Group Velocity and Phase Velocity (1A) Young Won Lim 5/26/12
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1 Group Velocity and Phase Velocity (1A)
2 Copyright (c) 211 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.
3 Wave Equation A(t, t) = A cos(k x ωt) A cos(k x ω t ) At the snapshot of the time t A cos(k x ωt ) At the fixed site of the distance x x 1 x 2 distance time t 2 t 1 Phase Velocity 3
4 Wavelength, Frequency A cos(k x ω t ) At the snapshot of the time t A cos(k x ωt ) At the fixed site of the distance x x 1 x 2 wavelength wave number distance λ = 2 π k k = 2 π λ time t 2 t 1 frequency period angular frequency angular frequency f = ω 2 π T = 2 π ω ω = 2π f ω = 2π T Phase Velocity 4
5 Wave Number, Angular Frequency A cos(k x ω t ) At the snapshot of the time t A cos(k x ωt ) At the fixed site of the distance x x 1 x 2 distance time t 2 t 1 wave number k = 2 π λ radians per unit distance angular frequency ω = 2π T radians per unit time Phase Velocity 5
6 Phase Velocity (1) A cos(k x ω t ) At the snapshot of the time t A cos(k x ωt ) At the fixed site of the distance x x 1 x 2 distance time t 2 t 1 wave number k = 2 π λ radians per unit distance angular frequency ω = 2π T radians per unit time Phase Velocity v p = λ T = 2π/ k 2π/ω = ω k v p = ω k Phase Velocity 6
7 Phase Velocity (2) Phase Velocity v p = ω k Acos(k x ω t) Given time t, ω t oscillations Corresponding distance x, the same oscillations k x = ω t v p = x t = ω k Phase Velocity 7
8 Phase Velocity, Group Velocity Phase Velocity v p = ω k Group Velocity v g = ω k Phase Velocity 8
9 Group Velocity Explanation (1) ω 1 > ω 2 k 1 > k 2 ω = (ω 1 +ω 2 ) 2 Δ ω = (ω 1 ω 2 ) 2 k = (k 1 +k 2 ) 2 Δ k = (k 1 k 2 ) 2 ω 2 ω 1 k 1 k 2 ω 1 = ω + Δ ω k 1 = k + Δ k ω 2 = ω Δ ω k 2 = k Δ k e j (k 1 x ω 1 t) e j (k 2 x ω 2 t) Phase Velocity 9
10 Group Velocity Explanation (2) ω 1 = ω + Δ ω k 1 = k + Δ k ω 2 = ω Δ ω k 2 = k Δ k ω 2 ω 1 k 1 k 2 e j (k 1 x ω 1 t) + e j(k 2 x ω 2 t ) = e j {(k + Δ k ) x (ω + Δ ω)t } + e j {(k Δ k ) x (ω Δ ω)t } = e j {(k x ωt ) + (Δ k x Δ ω t)} + e j {(k x ωt) (Δ k x Δ ωt )} = e j(k x ω t) {e j (Δ k x Δ ωt ) + e j (Δ k x Δ ω t) } = 2cos(Δ k x Δ ω t)e j(k x ω t) Envelope Phase Velocity 1
11 Group Velocity Explanation (3) ω 1 = ω + Δ ω k 1 = k + Δ k ω 2 = ω Δ ω k 2 = k Δ k ω 2 ω 1 k 1 k 2 e j (k 1 x ω 1 t) + e j(k 2 x ω 2 t ) = 2cos(Δ k x Δ ω t)e j(k x ω t) Envelope Δ k k 1, k 2 Small Wave number Long Wavelength Δ ω ω 1, ω 2 Small Frequency Long Period Envelope Velocity v g = Δ ω Δ k Group Velocity v g = d ω d k Phase Velocity 11
12 Fourier Transform (1) A periodic function f (θ) = k a k sin(k θ) + b k cos(k θ) a k = 1 +π π π f (θ) sin(k θ) d θ b k = π 1 +π π f (θ) cos(k θ) d θ A non-periodic function f ( x) = F (k) = 1 2π F (k) e j k x d k f (x) e j k x d x Phase Velocity 12
13 Fourier Transform (2) A non-periodic function f ( x) = F (k) e j k x d k F (k) = 1 2π f (x) e j k x d x Infinite number of sine waves Well-defined wavelength Well-defined k (wave number) Lots of sine waves of diff wavelengths Short wave packet A long wave packet A small spread in k Sharp peak Phase Velocity 13
14 Fourier Transform (3) A non-periodic function f ( x) = F (k) e j k x d k F (k) = 1 2π f (x) e j k x d x A long wave packet At some initial time t = t f (x,) = After time f (x,t ) = ω(k) t F ( x)e j k x d k F (x)e j (k x ω(k)t ) d k different wavelength components have different frequencies A small spread in k Sharp peak at k Taylor expansion to 1 st order ω(k) = ω + d ω d k (k k ) f (x,t ) = F (x)e j( k x (ω + d ω d k (k k ))t ) d k Phase Velocity 14
15 Fourier Transform (3) After time A long wave packet t f (x,t ) = F (x)e j (k x ω(k)t ) d k Taylor expansion to 1 st order ω(k) = ω + d ω d k (k k ) f (x,t ) A small spread in k Sharp peak at k = F (k )e j (k x (ω + d ω d k (k k ))t) d k = F (k )e j (k x + k x k x (ω + d ω d k ( k k ))t ) d k = e j(k x ω t ) F (k)e j ((k k ) x d ω d k (k k )t ) d k = e j(k x ω t ) F (k)e j (k k ( ) x d ω d k ) t d k Group Velocity v g = d ω d k ( d ω x d k t ) Phase Velocity 15
16 x = linspace(-1, +1, 1); y = zeros(1, 1); for k= 1.:.1:2. y = y + cos(4*k*(x-k)); end plot(x, y);
17 cos(x - t) t : time x : distance 1
18 cos(x - t) t : time x : distance 1
19 cos(x - t) t : time x : distance 1
20 cos(x) t : time x : distance 1
21 cos(-t) t : time x : distance 1
22 tx = ty = linspace(-1, 1, 51); [xx, yy] = meshgrid(tx, ty); tz = cos(xx-yy) ; mesh(tx, ty, tz); title("cos(x - t)"); xlabel("x : distance"); ylabel("t : time"); print -demf fig1.emf tx = ty = linspace(-1, 1, 51); [xx, yy] = meshgrid(tx, ty); bb = [zeros(26, 51); ones(25, 51)]; tz = cos(xx-yy).* bb; mesh(tx, ty, tz); title("cos(x - t)"); xlabel("x : distance"); ylabel("t : time"); print -demf fig2.emf tx = ty = linspace(-1, 1, 51); [xx, yy] = meshgrid(tx, ty); bb = [zeros(51, 26) ones(51, 25)]; tz = cos(xx-yy).* bb; mesh(tx, ty, tz); title("cos(x - t)"); xlabel("x : distance"); ylabel("t : time"); print -demf fig3.emf tx = ty = linspace(-1, 1, 51); [xx, yy] = meshgrid(tx, ty); aa = zeros(51, 51); aa(26,:) = ones(1, 51); tz1 = cos(xx - yy).* aa; mesh(tx, ty, tz1); title("cos(x)"); xlabel("x : distance"); ylabel("t : time"); print -demf fig4.emf
23 tx = ty = linspace(-1, 1, 51); [xx, yy] = meshgrid(tx, ty); aa = zeros(51, 51); aa(:, 26) = ones(51, 1); tz1 = cos(xx - yy).* aa; mesh(tx, ty, tz1); title("cos(-t)"); xlabel("x : distance"); ylabel("t : time"); print -demf fig5.emf
24 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 23 [3] Phase, Group, and Signal Velocity [4] R. Barlow, [5] P. Hofmann,
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