Acoustics of finite beams

We derive the expression for the most general possible finite acoustic beam, i.e., for a beam finite in two dimensions, which emerges from an arbitrarily shaped collimating aperture at an arbitrary angle and with an arbitrary amplitude profile. We investigate the spreading of such a beam for two amplitude profiles; a Gaussian profile and a flat-topped square profile. In both cases the spreading is determined as a function of the ratios between the collimating aperture sizes (σ or Ro) and three parameters; wavelength, axial distance from the aperture plane, and radial distance from the beam axis. The general applications of this analysis are numerous, particularly when one considers that almost all physically realizable acoustic sources project finite sound beams as opposed to waves of infinite extent. As a specific application of this general theory, we evaluate the scattering of a finite beam from a rigid sphere in an exact fashion for the two amplitude profiles of the finite beam mentioned above, and we finally compare plots for the bistatic cross sections to the results obtained in previous work of ours using the approximate Kirchhoff method.