ELEG413lec4 - ELEG 413 Lecture #4 Mark Mirotznik, Ph.D....

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Unformatted text preview: ELEG 413 Lecture #4 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Email: mirotzni@ece.udel.edu General Solution Case: Time Harmonic Rectangular Coordinates ~ ~ 2 2 = + E E ~ ~ 2 2 = + E E = Wave Number ~ ~ ~ ~ 2 2 2 2 2 2 2 = + + + E z E y E x E 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = + + + = + + + = + + + z z z z y y y y x x x x E z E y E x E E z E y E x E E z E y E x E Separation of Variable Solutions 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( z y x z y x z h z h y g y g x f x f + + = = + = + = + Solutions: z z j z z j y y j y y j x x j x x j e B e A z h e B e A y g e B e A x f 3 3 2 2 1 1 ) ( ) ( ) ( + = + = + =--- purely real purely imaginary complex x x j x x j e B e A 1 1 +- Traveling and standing waves Evanesent waves x x e B e A 1 1 +- Exponentially modulated traveling wave x x j x x x j x e e B e e A --- + 1 1 x x j x x x j x e e B e e A 1 1 +- or or ) sin( ) cos( 1 1 x D x C x x + General Solution Case: Time Harmonic Cylindrical Coordinates ~ ~ 2 2 = + E E ~ ~ 2 2 = + E E = Wave Number 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = + + + = + +- + + = + -- + + z z z z E z E E E E E E z E E E E E E z E E E ~ ~ ) ~ ( ~ ~ 2 2 2 = + - = + E E E E E In cylindrical coordinates: Cylindrical Coordinates: Wave Types Wave Type Wave Function Traveling Waves Standing Waves Evanescent Waves...
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ELEG413lec4 - ELEG 413 Lecture #4 Mark Mirotznik, Ph.D....

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