(2A) 1-D
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Wave Equation A( x, t) = A 0 cos(k x ωt) A 0 cos(k x ω t 0 ) At the snapshot of the time t 0 A 0 cos(k x 0 ωt ) At the fixed site of the distance x 0 λ distance T time x 1 x 2 t 1 t 2 (2A) 3
Wavelength, Frequency A 0 cos(k x ω t 0 ) At the snapshot of the time t 0 A 0 cos(k x 0 ωt ) At the fixed site of the distance x 0 λ T x 1 x 2 distance t 1 t 2 wavelength λ = 2 π k frequency time f = ω 2 π period T = 2 π ω wave number k = 2 π λ angular frequency ω = 2π f angular frequency ω = 2π T (2A) 4
Wave Number, Angular Frequency A 0 cos(k x ω t 0 ) At the snapshot of the time t 0 A 0 cos(k x 0 ωt ) At the fixed site of the distance x 0 λ T x 1 x 2 t 1 t 2 wave number k = 2 π λ angular frequency ω = 2π T radians per unit distance radians per unit time (2A) 5
Phase Velocity (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 (2A) 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 (2A) 8
Phase Velocity, Group Velocity Phase Velocity v p = ω k Group Velocity v g = ω k (2A) 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) (2A) 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 (2A) 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 (2A) 12
Group Velocity & 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 (2A) 13
Group Velocity & 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 (2A) 14
Group Velocity & 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 0 f (x,0) = 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 0 Taylor expansion to 1 st order ω(k) = ω 0 + d ω d k (k k 0 ) f (x,t ) = F (x)e j( k x (ω + d ω 0 d k (k k ))t ) 0 d k (2A) 15
Group Velocity & 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 0 + d ω d k (k k 0 ) f (x,t ) A small spread in k Sharp peak at k 0 = F (k )e j (k x (ω + d ω 0 d k (k k ))t) 0 d k = F (k )e j (k x + k x k x (ω + d ω 0 0 0 d k ( k k ))t ) 0 d k = e j(k x ω t ) 0 0 F (k)e j ((k k ) x d ω 0 d k (k k )t ) 0 d k = e j(k x ω t ) 0 0 F (k)e j (k k 0 ( ) x d ω d k ) t d k Group Velocity v g = d ω d k ( d ω x d k t ) (2A) 16
Dispersion Phase Velocity v p = ω k Group Velocity v g = ω k Dispersion : The angular frequency depends on the wave number (or wavelength) ω(k ) = ω(k) k ω(k ) = k c ω(k ) = h k 2 2 m Light in vacuum Free, non-relativistic quantum mechanical particle of mass m ω(k ) = 2 γ M sin k a 2 Acoustic branch of vibrations in a crystal (2A) 17
Linear Dispersion ω(k ) = k c Light in vacuum (2A) 18
Quadratic Dispersion ω(k ) = h k 2 2 m Free, non-relativistic quantum mechanical particle of mass m (2A) 19
Acoustic Phonon Dispersion ω(k ) = 2 γ M sin k a 2 Acoustic branch of vibrations in a crystal (2A) 20
10 5 0-5 -10-15 -10-5 0 5 10 x = linspace(-10, +10, 1000); y = zeros(1, 1000); for k= 1.0:0.1:2.0 y = y + cos(4*k*(x-k)); end plot(x, y);
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