Analysis of the electromagnetic field perturbed by a subsurface crack in a half space
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ANALYSIS OF THE ELECTROMAGNETIC FIELD PERTURBED BY A SUBSURFACE CRACK IN A HALF SPACE Z. T. Nazarchuk, A. Ya. Teterko, and V. I. Hutnyk
UDC 620.179.14
The rigorous method of integral equations is used to study the field of a longitudinal crack located in a conducting half space. The results of evaluation of the electric and magnetic components of the perturbed field are presented for different locations of the crack.
Consider a thin longitudinal cracklike defect located in a homogeneous isotropic conducting half space and parallel to the Oz-axis. A source of a two-dimensional E-polarized monochromatic electromagnetic field is placed in the upper half space. The basis ( x O y ) and local ( x1 O1 y1 ) coordinate systems are chosen as shown in Fig. 1. The problem of diffraction is reduced to the problem of finding solutions of the Helmholtz equation satisfying the conditions of continuity of the tangential components of the vectors of electromagnetic field on the interfaces of the media, the corresponding conditions imposed at infinity, and the Meixner-type conditions on the edges of the scatterers. By analyzing the exact integral equations defined on the boundary of the cross section of a cylindrical scatterer, one can show that, for a thin dielectric inclusion, the electric component of the total field weakly varies across its thickness [1]. In the zero-order approximation, it can be regarded as independent of the thickness of the defect and equal to a certain value attained in its median line L. Assume that L is a smooth nonclosed Lyapunov contour (this is the sole restriction imposed on the geometry of the scatterer). In this case, as shown in [1], the problem of diffraction is equivalent to the following integral equation: E(t0 ) − d(k22 − k32 ) ∫ E(t )G(2) (t, t0 ) ds = E * (t0 ) ,
(1)
L
where G
(2)
( t , t0 ) =
ν1, 2 =
1 i (1) H0 (k2 r ) + 4 2π
q 2 − k12, 2 ,
∞
∫ 0
exp [ ν2 ( y + y0 )]( ν2 − ν1 ) cos (q ( x − x0 )) dq , ν2 ( ν1 + ν2 )
Re ( ν1, 2 ) ≥ 0,
t = t ( s ) ≡ x ( s ) + i y (s ),
t0 = L,
where H0(1) (k2 r ) is a Hankel function of the first kind, E * (t0 ) is the known distribution of the longitudinal (along the Oz-axis) component of the electromagnetic field in the absence of the inclusion, E (t ) is the unknown distribution of the electric component of the field induced by the inclusion, s is the arc abscissa of the median line L of the section of the thin inclusion by the plane x O y, r is the distance between two points of the conKarpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 5, pp. 69–74, September–October, 2006. Original article submitted December 5, 2005. 1068–820X/06/4205–0649
© 2006
Springer Science+Business Media, Inc.
649
650
Z. T. N AZARCHUK , A. YA. T ETERKO,
AND
V. I. HUTNYK
tour L with affixes t and t0 and arc abscissas s and s0 , k1 , k2 , and k3 are the wave numbers of the upper and lower half spaces and the inclusion, respectively, d is the thickness of the inclusion
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