```2nd & 3th N.U.T.S. Workshops
Gulu University
Naples FEDERICO II University
4 – Waves
2nd & 3th NUTS Workshop ( Jan 2010)
Wave Motion
Waves are everywhere:
Earthquakes, vibrating strings of a guitar, light from the sun; a
Something moving, passing by, bringing a change and then going
away, sometimes without a trace…
radio waves, mexican wave, microwaves, annoying sound
waves of a physics lecture, crime wave rap music blaring out
of an audio system in a car, on the crest of a wave, ….
Waves Appl.
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Wave Definition
A wave is a travelling disturbance that
transports energy but not matter.


Mechanical waves require
•
•
•
Some source of disturbance
A medium that can be disturbed
Some physical connection between or mechanism through which
adjacent portions of the medium influence each other
All waves carry energy and momentum
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2nd & 3th NUTS Workshop ( Jan 2010)
Free Harmonic Oscillations & Sinusoidal Motion
y (t )  A sin( t   )  A sin( 2ft   )
y height of the object with respect to its equilibrium position;
A amplitude of the oscillations;
ω  2πf angular frequency (in rad/s);
f = 1/T regular frequency (in Hertz or cycles per second or s-1);
T period of oscillations (in seconds)
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2nd & 3th NUTS Workshop ( Jan 2010)
Wavelength
Waves are characterised by the same variables as the oscillation:
amplitude, frequency, period, energy…
But they have much more, because they propagate in space…
l
4- Waves
The two basic new
parameters are:
wavelength
and
wave speed
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2nd & 3th NUTS Workshop ( Jan 2010)
How to Calculate the Speed of a Traveling Wave?
Some poles that are placed a wavelength l apart.
The oscillations at the poles are always in phase.
Time taken by a crest to travel between two consecutive poles =
Period of the oscillation =T
(exactly the time between two consecutive crests at a same pole!).
l
The wave speed, v, can be calculated as:
v
λ
T
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 λf
2nd & 3th NUTS Workshop ( Jan 2010)
Wave Motion (a travelling wave)
l
NO direct connection between the wave speed:
v
T
and the speed of the oscillating material particles:
v y (t ) 
4- Waves
dy
λ
 A cos(t   )
dt
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2nd & 3th NUTS Workshop ( Jan 2010)
How to Calculate the Frequency of a Travelling Wave?
v
l
Blue light has shorter
v
f 
  lf wavelength than red light;
l
T
Sound and light are very different types of waves (cfr. later)
When they have the same wavelength:
which has the higher frequency?
Free Harmonic Waves:
Sinusoidal waves where the crests move with a constant speed,
while the material elements oscillate harmonically.
A sin( 2ft    .....) or
A cos( 2ft    .....)
What is missing here ??
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Transverse Waves
material
velocity
wave speed
Electric
Field
Transverse wave – material elements (medium) move (or
variable Electric and Magnetic Field change) perpendicularly
to the direction of wave propagation
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2nd & 3th NUTS Workshop ( Jan 2010)
Longitudinal Waves
In a longitudinal wave the particle displacement is parallel to the
direction of wave propagation.
The animation shows a one-dimensional longitudinal plane wave
propagating down a tube.
The particles do not move down the tube with the wave; they simply
oscillate back and forth about their individual equilibrium positions. Pick
a single particle and watch its motion!
The wave is seen as the motion of the compressed region (i.e. it is a
pressure wave), which moves from left to right.
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Transverse vs. Longitudinal Wave
Both propagate from left to right, but cause disturbances in
different directions, Dy and Dx.
wavelength, l
Dx(t )  Ax cos(t )
wavelength, l
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2nd & 3th NUTS Workshop ( Jan 2010)
Waves on a Spring
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2nd & 3th NUTS Workshop ( Jan 2010)
Harmonic Waves are not the only Possible Type of Waves!
A wave can also have the shape of a propagating impulse.
True for both transverse and longitudinal waves.
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2nd & 3th NUTS Workshop ( Jan 2010)
Wave Train
An harmonic wave and a pulse are extreme cases.
Intermediate case = a wave train = a finite duration sinusoidal
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2nd & 3th NUTS Workshop ( Jan 2010)
Mathematical Description of an Harmonic Wave
y
x
l
• Features to incorporate:
at any space point the wave produces harmonic oscillations as:
y(x) = Aycos(ωt+φ)
ω angular frequency , φ initial phase
If we “freeze” the wave in time, an harmonic function results in space:
y(x) = Aycos(kx+φ)
what is k ?
If we freeze the wave and move 1 wavelength λ along it, the same level of
disturbance y has to be found
Therefore, it must be kλ=2π so that
y ( x  l )  A y cos( k ( x  l )   )  A y cos( kx  2   )  y ( x )
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2nd & 3th NUTS Workshop ( Jan 2010)
Time and Space in Harmonic Waves
y ( t )  A y cos(  t   )

y ( x )  A y cos( kx   )
 phase
k l  2  k  2 / l
k is measured in m-1. What is its meaning?
tells us every how many times per meter it is going to happen
k / 2
to have a crest, freezing the time.
 / 2  f tells us every how many times per second there is a crest if the
angular frequency
position is frozen and the wave propagates
y
x
l
k is pretty much the same for space as  is for time!
k behaves like a spatial frequency and is usually called the “wave number”
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2nd & 3th NUTS Workshop ( Jan 2010)
Waves: Space and Time 1
x
l
y
y ( t )  A y cos(  t   )
  2 / T
T
is period in time
y ( x )  A y cos( kx   )
k  2 / l
l is period in space
How do we combine the two equations (in time and in space)?
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2nd & 3th NUTS Workshop ( Jan 2010)
Waves Time and Space 2
At one specific point in space, x0,    kx0
At one specific instant t0 ,   =  t0
  2 / T  2 f
k  2 / l
l 
v
; k 
f
2 f


v
   kv
v
y ( x , t )  A y cos( kx   t )  A y cos( kx  k v t )  A y cos[ k ( x  v t )]
A crest at a point, where
Position of the crest
k ( x  vt )  0
x  vt
It is moving with wave speed v !!!
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2nd & 3th NUTS Workshop ( Jan 2010)
Harmonic Waves: Summary
y ( x , t )  A y cos( kx   t )
equation of a harmonic wave
y ( x , t )  A y cos[ k ( x  v t )]
the same equation, in a form
emphasizing propagation
along x axis and wave speed v
y ( x , t )  A y cos[ k ( x  v t )]
what would this one stand for?
- v is changed to + v ,  the wave is propagating in the negative
x direction, from right to left according to usual convention
In this case location of a crest is given by cos[ k ( x  v t )]  1
x  vt  0
4- Waves

x   vt
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2nd & 3th NUTS Workshop ( Jan 2010)
An Example
The figure shows a simple harmonic wave at t = 0, and later
at t = 2.6 s. Write a mathematical description of the wave.
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2nd & 3th NUTS Workshop ( Jan 2010)
Wave Fronts
Wave front is a continuous
line or a surface connecting
nearby wave crests.
Plane waves
Wave fronts = flat surfaces for
waves propagating in one
Wave fronts = straight lines for
ripples on water surface at shore
line.
“Straight” waves
4- Waves
wave fronts
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2nd & 3th NUTS Workshop ( Jan 2010)
Spherical Wave Fronts
Wave fronts = spherical
surfaces for spherical
waves originating from a
point source and
propagating in 3D space.
Wave fronts = circles for
waves on water surface
originating from a point
source.
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Wave Intensity
Wave intensity is the wave power per unit area: I = P/A
Plane wave: its intensity
remains constant because the
wave front area remains
constant
Spherical wave: its intensity
decreases with the distance, r,
from the wave source
I = P/4r2
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Wave Interference
When two (or more) waves of the same kind propagate through the same
space region a composite wave results. (wave interference)
It is constructive, when the waves reinforce each other.
It is destructive, when they reduce each other’s amplitude.
Usually the disturbances (displacements) the waves produce are
added algebraically. This is called superposition principle.
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2nd & 3th NUTS Workshop ( Jan 2010)
Superposition of Pulses
Constructive Interference
Destructive Interference
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2nd & 3th NUTS Workshop ( Jan 2010)
Electromagnetic Waves
Electromagnetic waves
propagate along a
direction perpendicular
to electric and
magnetic field, with a
speed c=3x108 m/s in
the vacuum
wavelength and frequency are connected as: l 
4- Waves
c
f
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2nd & 3th NUTS Workshop ( Jan 2010)
Electromagnetic Spectrum
increasing frequency
4- Waves
increasing wavelength
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2nd & 3th NUTS Workshop ( Jan 2010)
E.M. Waves: Frequency and Wavelength in the Vacuum
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2nd & 3th NUTS Workshop ( Jan 2010)
Rays and Wave Fronts
In the wave formulation of
optics, the mathematical
model of thin beams (rays)
correspond to lines
perpendicular to the wave
fronts
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2nd & 3th NUTS Workshop ( Jan 2010)
Light Refraction: Frequency and Wavelength

As light travels from one medium to
another, its frequency does not change
•
•
Both the wave speed and the wavelength
do change
The wavefronts do not pile up, nor are
created or destroyed at the media
interface, so ƒ must stay the same
v1
f1  f 2
l1
l2

v1
v2
l1


v2
l2
c / n1
c / n2

n2
n1
So:
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2nd & 3th NUTS Workshop ( Jan 2010)
Refractive Index and Speed of the E.M. Wave
When an E.M. Plane wave
θ1
traveling through a
transparent medium
encounters an interface with
another transparent medium
θ1
θ2
the refraction phenomenon
can be described with the
mathematical model of rays
n2
θ2
corresponding to lines
perpendicular to the wave
 Refraction Index n = c1/c2
fronts (plane)
If c2 < c1 (medium 2 is more dense
n1

than medium 1) the refracted light
beams bends toward the normal to
the interface
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Wavelengths and Colours
We see colour when waves of different wavelengths enter our eyes!
Light with wavelength of 650 nm
appears red when it enters a viewers eye
Light with wavelength of 520 nm
appears green when it enters a viewers eye
Light with wavelength of 470 nm
appears blue when it enters a viewers eye
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Two Different Wavelengths and Colours
What happens when two or more waves with different
Light with both wavelengths 650 nm and
520 nm appears yellow when it enters
a viewers eye
Light with only wavelength 580 nm
ALSO appears yellow when it enters
a viewers eye (A DEEPER YELLOW
THAN FOR THE CASE ABOVE)
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2nd & 3th NUTS Workshop ( Jan 2010)
What is White Light?
Light which is a mixture of all
wavelengths of the visible light
spectrum
It appears WHITE when it
No single wavelength (monochromatic) wave appears white
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2nd & 3th NUTS Workshop ( Jan 2010)
Colours of NON Luminous Objects
The colour of an object depends on the range of visible light
wavelength that it absorbs
The absorbed range depends on the chemical properties of the
substances composing the objects
The light coming from the object and reaching the eye does not have
the absorbed range
The perceived colour depends on the NOT absorbed range
4- Waves
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2nd & 3th NUTS Workshop ( Jan 2010)
Color of the Sun from under Water




The Sun looks yellow, since its radiation intensity has a
maximum at l = 550 nm, which is yellow light.
Wavelength of this yellow light in water will be l’ = l / nwater
= l/1.33 = 413 nm, corresponding to violet light.
Is the Sun going to look violet from under water?
NO! The only thing that matters is the
wavelength inside your eye, which is
defined by n of eye vitreous humor.
f eye  f air
4- Waves
l eye  l air
n air
n eye
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