On the solutions of a class of dual integral equations occurring in diffraction problems

K. Eswaran
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引用次数: 49

Abstract

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a ‘standard form‘ of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions. The sufficient conditions for a given dual integral equations to be solvable in this manner are also determined. Hence, the method forms a complement to the Weiner-Hopf method. To illustrate the method solutions are obtained for a bench-mark problem : the diffraction of light by a finite perfectly conducting strip (or equivalently the diffraction of SH waves by a crack). Comparison with results obtained by others for low, intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies. Both the near field and the far field are obtained, the latter is shown to correspond to the Fraunhoffer diffraction pattern for high frequency. It is also shown that for the equivalent crack problem the stress intensity factor (SIF) fluctuates rapidly with changes in the angle of incidence for high frequencies, thus making the SIF especially sensitive to angle of incidence at high frequencies.
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衍射问题中一类对偶积分方程的解
结果表明,存在一类二维衍射问题,可以化为对偶积分方程的“标准形式”。这些衍射问题包括:电磁波通过有限条、有限缝的衍射,标量或矢量弹性波通过刚性条或裂纹的衍射等。通过构造紧支撑函数集与另一给定函数集双正交的技巧,给出了求解这类对偶积分方程的一般方法。给出了对偶积分方程用这种方法可解的充分条件。因此,该方法是对Weiner-Hopf方法的补充。为了说明该方法,得到了一个基准问题的解:光通过有限完全导电带的衍射(或等效的SH波通过裂纹的衍射)。与其它低频、中频和高频的结果比较,表明了该方法在整个频率范围内的实用性和准确性。得到了近场和远场,远场与高频的夫琅和费衍射图相对应。对于等效裂纹问题,应力强度因子(SIF)在高频时随入射角的变化而快速波动,从而使SIF对高频入射角特别敏感。
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