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Physics 42200 Waves & Oscillations Lecture 17 – French, Chapter 7 Spring 2013 Semester Matthew Jones Midterm Exam: Date: Time: Room: Material: Wednesday, March 6th 8:00 – 10:00 pm PHYS 203 French, chapters 1-8 Wave Propagation • We are considering the propagation of a disturbance on a continuous string, far from the ends: • For convenience, we can define = 0 to be the point where the pulse passes = 0. • Suppose that at this time the pulse has a Gaussian profile: 1 / ~ 2 • Let’s fix this up so that the dimensions are more physical… Wave Propagation • Gaussian pulse: = – Normalized to unit area: 1 2 – Characteristic width: 1 1 / 2 / / =1 = 2 – Change to dimensionless variables: Let = / , then = and = / Fourier Transform • The amplitudes of the different frequency components of the pulse are given by the Fourier transform: 1 = ( ) cos 2 1 / cos = 2 • From your table of integrals: ! • Substituting # = 1/2 cos " = # gives: $ /%! and " = 1 / & = 2 • The frequency components are also Gaussian Notes about Fourier Transforms • Fourier cosine transform: 1 = ( ) cos 2 • Fourier sine transform: 1 = ( ) sin ' 2 • The factor 1/ 2 is there for convenience. – What matters is how to relate ( ) and '( ) to the original function – With this definition, = = 1 2 2 * cos( cos( ) ) + + 1 2 2 * ' ' sin( sin( ) ) Notes about Fourier Transforms • For the Gaussian pulse, = 1 / 2 • The amplitudes of the frequency components are: = & / , ' =0 • When the pulse is narrow, ≪ 1, then the exponent in ( ) is large for a large range of – Since - = / , a narrow pulse has a wide range of frequency components. • Conversely, a wide pulse has a narrow range of frequencies. Another Example • Light is characterized by an oscillating electric and magnetic field. • When we treat photons as particles, we think of them as being localized at a point in space that moves at the speed of light. – Localized at = 0 implies a wide range of frequencies – A narrow range of frequencies implies that the photon is not localized in space… • How, then, should we think about photons? Another Example • A photon can be described as a localized oscillation: E(t) t -T At = 0, . At = 0,. =/ =/ +T .* cos - when 4 5 0otherwise .* cos when 4 85 0otherwise ′ = .* 2 :; :; cos cos < Another Example ′ = .* :; cos cos < 2 :; • Trigonometric identity: 1 cos = cos > = cos = − > + cos = + > 2 < 85 < 85 sin − sin + . * < = + − ′ + ′ 2 Another Example A = 48@ 2 = 1.5718@ = A (8@) < (8@ ) Frequency Representation • Why would we want to represent a function in terms of its frequency components? – Both representations contain the same information • Physical properties can depend on the frequency • Examples: – Maximum frequency for discrete masses G -F = 2-* sin 2(H + 1) -I! = 2-* – Speed of light depends on wavelength: = 8/G A Time Dependence • The time-dependent description of the propagating pulse is, in the case where ' = 0, 1 ( ) cos −J , = 2 - = ( )/ • Phase velocity is = - but this is the speed at which the individual harmonic waves move. • How fast does the pulse move? Group Velocity • Consider two harmonic waves with wavenumbers which have frequencies - and • The phase velocity of each wave is =- / =- / ≠ • Superposition: J , = cos − - + cos −• Trigonometric identity: =+> =−> cos = + cos > = 2 cos cos 2 2 Δ ΔJ , = 2 cos − - cos − 2 2 and Group Velocity J , = 2 cos ΔΔ − - cos − 2 2 Fast component with velocity = -/ Slow component with velocity ′ = Δ-/Δ We generalize this to define the group velocity as N = which we evaluate at the average frequency, . Example Group velocity same as phase velocity. A = 58@ ΔA = 0.18@ =0 = 1GO (8@) Example - = = 30 8@⁄GO Δ= 158@/GO Δ A = 58@ ΔA = 0.18@ =0 = 1GO (8@) Example from Wikipedia • Green points are at the minima between the pulses and move with the group velocity. • The red point is constant in phase and moves with the phase velocity. • The group velocity determines the rate of energy transfer. Dispersion • Different frequencies propagate with different speeds. • For light, the speed depends on the index of refraction: - 8 = = G • This is the phase velocity. • The index of refraction can be expressed as a function of wavelength: 8 8 2 -= = = G( ) G A A • Let’s calculate the group velocity… Dispersion • Frequency, expressed as a function of wavelength: = ( ) • Group velocity: = + N = • Phase velocity: = G 8 N = G 8 1− G G =8 G−A A G 8 =− G G Optical Dispersion