Herglotz波函数、Vekua变换和封闭方法
The Herglotz wave function, the Vekua transform and the enclosure method
摘要：
本文给出了作者提出的封闭方法在二维典型逆障碍和裂纹散射问题中的应用。给出了在固定波数和最多两个入射方向下，从散射场远场图中提取未知多边形声硬障碍物和分段线性裂纹的凸包的显式公式。本文的主要创新点是：将封闭方法与Herglotz波函数相结合，利用Vekua变换的思想显式构造Herglotz波函数中的密度。利用这种构造方法，可以避免对提取公式中的波数作任何限制。对有限角度下远场方向图的情况也作了尝试。
英文摘要：
This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given.
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