VIEWS: 317 PAGES: 4 CATEGORY: Education POSTED ON: 5/14/2010 Public Domain
313 313 Lecture 02 2/9/07 Waves Wave motion: mechanism by which disturbance can travel from one point to another Wave motion I • 1-D Waves Displacements cause neighboring parts to be • Harmonic waves displaced – can model system as coupled • Phase and phase velocity oscillators • Intro to Fourier analysis Spiral density wave in M-51 Energy, momentum, information, can travel without any material object making journey Most of the energy in the universe is transported by waves Since E=mc2, in principle could transport objects by waves too! Common example: pulse or waveform on taut string propagates Reading: Chapter 2, start of 3 in Hecht without any parts moving in direction of pulse Homework set 1: Chapter 2: 5, 8, 21, 22, 23, 32, 38, 39, 42 Due Friday 2/16 313 313 Three Types of Waves Transverse & Mechanical waves Electromagnetic waves Longitudinal • governed by Newton’s laws • governed by Maxwell’s equations • exist within a material medium • do not need a medium to exist Simplest mechanical wave is one sent • only travel at one speed in vacuum along a stretched string or spring • 1D example: waves on a string examples: gamma rays, x-rays, UV light, • 2D: water waves visible light, microwaves, radio waves: If disturbance is short it creates a • 3D: sound waves all the same! pulse, if not a long waveform can emerge Matter waves • associated with String motion vertical while waveform is in direction of spring: transverse wave small, fundamental particles • governed by laws of Spring motion in direction quantum mechanics of wave: longitudinal wave • although not apparent with large objects, modern technology relies on their properties 313 313 1-D Traveling Waves Example A pulse of the form y = Ae − bx is formed on a rope, where A and b are 2 Consider a stationary 1-D pulse of shape y ( x ) = f ( x ) constants and x is in centimeters. Sketch the pulse, then write an equation Pulse y ( x, t ) = f ( x − vt ) must have same profile that represents the pulse moving in the negative direction at 10 cm/s. but move to right at velocity v Pulse y = f ( x + vt ) must move to left at velocity v y ( x, t ) = Ae − b( x +10t ) 2 y ( x, t ) = f ( x ± vt ) is a general expression for a 1-D traveling wave y ( x, t ) or ψ ( x, t ) is the wavefunction 313 313 1-D Wave Equation Harmonic Waves A traveling wave obeys an important partial differential equation Harmonic waves involve sine or cosines: Define x′ = x ∓ vt and define traveling wave by: ψ ( x, t ) = f ( x′ ) y ( x, t ) = A sin ⎡ k ( x ± vt ) ⎤ ⎣ ⎦ df df dg ∂x′ ∂x′ Chain rule = ⋅ Partial derivatives of x′ are = 1 and = ∓v dx dg dx ∂x ∂t Show ψ ( x, t ) = sin [ k ( x − vt ) ] is a solution of the differential wave equation. ∂ψ ∂f ∂x′ ∂f ∂ψ ∂f 1st spatial derivative: = = ⇒ = ∂x′ ∂x′ ∂x ∂x′ ∂x ∂x′ ∂x ∂x′ x′ = k ( x − vt ) =k = − kv ∂x ∂t ∂ψ Wave Equation ∂ 2ψ ∂ ⎛ ∂ψ ⎞ ∂ ( ∂ψ / ∂x ) ∂x′ ∂ ⎛ ∂f ⎞ ∂ f 2 ∂ 2ψ ∂ 2 f = k cos ⎡ k ( x − vt ) ⎤ ⎣ ⎦ = ⎜ ∂x ∂ 2ψ 1 ∂ 2ψ ⎟= = ⎜ ′ ⎟ = ′2 ⇒ = Combining results: = 2 2 ∂x 2 ∂x ⎝ ∂x ⎠ ∂x′ ∂x ∂x′ ⎝ ∂x ⎠ ∂x ∂x 2 ∂x′2 ∂ψ2 ∂x 2 v ∂t = − k 2 sin ⎡ k ( x − vt ) ⎤ = − k 2ψ ⎣ ⎦ ∂ 2ψ 1 ∂ 2ψ ∂x 2 −1 = − k 2ψ = 2 k 2v 2ψ ∂ψ ∂f ∂x′ ∂f ∂ψ ∂f ∂x 2 v 2 ∂t 2 ∂ψ v 1st time derivative: = = ∓v ⇒ = ∓v = − kv cos ⎡ k ( x − vt ) ⎤ ⎣ ⎦ ∂t ∂x′ ∂t ∂x′ ∂t ∂x′ 1-D Wave Equation ∂t ∂ψ2 = − k 2v 2 sin ⎡ k ( x − vt ) ⎤ = − k 2v 2ψ ⎣ ⎦ ∂ 2ψ ∂ ⎛ ∂ψ ⎞ ∂ ( ∂ψ / ∂t ) ∂x′ ∂ ⎛ ∂f ⎞ 2 ∂ f 2 ∂ 2ψ ∂2 f ∂t 2 = ⎜ ⎟= = ⎜ ∓v ′ ⎟ ( ∓v ) = v ⇒ = v2 2 ∂t 2 ∂t ⎝ ∂t ⎠ ∂x′ ∂t ∂x′ ⎝ ∂x ⎠ ∂x′2 ∂t 2 ∂x′ 313 313 Harmonic Waves II Phase and phase velocity Harmonic waves involve sine or cosines: The argument of the trig function is the f ( x, t ) = A sin ⎡ k ( x ± vt ) ⎤ ⎣ ⎦ phase ϕ A is the wave amplitude If ψ ( x, t ) = A sin ⎡ k ( x ± vt ) ⎤ , ϕ = k ( x ± vt ) ⎣ ⎦ 2π We get a spatial periodicity λ if k ≡ λ Must be dimensionless, and please work with radians! k : angular wavenumber or propagation number ω 2π ω v= fλ = = 1 2π k k κ≡ : wavenumber or spatial frequency Constant phase corresponds to fixed point on wave λ Speed at which wavefront moves is phase velocity − ( ∂ϕ / ∂t ) x ⎛ ∂x ⎞ ω We get a time periodicity f if v = λ f ⎜ ⎟ = = ± = ±v ⎝ ∂t ⎠ϕ ( ∂ϕ / ∂x )t k f is ordinary frequency ⎛ π⎞ − ( ∂ψ / ∂t ) x 1 We can switch trig functions by adjusting phase: sin ⎜ θ + ⎟ = cos θ ±v = T= is period ⎝ 2⎠ ( ∂ψ / ∂x )t f ω ≡ 2π f is angular frequency Often need to calculate a waves initial phase angle ϕ0 ⎡ ⎛ x t ⎞⎤ If y ( x, t ) = A sin ⎡ k ( x ± vt ) + ϕ0 ⎤ and y = y0 when x = 0 and t = 0, Equivalent: y ( x, t ) = A sin ⎢ 2π ⎜ ± ⎟ ⎥ ⎣ ⎦ ⎣ ⎝ λ T ⎠⎦ ⎛y ⎞ A sin ϕ = y0 ⇒ ϕ0 = sin −1 ⎜ 0 ⎟ y ( x, t ) = A sin ( k x ± ω t ) ⎝ A⎠ 313 313 Traveling Wave Applet Harmonic wave example Instructions on how to run or download any applets used The equation of a transverse wave along a string is y = 6.0 sin ( 0.020π x + 4.0π t ) during lecture are on class with x, y in cm and t in seconds. What are the characteristics of the wave? website Amplitude A = 6 cm You can get an idea about how fast the wave is traveling by measuring the time it takes angular wavenumber 2π k = 0.02π cm −1 λ= = 100 cm the wave to get from one end wavelength k of the string to the other. angular frequency ω ω = 4π f = = 2 Hz One at a time, adjust the frequency 2π amplitude, frequency, and ω string thickness to determine wavespeed v=λf = = 200 cm / s which of these affects the k wave speed. Initial phase angle ϕ0 = 0 How is the wavelength related to the wave's speed and the frequency? Does What is the displacement when x = 3.5 cm and t = 0.26 s ? the wavelength depend on the amplitude of the wave? Focusing on the one piece of the string in red, what kind of motion does this y (3.5, 0.26) = 6.0 cm sin ⎡( 0.020π cm -1 ) ( 3.5cm ) + ( 4.0π sec −1 ) ( 0.26sec ) ⎤ = − 2.03 cm ⎣ ⎦ piece follow? Is this true for all parts of the string? 313 313 Why Harmonic Waves? Fourier Analysis applet – www.falstad.com Harmonic motion is common in nature The sine and cosines functions form a complete set so any general (anharmonic) wave can be represented as a sum of harmonic waves Fourier's theorem: a function f ( x) having a spatial period λ can by synthesized by a sum of harmonic functions whose wavelengths are integral submultiples of λ ⎛ 2π ⎞ ⎛ 2π ⎞ f ( x ) = C0 + C1 cos ⎜ x + ϕ1 ⎟ + C2 cos ⎜ x + ϕ2 ⎟ + … ⎝ λ ⎠ ⎝λ/2 ⎠ Can rewrite using trig identity Cm cos ( m k x + ϕ m ) = Am cos ( m k x ) + Bm sin ( m k x ) : A0 ∞ ∞ f ( x) = + ∑ Am cos m k x + ∑ Bm sin m k x 2 m =1 m =1