annular superposition method achieves 10% peak error with the application of 20 abscissas, the Rayleigh-Sommerfeld approach requires 70 abscissas, and the generalized King integral requires 56 abscissas. Since the computation times are similar for each integral evaluated with the same number of abscissas, (b) demonstrates the present method’s speed advantage compared to the Rayleigh-Sommerfeld approach and the generalized King integral at all error levels considered.
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