Adv Electromag Waves
Adv Electromag Waves ECE 6341
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This 14 page Class Notes was uploaded by Karolann Wiegand on Saturday September 19, 2015. The Class Notes belongs to ECE 6341 at University of Houston taught by David Jackson in Fall. Since its upload, it has received 97 views. For similar materials see /class/208289/ece-6341-university-of-houston in Electrical Engineering at University of Houston.
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Date Created: 09/19/15
ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 13 Plane Wave Expansion The goal is to represent a plane wave in cylindrical coordinates as a series of cylindrical waves to help us do scattering problems y E 2 E0 W0C p w 6 1quot x pcos Generating function Schaum s Outline Eq 2416 Plane Wave Expansion cont Let azkp t je 1 1 t jej j 39e j je J 2jcos t 6 Hence 6taek7p 2jcos e jk0cos e jkx 2010 Jnkpjn ejn n oo Plane Wave Expansion cont JacobiAnger or Expan3on Generalization to be a solution of the scalar x Helmholtz equation 1 This must be true for the field 2 where kp 2k k2 k2 Alternative Derivation Let 6 1 Z aank0ej 712 00 Multiple by e39JWand integrate over 5 0 27 Note that 27 2 7239 m n Ie JWergf 3 0 0 min 27 Hence Jejkxejm d 27mme kp 0 Alternative Derivation 27 Jejbcejm d 2 amek0 0 2 1 zjejbcejm d 2mltkpgt 0 Hence a 27 am Je Jkpcos e jM d 2mltkpgt 0 or Alternative Derivation cont Identity adapted from Schaum s Eq 2499 2 vL72quot 6 xCOS m d x m Hence or Scattering by Cylinder A TMZ plane wave is incident on 3 PEG cylinder kxk0056i kzzksinel jkxxkzz H lziHoe Scattering by Cylinder cont Let A1 e jkxxkzz To find A 11 i u 6x i kx A1 eJ39kxxkzz u Scattering by Cylinder cont Hence H0 ikxAl u or For 0 2 a denote 44 Scattering by Cylinder cont To solve for A first put A into cylindrical form JacobiAnger identity where kp zkx 21600561 Assume the following form for the scattered field Scattering by Cylinder cont At pa E220 E O 2 1 k2AZ 10415 62 2 1 iaAZ E jams 0 6 62 Both will be satisfied if AZ a z 0 Scattering by Cylinder cont Hence A a Z A a Z This yields or a Then we have Scattering by Cylinder cont Note we were successful in solving the scattering problem using only a TMz scattered field This is because the cylinder was perfectly conducting For a dielectric cylinder the scattered field must have BOTH AZ and F2
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