September 2003©2003 by H.L. Bertoni1 VIII. Further Aspects of Edge Diffraction Other Diffraction Coefficients Oblique Incidence Spherical Wave Diffraction by an Edge Path Gain Diffraction by Two Edges Numerical Examples

September 2003©2003 by H.L. Bertoni2 Other Diffraction Coefficients Felsen’s Rigorous Solution for Absorbing Screen (  = 0 ) Conducting Screen     Reflected plane wave Incident plane wave RSB ISB

September 2003©2003 by H.L. Bertoni3 Comparison of Diffraction Coefficients D 1 : Kirchhoff -HuygensD 3 : Conductor for TE polarization D 2 : FelsenD 4 : 90  conducting wedge for TM polarization angle,  D1D1 D2D2 D3D3 D4D4 D  2  k RSB   ISB 90 o wedge  RSB  ISB Edge

September 2003©2003 by H.L. Bertoni4 Diffraction for Oblique Incidence Diffracted rays lie on a cone whose angle is the same as that between incident ray and edge. All waves have wave number k sin  along edge k cos  in (x,y) plane Replace k for normal incidence by k cos   y x z

September 2003©2003 by H.L. Bertoni5 Diffraction of an Incident Spherical Wave (for paths that are nearly perpendicular to the edge) Field incident on the edge Diffracted cylindrical wave dipole r  dA r0r0 00  dA  In the horizontal plane, rays spread as if they came from a point r 0 behind the edge.

September 2003©2003 by H.L. Bertoni6 Top and Side Views of the Diffracted Rays Dipole r 0   r W(  ) W(r) Dipole r 0   r L(  )  L(r) Top View Side View

September 2003©2003 by H.L. Bertoni7 Diffracted Field Amplitude Must Conserve Power in a Ray Tube dipole r  dA r0r0 00  dA 

September 2003©2003 by H.L. Bertoni8 Path Gain for Diffracted Field

September 2003©2003 by H.L. Bertoni9 UTD Diffraction for Perpendicular Incidence of Rays From a Point Source

September 2003©2003 by H.L. Bertoni10 Example of Path Gain for Diffracted Field   30° 2 m 20 m17.3 m 12 m f = 900 MHz,  =1/3 m, k =6  m -1

September 2003©2003 by H.L. Bertoni11 Diffraction of Point Source Rays Incident Oblique to the Edge dipole r  rcos  dA r0r0 00    dA     z

September 2003©2003 by H.L. Bertoni12 Field incident on the edge Diffracted cylindrical wave Diffraction of Point Source Rays Incident Oblique to the Edge - cont.

September 2003©2003 by H.L. Bertoni13 Path Gain for Paths Oblique to the Edge

September 2003©2003 by H.L. Bertoni14 UTD Diffraction for Oblique Incidence of Rays From a Point Source

September 2003©2003 by H.L. Bertoni15 Example of Diffraction on Oblique Paths Cordless telephones over a brick wall-perspective view Rx Located at (4,-1,15) z -7  ’ y Tx Located at (-7,-1.5,0) zwzw x r o  = 90 o -  r 15  ’’  = 90 o - 

September 2003©2003 by H.L. Bertoni16 Example of Diffraction on Oblique Paths Cordless telephones over a brick wall-end view y -7  4 x    Tx  ’ Rx (-7,-1.5) (4,-1) Band S|F(S)| 450 MHz2/ MHz1/33.015≈1 2.4 GHz1/

September 2003©2003 by H.L. Bertoni17 Diffraction on Oblique Paths - cont. Cordless telephones over a brick wall Band S|F(S)|PGL dB 450 MHz2/ x MHz1/33.015≈14.50x GHz1/ x

September 2003©2003 by H.L. Bertoni18 Diffraction by Successive, Parallel Edges --Top and Side Views-- Top View Side View  r W(  W(r) Dipole r 0 r 1 r 0 r 1   r     )  L(r)

September 2003©2003 by H.L. Bertoni19 Diffraction of Vertical Dipole Fields by Successive, Parallel Edges r1r1 r0r0    r cylindrical wave near edge Assume the second edge is not near the shadow boundary of the fist edge.

September 2003©2003 by H.L. Bertoni20 Path Gain for Diffraction at Parallel Edges     30° 17.3 m 2 m   30° 2 m 60 m17.3 m 20 m f = 900 MHz  =1/3 m k =6  m -1

September 2003©2003 by H.L. Bertoni21 f = 450 MHz  =2/3 m k =3  m -1  tan -1 (10/5) =  rad 5 m 2 m 20 m 5 m 11.2 m 12 m Walk About Transmission Over a Building

September 2003©2003 by H.L. Bertoni22 Diffraction of Dipole Fields by Successive Perpendicular Edges r1r1  r0r0   r  cylindrical wave near edge

September 2003©2003 by H.L. Bertoni23 Path Gain for Perpendicular Edges 30° 60 m -30° 12 m 2 m 12 m 20 m f = 900 MHz  =1/3 m k =6  m -1