# 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
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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