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Class Note for ECE 6341 with Professor Jackson at UH 3


Class Note for ECE 6341 with Professor Jackson at UH 3

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This 18 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 16 views.

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Date Created: 02/06/15
ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 25 Wave Transformation Incident wave jkz Wx9yaz e scalar function eg pressure x E39E w e jkz e jkrcos6 00 Notes I no variation gt m 0 Z Z anjn I COS must be finite at the origin n20 must be finite on the z axis Wave Transformation cont Multiply both sides by Pm COS 98i116 and integrate Orthogonality 2 IPHCOS9PmCOS9Si1 19d9 2n 1 0 0 n m Eq641 nm Hence am2m2ljjmkr Jefkmsapmmsa sinada We can now relabel m gt n Wave Transformation cont Let xkr ucos du sin9d9 We then have 2 1 an2nljjnltxgt Je Pnltugtdu 1 1 2 jeW Pnudu 1 Wave Transformation cont The coefficients are therefore determined from an 2 jjnoc er Pnltugtdu 2n1 4 To find the coefficients take the limit as x gt 0 Wave Transformation cont Recall that 7r xU Jnx Jn x JUx xltlt1 2x 1 2 FU 1 Note Therefore as x gt 0 we have Fn 1 n 1 o 72 xn5 Jnx n1 271 n1l 2 Wave Transformation cont or xn mow 1 2 11 n 1 2 As x gt 0 we therefore have an 2 jjnoc W Pnltugtdu 2n1 4 1 an 2 j Ie 1x udu 2quot 2 Fn1j 1 Wave Transformation cont Note if we now let x gt 0 we get zero on both sides unless n 0 Solution take the derivative with respect to x n times before setting x 0 Myer 1 2 1Fn12 71 d quot11004 dx 2 1Fn1 Wave Transformation cont Denote In E udu 0 Wave Transformation cont Hence 1 a In u P udu 1 1 1 dquot n 2nnd quotLiz 1 du Rodriguez s formula 71 u I Wave Transformation cont Therefore I 2 1 quot 2 n39 Wave Transformation cont or Schaum s outline Mathematical Handbook Eq 1524 1 n FFn1 1 x2 dx2 1 0 2r j n1 2 Wave Transformation cont Hence Wave Transformation cont We then have Note J Wave Transformation cont Nowuse Fn1n so Hence Acoustic Scattering so xy z pressure of sound wave Acoustic Scattering cont Wi e jkz 27 k 7 x13 2 xi of sound wave wVws aws Acoustic Scattering cont We have Mi 2 2a n0 where an 2n 1 Choose ws Z Z hf P COS n0 Hence


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