An exact expression for the Lommel-diffraction correction integral

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 [Abh. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. 15, 233 (1886)]. Lommel's expression gives the piston field in terms of trigonometric functions and Lommel functions of two variables. 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, and approximation formulas for these roots are given. It is also shown that the same expression can be obtained by taking the limit as ka → ∞ (k is the wavenumber and a is the piston radius) of Williams's exact integral expression [J. Acoust. Soc. Am. 23, 1–6 (1951)] for the diffraction correction. Finally, it is shown both analytically and by comparison with numerical values for Williams's exact expression that this simple closed-form expression is a good approximation for the diffraction correction at all distances from the source provided that (ka)1/2 ≫ 1.