Spatial averaging in the beam of a piston transducer
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This paper presents a theoretical analysis of the spatially averaged free-field responses of phase-sensitive and phase-insensitive receivers centered in the beam of a harmonically excited piston transmitter. The responses of unfocused circular plane piston receivers are analyzed, and both unfocused and spherically focused piston transmitters are considered. A set of closed-form expressions figures prominently in the analysis. The expressions are based on the Lommel diffraction formulation which is, in turn, based on the Fresnel approximation. Although approximate, the expressions allow for quick and easy estimation of phase-sensitive or phase-insensitive unfocused piston receiver responses. It is shown that the spatial averaging effects associated with phase-sensitive and phase-insensitive receivers are virtually identical when γ ≤ 0.1, where γ = b/a is the ratio of receiver radius b to transmitter radius a. In addition, numerical results obtained from the closed-form expressions are compared with previously reported results. The comparisons indicate that the approximate results are valid from the m = 3 maxima forward under the assumption of linear propagation when ka > 58, where k is the circular wave number. Finally, it is pointed out that the closed-form expressions may prove useful in the estimation of the potential for bioeffects associated with diagnostic ultrasound.