September 2003©2003 by H.L. Bertoni1 VII. Diffraction by an Absorbing Half-Screen Kirchhoff-Huygens Approximation for Plane Wave Diffraction by an Edge Geometrical Theory of Diffraction Uniform Theory of Diffraction

September 2003©2003 by H.L. Bertoni2 Plane Wave Illumination of an Absorbing Half-Plane z x y r  dy dz ( x, y, 0 )

September 2003©2003 by H.L. Bertoni3 Evaluation of the Integration Over z The Contribution to the integral come from a small region about z = 0

September 2003©2003 by H.L. Bertoni4 Evaluation of the Integration Over y We distinguish two cases that are most easily solved 1) Well inside the region y > 0 illuminated by the plane wave. 2) Well inside the shadow region y < 0.  R  X  y’ y  R   X y’ y

September 2003©2003 by H.L. Bertoni5 Inside the Illuminated Region y > 0 y y Interrupted cancellation Cancellation of alternate half cycles

September 2003©2003 by H.L. Bertoni6 Evaluating y Integral for y > 0

September 2003©2003 by H.L. Bertoni7 Evaluating y Integral for y > 0 - cont.

September 2003©2003 by H.L. Bertoni8 Evaluating y Integral for y > 0 - cont. Y    x Incident plane wave

September 2003©2003 by H.L. Bertoni9 Inside the Shadow Region y < 0 y y Interrupted cancellation Cancellation of alternate half cycles

September 2003©2003 by H.L. Bertoni10 Evaluating y Integral for y < 0

September 2003©2003 by H.L. Bertoni11 Geometrical Theory of Diffraction (GTD ) (  ) y x Shadow boundary Diffracted Cylindrical wave Incident Plane wave

September 2003©2003 by H.L. Bertoni12 GTD Valid Outside Transition Region Fresnel zone Shadow boundary Incident plane wave y x

September 2003©2003 by H.L. Bertoni13 Example of Shadowing at Building Corners Building y x From Transmitter 2 m

September 2003©2003 by H.L. Bertoni14 Uniform Theory of Diffraction for Small y

September 2003©2003 by H.L. Bertoni15 Evaluating End Point Integrals

September 2003©2003 by H.L. Bertoni16 Evaluating End Point Integrals - cont.

September 2003©2003 by H.L. Bertoni17 Evaluating End Point Integrals - cont.

September 2003©2003 by H.L. Bertoni18 Approximation for F(s)

September 2003©2003 by H.L. Bertoni19 Field at the Shadow Boundary

September 2003©2003 by H.L. Bertoni20 Value of F(s) for Large s

September 2003©2003 by H.L. Bertoni21 Variation of Field Near Shadow Boundary Y  x x= 30 m Received Signal(dB) y(meters) WFWF -W F