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# Note 13 for ECE 6341 with Professor Jackson at UH

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This 14 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
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 7 6 170C H I AVAVAVAVA x x pcos Generating function Schaum s Outline Eq 2416 Plane Wave Expansion cont Let azkp tz jej 1 mi 1 mi o j t e e e 2COS t 61 Hence 1 kp J 00 n n ex Ie2 e 1kpcos e 1105 Z 61 I l OO Plane Wave Expansion cont JacobiAnger or Expan3on Generalization to be a solution of the scalar This must be true for the field 2 Helmholtz equation where kp 2k k2 k2 Alternative Derivation Let 6 1 Z aank0ej Multiple by e39fmand integrate over 5 0 27 Notethat 2 m W 27 mn ye e d 0 0 min 27 Hence Jejkxejm d 27mme kp 0 Alternative Derivation 27 j e 1bCe jm bdgz 27amek0 0 27 Hence am 2 Jejhejm d 1 27 or a JeJkpcos eJm d 27Jmltkpgt 0 Alternative Derivation cont Identity adapted from Schaum s Eq 2499 2 07 excos m d 2o 7ZJm x m J Hence Scattering by Cylinder A TMZ plane wave is incident on 3 PEG cylinder kx kcos 9i kzzksin i R H lt gt jkxxkzz H 0 e Scattering by Cylinder cont Let A1 e jkxxkzz To find A u 6x 1 Jkx A1 e jltkxxkzz Scattering by Cylinder cont Hence H0 ik A1 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 11 Scattering by Cylinder cont EZO At 0a E O jw g 622 1 i 6212 jams 0 6 62 2 1 6 E Both will be satisfied if AZ 61 Z 0 Scattering by Cylinder cont Hence A a Z A a Z This yields J kpa 2 an H 2 kpa I l J kpa gnawpa or an 2 Then we have 14 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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