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 compared with results based on the closed-form time-domain or impulse-response formalism. It will be shown that the proposed frequency-domain formalism has theoretical and practical value. Finally, specific computational considerations are discussed as necessary.

Publication Date



Copyright © 1999 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.

The following article appeared in

C. J. Daly and N. A. H. K. Rao, “The arccos and Lommel formulations—Approximate closed-form diffraction corrections,” J. Acoust. Soc. Am. 105, 3067–3077 (1999).

and may be found at https://doi.org/10.1121/1.424636.

The authors thank Emil Wolf, Wilson Professor of Optical Physics at the University of Rochester, New York, for his assistance. He discussed certain aspects of this work with Captain Daly, provided references, and pointed the way to others. The elegant results he derived in the 1950’s made this paper possible. The authors also thank the reviewers for their insightful comments and helpful suggestions.

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Document Type


Department, Program, or Center

Chester F. Carlson Center for Imaging Science (COS)


RIT – Main Campus