0018-926X (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2014.2368114, IEEE Transactions on Antennas and Propagation 1 High-frequency Asymptotics for the Radar Cross-Section Computation of a Prolate Spheroid with High Aspect Ratio Ivan V. Andronov, and Raj Mittra Abstract—The problem of high-frequency diffraction by elon- gated bodies is discussed in this work. The asymptotics are governed by the elongation parameter, which is the ratio of the longitudinal wave dimensions of the body to its cross-section. The cases of axial incidence and that of incidence at a grazing angle to the axis are considered, and the asymptotics of the far field amplitude are developed. Comparisons with numerical results for a set of test problems show that the leading terms of the new asymptotics provide good approximation with respect to the rate of elongation in a uniform manner. Effects of strong elongation on the RCS are discussed. Index Terms—Electromagnetic diffraction, high frequency asymptotics, parabolic wave equation, strongly elongated body. I. I NTRODUCTION M ETHODS of high-frequency diffraction remain an im- portant tool for the analysis of wave phenomena in many applications. Classical results of Fock, Keller and others are restricted to geometries with a single large parameter kρ, which measures the characteristic size in wavelengths. The condition of validity of these asymptotic expansions requires that all other quantities describing the problem should not compete with this large parameter. In some cases, this imposes conditions that are too restrictive on the frequency, by requiring it to be very high. One such case is the diffraction by elongated bodies. For strongly elongated bodies, there is another large param- eter besides the usual large parameter kb; namely, the aspect ratio Λ of the length to the transverse size. The asymptotic formulas of high-frequency approach are applicable only if the parameter Λ does not compete with the asymptotic parameter kb. Analysis of [1] shows that the parameter Λ, which we define for spheroids as the ratio of the major semiaxis b to the minor semiaxis a, starts to manifest itself in the leading order term of the creeping waves asymptotics, when Λ= O ( (kb) 1/5 ) and completely changes this asymptotics if Λ= O  (kb) 1/2  . (1) Recent attempts to reconstruct the high-frequency asymptotics in the latter case [2] show that the basic strategy of the usual approach needs to be changed. Such a new approach has appeared in [3] and [4], and is currently under further development. The approach yields the I. V. Andronov is with University of Saint Petersburg, Russia, e-mail:iva---@list.ru R. Mittra is with Pennsylvania State University and with University of Central Florida, USA, e-mail:rajmittra@ieee.org new asymptotic formulas, which contain the ratio χ = kb/Λ 2 , referred to herein as the elongation parameter. It is shown in Ref. [5] that the new asymptotic formulas reduce to well- known Fock’s asymptotics for the field in the penumbra when χ becomes asymptotically large. Numerical computations in [5] have demonstrated the accuracy of the new asymptotic formulas for the induced currents on the spheroid for the case of axial incidence. Accounting for the backward wave, whose asymptotics have been constructed in Ref. [6], leads to a close agreement between the asymptotic and numerical results for the currents in the middle part of the spheroid. Generalization of the approach for plane wave incidence at an angle to the axis has been achieved in [7], where the leading order approximation for the longitudinal component of the induced current was derived under the assumption that the angle of incidence ϑ is small in a manner that β ≡ √ kb ϑ = O(1). Using these asymptotic results for the induced currents, the far field amplitude of the scattered field can be found by using the Stratton-Chu formula [8]. A simpler case of axial incidence was reported in [9]. Here we derive the far field asymptotics for the case of skew incidence. It is worth noting that purely numerical methods may fail if the body is sufficiently large. Usually in this case high- frequency asymptotics of hybrid methods have been used. However, if the body is highly elongated, classical asymptotic methods may require that the frequency be very high. There may be a gap between frequencies at which numerical methods work, and those for which asymptotic approaches are viable. Our new asymptotic formulas aim to fill in this gap, which is the practical goal of our research. II. THE PROBLEM OF DIFFRACTION AND THE FIELD NEAR THE SURFACE We use the spheroidal coordinates (ξ,η,ϕ) that are related to the cylindrical ones (r,z,ϕ), where the z-axis is also the axis of the body (see Fig. 1), as follows z = pξη, r = p  ξ 2 − 1  1 − η 2 , (2) where p = √ b 2 − a 2 is the semi-focal distance. If the body were not elongated, we would use the Fock asymptotics given in [10], that are governed by the large parameter m =(kρ/2) 1/3 , where ρ is the characteristic radius of the surface curvature. The size of Fock domain is on the order of 1/m, in the direction of the wave incidence, while it is on the order of 1/m 2 along the normal direction. We observe