Wave (1B) 3-D Wave
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Maxwell Equations E = μ H t H = + ϵ E t (ϵ E ) = 0 (μ H ) = 0 = x i + x y i + y z i z 2 = 2 x + 2 2 y + 2 2 z 2 2 E = 1 2 E s( x,t) 2 s c 2 t 2 x + 2 s 2 y + 2 s = 1 2 s 2 z 2 c 2 t 2 Wave (1B) 3
Wave Equation in Cartesian Coordinates Assume a separable solution s( x, y, z,t) = Ae j(ω t k x x k y y k z z) = Ae jk x x e jk y y e jk z z e j ω t = f (x) g (x)h(x) p(t) 2 s x + 2 s 2 y + 2 s = 1 2 s 2 z 2 c 2 t 2 k x 2 + k y 2 + k z 2 = ω2 c 2 k x 2 s( x, y, z,t ) + k y 2 s(x, y, z, t) + k z 2 s(x, y, z, t) = ω2 s(x, y, z, t) c 2 Wave (1B) 4
Monochrome Plane Wave (1) s( x, y, z,t) = Ae j(ω t k x x k y y k z z) Fixed point ( x, y, z) = (0, 0,0) s(0,0,0,t) = Ae j ω t = Acos ω t + Asin ω t Monochrome Wave s( x, y, z,t) = Ae j(ω t k x x k y y k z z) Fixed time t = t 0 s( x, y, z,t 0 ) = A e j(ω t 0 [k x x+k y y +k z z] ) points (x, y, z) such that k x x + k y y + k z z = C Plane Wave planes of constant phase s( x, y, z,t 0 ) = A e j(ω t 0 k x x k y y k z z) has the same value A e j (ω t 0 C ) Wave (1B) 5
Monochrome Plane Wave (2) s( x, y, z,t) = Ae j(ω t k x x k y y k z z) s( x,t) = Ae j(ω t k x ) planes of constant phase k x = C If truly a propagating wave planes of constant phase move by δ x as time advances by δ t ω(t+δt) k ( x+δ x) = C ω t k x = C s( x+δ x,t+δ t) = s( x,t) ω(t+δt ) k (x+δ x) = ω t k x δ x ω δt k δ x = 0 s( x+δ x,t+δt) s( x, t) k Wave (1B) 6
Monochrome Plane Wave (3) constant phase planes of constant phase k x = C perpendicular to k Plane Equation n (r r 0 ) = 0 ω(t+δt) k ( x+δ x) = C Normal Vector Position vector of a point in the plane δ x ω t k x = C n r 0 k x C = 0 s( x+δ x,t+δt) s( x, t) k Wave (1B) 7
Monochrome Plane Wave (4) constant phase planes of constant phase k x = C perpendicular to k If truly a propagating wave planes of constant phase move by δ x as time advances by δ t s( x+δ x,t+δ t) = s( x,t) ω δt k δ x = 0 δ x in the same direction k : minimum δ x The direction of propagation ζ 0 = k k If in the same direction k δ x = k δ x Wave (1B) 8
Monochrome Plane Wave (5) constant phase planes of constant phase k x = C perpendicular to k s( x+δ x,t+δ t) = s( x,t) ω δt k δ x = 0 δ x in the same direction k : minimum δ x The direction of propagation ζ 0 = k k If in the same direction k δ x = k δ x ω δt = k δ x ω k = δ x δt k x 2 + k y 2 + k z 2 c 2 = ω2 k 2 = ω2 c 2 c = ω k k 2 = ω2 c 2 c = δ x δt The speed of propagation of the plane wave Wave (1B) 9
Wave Number, Angular Frequency wave number k = 2 π λ angular frequency ω = 2π T How many λ in 2π (rad / m) How many T in 2π (rad / sec) 3-dimensional space ω δt k δ x = 0 δ t T = 2π ω δ x λ = 2 π k period wavelength wave number vector spatial frequency variable Its magnitude represents the number of cycles (in rad) per meter of length that the monochromatic plane wave exhibits in the direction of propagation. Wave (1B) 10
Wavelength, Frequency s( x,t) = Ae j(ω t k x ) (ω t k x) = ω (t ( k ω) x) position vector temporal angular frequency 3-d spatial frequency slowness vector α s( x,t) = Ae j(ω(t α x)) = [ω(t α x)] Function of a single variable s(u) = A e j (ω u) Slowness Vector α = k ω s(t α x) = Ae j(ω (t α x )) Speed Vector (Phase Velocity) = s( x,t) v p = ω k Wave (1B) 11
Summary Plane Wave s(t α x) = Ae j(ω (t α x )) = s( x,t) Sinusoidal Plane Wave sin(ω(t α x)) = sin(ω t k x) Slowness Vector Wavenumber Vector α = k ω k = ω α α = 1 c k = 2π λ Frequency and wavelength α = k ω c = 1 α = ω k c = ω λ 2 π Wave (1B) 12
Maxwell Equations A(t, t) = A 0 cos(k x ωt) Wave (1B) 13
Maxwell Equations A(t, t) = A 0 cos(k x ωt) Wave (1B) 14
References [1] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] http://www.mathpages.com/, Phase, Group, and Signal Velocity [4] R. Barlow, www.hep.man.ac.uk/u/roger/phys10302/lecture15.pdf [5] P. Hofmann, www.philiphofmann.net/book_material/notes/groupphasevelocity.pdf [6] D.H. Johnson, et. al., Arrary Signal Processing: concepts and techniques