Some Diffraction Problems Involving Conical Geometries and their Rigorous Analysis

D. Kuryliak
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引用次数: 4

Abstract

The wave diffraction from the hollow finite and truncated perfectly conducting (rigid., soft) conical scatterers is considered. It is supposed that conical surfaces have zero thickness. The diffraction problems are formulated in the spherical coordinate system as the boundary value problems for the Helmholtz equation with respect to the scattered scalar potentials. The diffracted field is given by expansion in the series of eigenfunctions for subregions formed by the scatterers. Due to enforcement of the conditions of continuity together with the orthogonality properties of the Legendre functions the diffraction problems are reduced to infinite system of linear algebraic equations (i.s.l.a.e.). Usage of the analytical regularization approach transforms i.s.l.a.e. to the second kind and allows to justify the truncation method for obtaining numerical solution in the required class of sequences. These systems are proved to be regulated by a couple of operators, which consist of the convolution type operator and the corresponding inverted one. The elements of the inverted operator can be found analytically using the factorization technique.
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一些涉及圆锥几何的衍射问题及其严密分析
波的衍射从空心有限和截断完全导电(刚性)。考虑了软锥形散射体。假设圆锥表面的厚度为零。在球坐标系中,衍射问题被表述为关于散射标量势的亥姆霍兹方程的边值问题。衍射场是由散射子形成的子区域的本征函数级数展开得到的。由于连续性条件和勒让德函数的正交性的实施,衍射问题被简化为线性代数方程组的无穷系统。解析正则化方法的使用将i.l.a.e.转换为第二类,并允许证明截断方法在所需的序列类中获得数值解。证明了这些系统是由一对由卷积型算子和相应的逆算子组成的算子所调控的。利用因式分解技术可以解析地求出倒算子的元素。
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