Charles J. Daly

Time- and frequency-domain descriptions of spatially averaged one-way diffraction for an unfocused piston transducer

Time-domain and frequency-domain expressions describing spatially averaged effects of one-way diffraction in the case of an unfocused piston transmitter and finite receiver are of theoretical and practical interest. Here, a time-domain description based on the arccos diffraction formulation and a frequency-domain description based on the Lommel diffraction formulation are derived. Numerical results obtained from the two descriptions are then compared. It is shown that the two descriptions show satisfactory agreement for finite receivers of practical interest. Two mathematical lemmas are also provided.

Spatial averaging in the beam of a piston transducer

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 gamma < or = 0.1, where gamma = 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.

A Spatially Averaged Impulse Response for an Unfocused Piston Transducer

Fourier–Bessel theory is used to derive a closed-form solution for the spatially averaged velocity-potential impulse response associated with one-way diffraction from an unfocused piston transducer of radius a. The derivation provides additional insight into the problem of diffraction from an unfocused piston transducer.

The arccos and Lommel formulations—Approximate closed-form diffraction corrections

A closed-form frequency-domain formalism for spatially integrated diffraction corrections is proposed. Spatially integrated diffraction corrections are necessary when trying to characterize material with ultrasonic probing. In the case of piston transducers and point receivers, the Lommel diffraction formulation is used when the excitation is monochromatic, and the arccos diffraction formulation is used when the excitation is impulsive. The Lommel and arccos formulations are usually treated separately; here, they are connected. Specifically, the arccos diffraction formulation and Lommel diffraction formulation are shown to form an approximate Fourier transform pair. Since the Lommel formulation is amenable to closed-form spatial integration, Lommel functions are used to derive diffraction corrections for unfocused piston transducers operating in receiveonly (one-way) mode or transmit/receive (two-way) mode. Results obtained from the proposed closed-form frequency-domain formalism are qualitatively compare...

Scalar diffraction from a circular aperture

List of Figures. Preface. Acknowledgments. 1. Introduction. 2. Literature Review. 3. Two Diffraction Formulations. 4. Spatially Averaged One-Way Diffraction. 5. Spatially Averaged Two-Way Diffraction. 6. Experimental Investigation. 7. Analytical Investigation. 8. Recommendations for Further Research.

Recommendations for Further Research

This monograph proposed a theory of spatially averaged diffraction correction for ultrasonic piston transducers operating in pulsed mode. A good portion of the theory has been cast in closed-form and verified, both analytically and numerically. Indeed, we even provided a few cases of indirect experimental validation of the theory. Additionally, auto-convolution diffraction corrections were applied experimentally. Despite this, much more work needs to be done.

Two Diffraction Formulations

This chapter establishes and verifies the Fourier equivalence of the arccos and Lommel diffraction formulations as an approximate Fourier transform pair. This relationship is important because it serves as the mathematical foundation for a proposed frequency-domain formalism of spatially averaged diffraction corrections for ultrasonic piston transducers. Although the development is cast in terms of ultrasonic propagation, the results are applicable to any physical problem involving scalar diffraction from a circular aperture. Some of the material in this chapter first appeared in References [23] and [25], and it is reprinted here with permission.

Spatially Averaged one-way Diffraction

The previous chapter established the Fourier equivalence of the arccos and Lommel diffraction formulations as an approximate Fourier transform pair. In this chapter, we exploit this Fourier equivalence and derive a set of general, closed-form, frequency-domain expressions describing spatially averaged one-way diffraction for unfocused piston transmitters and receivers. The expressions derived are general in the sense that the area of the receiver may be less than, equal to, or greater than that of the transmitter. In the time-domain, we present a novel derivation of a closed-form description of one-way diffraction with a finite receiver. Both the time-domain and frequency-domain derivations are followed by discussions and analysis which serve to unify and extend existing theory. Results obtained from the time-domain expressions are compared with those obtained from the frequency-domain expressions. Portions of the material in this chapter first appeared in References [23]–[25] and is reprinted here with permission. Finally, some focused frequency-domain results are derived in Chapter 7.