Three Types of Waves by nye15450

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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




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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
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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




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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′
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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⎠




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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?
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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

								
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