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Unformatted text preview: hapter 2 Chapter 2 ave otion III Wave Motion III 3D Waves: Plane Waves (simplest 3D waves) All the surfaces of constant phase of the disturbance form parallel planes that are generally perpendicular to the propagation direction nit vectors An equation of a plane that is perpendicular to ( ) k j i ˆ ˆ ˆ z y x k k k k + + = G G Unit vectors a const r k = = ⋅ G All possible coordinates of vector r are on a plane ⊥ k Can construct a set of planes over which ψ varies in space harmonically: ( ) ( ) r k A r G G G ⋅ = sin ψ G G G s r ( ) ( ) r k A r cos ψ or ( ) r k i Ae r G G G ⋅ = ψ or 2 Plane Waves ( ) ( ) r k r G G G ⋅ = sin ψ The spatially repetitive nature can be expressed as: G ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = k k r r G G λ ψ ψ In exponential form: ( ) ( ) k i r k i k k r k i r k i e Ae Ae Ae r λ λ ψ G G G G G G G G ⋅ + ⋅ ⋅ = = = / or that to be true: 2 π i For that to be true: 1 = e π λ 2 = k λ π 2 = k Vector k is called propagation vector 3 Plane Waves ( ) r k i Ae r G G G ⋅ = ψ This is snapshot in time, no time dependence o make it move we need to add time dependence the same way as To make it move we need to add time dependence the same way as for a onedimensional wave: ( ) i k r t t A e ω ⋅ G G ∓ G lane wave equation ( ) , r t A e ψ = Plane wave equation Surfaces joining all points of equal phase are called wavefronts . xample: Example: Wavefronts of 2D circular waves on water surface uperposition where waves (superposition where waves overlap) Plane Wave: propagation velocity Can simplify to a 1D case assuming that the wave propagates along x : G G G i ˆ  r G ( ) t kx i e ω ∓ G ( ) ( ) t r k i Ae t r ω ψ ∓ = , ( ) Ae t r ψ = , e have shown that for 1 ave e phase elocity is: We have shown that for a 1 D wave the phase velocity is: ω ± = v That is true for any direction of k + propagate with k k propagate opposite to k More general case: see page 26, Hecht Example: 2 Plane Waves...
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This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.
 Spring '08
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