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An Exact Expression for Lommel's Diffraction Integral

### Abstract

A number of authors have obtained diffraction corrections for a circular piston source by numerical or graphical integration of an approximate expression for the piston field attributable to E. Lommel [Abhandl. Bayer. Akad. Wiss. 15, 233 (1886)]. Lommel's expression gives the piston field in terms of trigonometric functions and Lommel functions of two variables, the latter being defined as infinite sums involving cylindrical Bessel functions. It is shown here that the required integral of Lommel's expression can be evaluated analytically to obtain a simple closed form expression for the diffraction correction. The extrema of this expression are obtained as roots of simple transcendental equations. It is also shown that the same expression can be obtained by taking the limit as ka → ∞ (where k is the wavenumber and a is the piston radius) of Williams's exact integral expression for the diffraction correction [J. Acoust. Soc. Am. 23, 1 (1951)]. Finally, it is shown both analytically and by numerical integration of Williams's exact expression that the simple closed form expression is a good approximation for all distances from the source provided that (ka) ≫ l.

© 1974 Acoustical Society of America