Bessel beam expansion of linear focused ultrasound

Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247-252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464-1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979-2991 (1993)] and O'Neil [J. Acoust. Soc. Am. 21, 516-526 (1949)] on axis and with direct integration of a Kirchhoff integral both on and off axis. This type of expansion will have applications to scattering problems.