A radial propagator for axisymmetric pressure fields

The concept of a propagator is central to the angular spectrum formulation of diffraction theory, which expresses the pressure field diffracted by a two-dimensional aperture as a superposition of a continuum of plane waves. In the conventional form, an exponential term, known as a propagator, is multiplied by the wavenumber spectrum obtained from a two-dimensional spatial Fourier transform of the aperture boundary condition, to obtain the wavenumber spectrum in a plane parallel to the boundary, offset by some distance specified in the propagator. By repeated use of this propagator and Fourier inversion, it is possible to completely construct the homogeneous part of the pressure field in the positive half-space beyond the planar boundary containing the aperture. Drawing upon preceding work relating the boundary condition to the axial pressure Pees, J. Acoust. Soc. Am. 127(3), 1381-1390 (2010)], it is shown in this article that when the aperture is axially symmetric, an alternative type of propagator can be derived that propagates an axial wavenumber spectrum away from the axis of the aperture. Use of this radial propagator can be computationally advantageous since it allows for field construction using one-dimensional Fourier transforms instead of Hankel transforms or two-dimensional Fourier transforms.