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: [email protected] General Solution Case: Time Harmonic Rectangular Coordinates ~ ~ 2 2 = + ∇ E E με ϖ ~ ~ 2 2 = + ∇ E E β με ϖ β = Wave Number ~ ~ ~ ~ 2 2 2 2 2 2 2 = + ∂ ∂ + ∂ ∂ + ∂ ∂ E z E y E x E β 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 β β β 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 β β β β β β β + + = = + ′ ′ = + ′ ′ = + ′ ′ 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 β β β β β β 3 3 2 2 1 1 ) ( ) ( ) ( + = + = + =--- β purely real β purely imaginary β complex x x j x x j e B e A β β 1 1 +- Traveling and standing waves Evanesent waves x x e B e A α α 1 1 +- Exponentially modulated traveling wave x x j x x x j x e e B e e A β α β α--- + 1 1 x x j x x x j x e e B e e A β α β α 1 1 +- or or ) sin( ) cos( 1 1 x D x C x x β β + General Solution Case: Time Harmonic Cylindrical Coordinates ~ ~ 2 2 = + ∇ E E με ϖ ~ ~ 2 2 = + ∇ E E β με ϖ β = 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 β φ ρ ρ ρ ρ ρ β φ ρ ρ φ ρ ρ ρ ρ ρ β φ ρ ρ φ ρ ρ ρ ρ ρ φ ρ φ φ φ φ ρ φ ρ ρ ρ ρ ~ ~ ) ~ ( ~ ~ 2 2 2 = + × ∇ × ∇- ⋅ ∇ ∇ = + ∇ E E E E E β β In cylindrical coordinates: Cylindrical Coordinates: Wave Types Wave Type Wave Function Traveling Waves Standing Waves Evanescent Waves...
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This note was uploaded on 12/13/2011 for the course ELEG 413 taught by Professor Mirotznik during the Spring '11 term at University of Delaware.

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

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