Wave-number sampling at short range

Spectral solutions can be obtained efficiently at long ranges by using a quadrature scheme that involves perturbing the integration contour off the Peal line [F. B. Jensen et al., Computational Ocean Acoustics (American Institute of Physics, New York, 1994), pp. 231-240]. The efficiency of this implementation of the spectral solution is investigated for short-range problems. Examples are presented to illustrate that the self-starter [J. Acoust. Sec. Am. 92, 2059-2074 (1992)] is about ten times faster at ranges on the order of ten wavelengths. The accuracy of the spectral solution and the self-starter are directly related to the accuracy of rational approximations. The rational function associated with the spectral solution is based on the numerical evaluation of the spectral integral. The rational function associated with the self-starter is based on the analytic evaluation of the spectral integral. The relationship between these approaches is analogous to the relationship between the finite-difference and split step Padi [J. Acoust. Sec, Am. 93, 1736-1742 (1993)] solutions of the parabolic equation. [S0001-4966(99)06211-6].