Scattering and the Parabolic Approximation for Slender Bodies

Abstract

The parabolic approximation theory for scattering by long thin transparent bodies, developed by Mei and Tuck in 1980 [SIAM J. Appl. Math., 39 (1980), pp. 178–191], is analysed critically by both a formal matched asymptotic expansion and a numerical scheme based on the boundary integral equation method. The existence of a special range of parameters, for which the theory is formally valid, is verified, and the use of the parabolic approximation for parameters outside this range is examined. It is shown that caution must be used in thus employing the parabolic approximation, with breakdown occurring once resonance, or a steep longitudinal gradient in amplitude, appears.The theory is also applied to opaque bodies, revealing the existence of a range of parameters for which the parabolic approximation is useful. The effect on the theory of curved ends is described and a wave perturbation in the numerical results is discussed.