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# Class Note for ECE 6341 with Professor Jackson at UH

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This 24 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 25 views.

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Date Created: 02/06/15

ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 8 Cylindrical Wave Functions Helmholtz equation VZW kzw O Wp z A2 0er 82 18 182 82 80 pap 0 8gb 82 jk2w0 Separation of variables Mar52 Rpltlgt ZZ Substitute into previous equation and divide by S Cylindrical Wave Functions apz pap p2 M 822 lIp z Rplt1gt Zz 2 2 2 8w 18w18w8wjk2w20 2 2 2 6 IECDZia RCDZLZa CEDRZa R jk2RCDZO 60 p p 0 6 62 RH 1R 1 DH Zquot 2 2 k 0 R p R p I Z Cylindrical Wave Functions RH 1R 1 DH Zquot 2 2 k 0 1 R 0 R 0 CI Z Z N 2 R H 1 R 1 1 or k 2 Z R 0 R 0 CD H J V fZ gm Hence constant kZ2 Cylindrical Wave Functions Z N 2 Hence kz Z Next to isolate the dependent term multiply Eq 1 by 0 2 02 R 1R 121 kZ2 k2p20 R pR 0 CI Cylindrical Wave Functions Hence 1quot 1R Rn pzk2kj q p R R Y I V f gm Hence constant 2 12 CD SO Cylindrical Wave Functions From Eq 2 we now have z202 k2k2 i R H Z pR R The next goal is to solve this equation for Rp First multiply by R and collect terms szquot pR39 p2 k2 kZZR 02R O Cylindrical Wave Functions Define k 2 kZ2 Then szquot pR39 kpp2 2R O k Next define x pp RC0 300 Note R j R y 39Xkp p p and Rquotp yquotxk Cylindrical Wave Functions Then we have xzyquot xy39 x2 02 O Bessel equation of order 1 Two independent solutions JD x YD x Hence Properties of Bessel Functions 0 n o H J x Ob 2 gt4 BesselFunc onsconL 0 f Hs 0 Rx ff n 1 112 1 12 BesselFunC onscont Zero Properties JD x w Ax YuxBx U 0 0 YoxClnx U For order zero the Bessel function of the second kind behaves as In rather than algebraically 13 BesselFunc onscont WC JuxJux Two linearly independent solutions Bessel equation is unchanged by U gt U Note JU x is a always a valid solution These are linearly independent when V is not an integer JD x w 14le JU x w Age V BesselFunc onscont JU x coswr JU x Y x E sinu7r This definition gives a nice asymptotic behavior as x gt oo BesselFunc onscont symmetry property Jn X 1 J 6 no longer linearly independent Kn x 1 Yn x In this case Y x limYUx U gtn BesselFunc onscont Frobenius solutionT The order 1 is restricted to z i n n12 The order 1 is not an integer in this equation TFerdinand Georg Frobenius October 26 1849 August 3 1917 was a German mathematician best known for his contributions to the theory of differential equations and to group theory Wikipedia 16 BesselFunc onscont From the limiting definition we have as V 972 Where CDp1i pgt0 CDOO p BesseIFunc onscont From the series solution and the symmetry property we have that JUxx 1 200 Ui 1 2 3 1 2 n Jnx 1 x n012 18 BesselFunc onsconL IIIIII IMm3 EJw wawruw6 7239 YUx iU lgj UgtO Ynx ln 1gj 7r x 7r x n123 YUXC9SU7T 1 gj U Ult0 1 smwr Fu1 x 1 2 n u 72 2n 12 Ynx 1 jn 1j Ui n n123 BesselFunc onsconL Asymptotic Formulas JUx icosx 077r j 7m 4 no isinx 07 j 7m 4 Hankel Functions As x gtoo 7239 jx uZ H 1 x N i 6 2 1 Incoming wave U 7m H2x i e Jx u 72x Outgoing wave Fields In Cylindrical Coordinates 23 TM Fields HZ 0 1 529 H ial jau8 6062 p 10 M 1 62 H L0l jawap 6 0p 24 TEZ Fields 1 5 1 62 E ial Hp 1 p 8p 6 Jw g 5052 1 62 E lal H W 8 60 jw gp a aZ

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