1. What’s a wave?
(a) Sinusoidal waves
i. Frequency, period, wavelength. De…nition of Hertz.
ii. Speed of propagation
(b) Modes: transverse, longitudinal
2. Waves and forces
(a) Energy
(b) Re‡ections
(c) Interference
(d) Standing waves
(e) Resonance
(f) Refraction
3. Electromagnetic radiation
(a) Properties of light
(b) Polarization
(c) Refraction
(d) Re‡ection
(e) Interference
(f) Di¤raction
4. Mechanical waves. Sound
(a) Re‡ection and absorption
(b) Di¤raction
(c) Refraction
(d) Doppler shift
1
1
What’s a wave?
“It ain’t what people don’t know that hurts them, it’s what they do know that
ain’t so” Mark Twain.
Waves are not earth, air, water, etc. but they travel in these media, a wave is
the propagation of a disturbance through a medium without actual displacement
of the medium itself.
Waves show a behavior that can be described mathematically, i.e. moving
waves carry energy from one point to another. When waves are absorbed or
re‡ected from an object they push on it, i.e. they have momentum. Also, it
takes time for a wave to travel, therefore they have velocity.
Waves possess the principle of linearity. To understand this concept throw
two pebbles in a pond and watch how the waves rather than destroying each
other, actually pass through each other. The pattern formed on the water
surface is that of two independent sets of expanding circles.
If you don’t have a pond nearby you can certainly listen to two people talking
to each other, you’ll notice that the sound waves don’t rebound from each other,
rather they pass through one another. In other words, waves that do not a¤ect
each other are linear waves; the total of two waves is the sum of the waves as
they exist separately.
1.1
Sinusoidal waves
Pick up a guitar and pluck a string. The vibration thus generated is a wave
that varies in a very simple way in space and time. Assume that we want to
look at this wave at time t = 0 :
1
0.5
0
0
50
100
150
200
250
300
350
Degrees
-0.5
-1
Fig. 1 Sinusoidal wave
The maximum distance that the wave deviates from the horizontal axis is known
as the amplitude, and is a measurement of the strength of the wave at that
particular time and space. The value along the vertical axis may represent the
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displacement of the string in this particular case, or the strength of an electric
or magnetic …eld, or a voltage, or a current, or the pressure or velocity of a
sound wave, etc.
The wave shown in Fig. 1 is described by the following equation:
y = A sin x
(1)
Where y represents the variation along the vertical axis (dependent variable),
A represents the maximum amplitude of the wave (in the case of Figure 1
A = 1); and x is the independent angular variable, whose values range between
0 x 360 ; or 0 x 2 rad
1.1.1
Frequency, period, wavelength. De…nition of Hertz
The wave represented in Fig. 1 repeats itself for x = 360 = 2 rad: In Figure
2 we show some other waves:
1
0.5
0
0
50
100
150
200
250
300
350
Degrees
-0.5
-1
Figure 2. A few sinusoidals
What is di¤erent in these three waves? Remember that the interval of the plot
is for values of x that range between 0 and 360 : Note that within this interval
the red wave repeats itself once. The green wave repeats itself two times and
the blue wave repeats itself three times.
Suppose that it takes for a point one second to travel from beginning to end
of the plot. This means that a point traveling along the red wave will take one
second to complete one cycle. A point on the green wave will complete two
cycles in one second, and a point on the blue trace will complete three cycles in
one second. The number of oscillations per second is called the frequency (f )
of the wave. The unit of frequency is a Hertz, Hz; and is equal to 1= s (one
oscillation per second):
Suppose that as we stand at one place and watch the wave pass by, we
measure the time between repetitions. This time is called the period (T ) of the
wave and is the inverse of the frequency:
T =
3
1
f
(2)
On the other hand, we can measure the distance along the direction of propagation, in which the wave repeats itself, i.e. the distance between one peak to
the next, or between one valley to the next, in other words, between equivalent
points. This is the wavelength ( ) ; and is measured in units of length.
Going back to the example in Fig. 2, we have that the red wave repeats
itself when x = 2 ; this means that its wavelength r = x = 2 : The green
wave repeats itself for 2x = 2 ; or g = x = 22 = : The blue wave repeats
itself for 3x = 2 ; or b = x = 23 : In general, a wave will repeat itself k times
for kx = 2 ; or k = 2 : k is known as the wave number, and corresponds to
the number of waves contained in the interval 2 :
1.1.2
Speed of propagation
According to the previous section, if we know the number of cycles a wave
completes in one second and the length of the wave, we can then calculate the
speed at which it propagates as
speed
v
= f requency
1
= f
=
s
wavelength
m
m=
s
(3)
Example 1 Attach the end of a rope to a wall or a …xed point. Grab the other
end and oscillate it up and down, as in Fig. 3, in such a way that you produce
three oscillations per second. Ask a friend to measure the wavelength. Let’s
assume that he comes up with a measurement of 2 m: The speed at which the
wave is propagating is then
speed
v
= f requency wavelength
= 3 Hz 2 m = 6 Hz m = 6 m= s
A
A
λ
Figure 3. Creating a wave on a string
4
1.2
Modes of Oscillation
When you abruptly close the door in a room that has the windows closed, you’ll
notice that the curtains on the other side of the room move. This is because a
wave has travelled through the air in the room moving the air molecules in the
same direction as the wave moves. This is an example of a longitudinal wave.
Example 2 Get yourself a Slinky toy and place it on a smooth surface. Hold
one end of the spring with one hand and stretch it with the other, grab a few
coils together and then release them. You’ll notice that they snap back creating
a disturbance that moves along the spring.
Example 3 Another variation of the same experiment may be accomplished
if you simply jerk back and forth one end of the spring in a regular, periodic
fashion. By doing this you’ll see a series of disturbances moving along. The wave
will consist of regions of compression (where the coils are pushed together) and
regions of expansion (where the coils are stretched apart).
rare fa ctio n or e xpan sio n
co m pre ssio n
Figure 4. Longitudinal wave in a Slinky
The observation from the previous examples explains why sometimes a longitudinal wave is also know as a compressional wave.
Example 4 Another good example of a longitudinal wave is when you hit a
metal bar …xed at one end. The bar rings due to the longitudinal waves that
travel back and forth.
Let’s now try something di¤erent:
Example 5 Once again stretch your Slinky on a table with one end …xed. Shake
the free end of the spring with a periodic motion, from side to side, perpendicular
to the length of the spring. In this new pattern, the molecules of the spring move
perpendicularly to the direction of the wave’s motion. This is a transverse wave
and is shown is Fig. 5.
Figure 5. Transverse wave in a Slinky
Example 6 Transverse waves are generated when you pluck the string of a
guitar.
5
Transverse waves move through matter as long as there are forces that tend
to restore the molecules to their initial position. In liquids and gases molecules
don’t usually return to their original positions because of the fast rearrangement
of the surrounding molecules, thus transverse waves do not propagate easily in
these media. However, transverse waves do propagate in solids.
2
2.1
Waves and Forces
Energy
To better understand the concept of energy let’s go back to our rope in Figure
3. The work that is done in shaking the rope provides the energy that the wave
carries with it. Work is de…ned as force times distance, which means that the
amount of energy carried by the wave depends on how far we pull the rope from
the resting position. In other words, energy depends on wave amplitude. It is
also true that for the same amplitude, more energy is required to generate a
faster oscillating wave than a slowly varying one.
Example 7 Once again, attach the rope to a wall and shake fast the free end up
and down with a constant amplitude movement. Since you are shaking fast, you
are delivering more energy per second. Next, with the same amplitude, shake the
rope in a lazy way. You’ll notice that to make short-wavelength, high-frequency
waves is more tiring that making long, slow-frequency waves.
2.2
Re‡ections
Hopefully you are still holding the rope, since there is something else we may
try while you are at it. Make sure that the rope is tightly stretched out, give it
just one hard ‡ip and close your eyes. After a short wait you should feel some
movement at your hand. The wave has re‡ected from the …xed end of the rope!
It is very possible that the jerk that you feel is not as strong as the original
‡ip, this is because some of the energy has dissipated as heat due to friction
within the material of the rope. Another fraction of the energy was transmitted
to the wall, or wherever you attached the rope. As a matter of fact, heat
dissipation is the reason why a wave propagating through matter eventually
dies.
Since the wall at the end of the rope is rigid, you may have noticed that the
wave returned upside down to you.
If you continue ‡ipping the end of the rope back and forth with a regular
motion you’ll create a train of waves. At some point in time the rope will support
two waves at once, one traveling away from you and another one traveling
towards you. When the waves meet, the particles of the rope will feel a pull
from each wave according to the net force they receive.
6
2.3
Interference
The result of this interaction is known as interference. Figure 6 shows a constructive interference example:
Figure 6. Constructive interference
Two waves of the same amplitude, traveling in opposite directions meet at one
point, the amplitude of the displacement of the particles in the rope at that
point will be twice as high as the individual amplitudes.
If a peak of a wave traveling in one direction meets at one point the valley
of another wave traveling in opposite direction, the particles on the rope at
that position won’t su¤er any vertical displacement at all. This is known as
destructive interference and is shown in Figure 7.
Figure 7. Destructive interference
Note as the two waves pass the same location there is no noticeable e¤ect on
the medium, however after this point the waves pass each other with the same
forms as before.
In general, waves of di¤erent amplitudes and frequencies will interfere with
each other as they pass through the same point in the medium.
7
2.4
Standing waves
This is going to be a little trickier. The idea is for you to move your end of the
rope with a few di¤erent frequencies. For some frequencies, the re‡ected wave
will interfere with the wave you are sending out to produce a steady, stationary
pattern. In other words, the rope moves up and down in a pattern that doesn’t
move along the rope.
Example 8 If the rope proves to be too tricky, you may try this with a rubber
band. Stretch it between your hands and disturb one of the lengths. You’ll
immediately see patterns of standing waves being generated depending on the
frequency at which you strike the band. These patterns will be similar to those
shown in Figure 8.
Antinodes
Nodes
Figure 8. Standing waves
In Figure 8 there are points where the waves always interact destructively, these
points are called nodes. Note that midway in between the nodes interaction
between the waves is always constructive and therefore the amplitude is at its
maximum, these locations are known as antinodes.
2.5
Resonance
The frequencies that give rise to standing waves are known as the natural frequency of the medium. In other words, any medium that supports waves has
a natural frequency of oscillation. If the disturbance applied to the medium is
at the natural frequency a standing wave will be produced. Energy is added to
the standing wave as the disturbance continues and its amplitude will increase.
This is resonance, and the natural frequencies are called resonant frequencies.
Example 9 Cut a few long, thin strips of aluminum foil and hang them from
one end down the side of a ‡at surface. Whistle as loudly as you can. The
8
strips will resonate in response and you may hear a buzz that is produced by the
vibration of the strips against the ‡at surface.
2.6
Refraction
When a wave comes to a region where its speed changes, its direction of travel
changes as well, unless it enters that region straight on, in which case there is
no change of direction.
If you live in Rochester, N.Y., or any snowy place, for that matter, you know
very well what refraction is all about when you are out driving in the winter.
Suppose that you are driving along a road and the right wheels of your car drive
on the snow. They will get extra resistance to their motion and this will make
the car swerve to the right. In other words, your car will turn into the medium
where it travels more slowly. This is shown in Fig. 9.
Incident beam
Incident beam
θi
θi
c1 slow medium
c2 fast medium
Figure 9. Refraction.
c1 fast medium
θt
i
c2 slow medium
Refracted
beam
θt
is the angle of incidence and
transmission
Refracted
beam
t
is the angle of
The sea also o¤ers a good example of refraction. Have you ever been to a
beach where the water is very shallow? You may have noticed that the waves
come parallel to the beach. That is because the rising ocean ‡oor slows the
waves and they turn into an area where they move more slowly. On the other
hand, on those beaches frequented by surfers you’ll notice that the waves come
onto the beach at a greater angle. The ocean ‡oor in these beaches falls o¤
rapidly and refraction cannot turn the waves straight towards the beach.
The amount of refraction is predicted by Snell’s law, which relates the transmitted beam direction to the incident beam direction and the speeds of propagation in the two media forming an interface. See Fig. 9 again for reference.
sin
sin
t
i
=
c1
c2
(4)
In this equation c1 is the speed of propagation on the incident side of the interface and c2 is the speed of propagation on the transmitted side. i is the
incident angle and t the transmitted, or refracted angle.
9
3
Electromagnetic radiation
3.1
Properties of light
“Everything in the future is a wave, everything in the past is a particle” Lawrence
Bragg.
Most of what we see is thanks to the sun since it is the source of most of the
light around us. When we read, watch TV, or walk outdoors the eyes detect a
narrow range of electromagnetic radiation called visible light. Figure 10.
Γ rays
IR
UV
TV, FM
long wave
λ
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
10
AM
X rays
µwaves
Visible
radio
Figure 10. Electromagnetic spectrum
3.1.1
Light rays
Suppose you are in a movie theater and instead of looking at the screen you
turn around and look towards the projection booth. What do you see? Because
of dust and other particles in the air you’ll notice the path of the light beam of
the projector. The edges are considerably sharp, suggesting that light travels
in straight lines.
In…nitely thin beams of light do not exist, however the idea of a ray is very
useful for tracing the path the light follows through space.
Example 10 Get a piece of cardboard and punch a little hole on it. Get outdoors on a bright-starry night and hold the cardboard parallel to the ‡oor. You’ll
see on the ‡oor an image of the moon. If there is a full moon you’ll see a nice
bright circle, if there is a crescent moon, you’ll see a crescent image!
This is the physical principle in a pinhole camera. The rays of light traced
from the object through the pinhole show you that the image will be inverted.
The image will be somewhat blurred as well since the pinhole is not in…nitesimally small. Of course, the smaller the pinhole the sharper the image, however,
smaller pinholes will let in less light therefore producing very faint images.
In the XVI century a 15 year old boy, Giambattista della Porta darkened
an entire room and allowed light from outside through a pinhole to create an
image on the wall. He called his room a camera obscura. In the XIX century J.
N. Niepce used a camera obscura to project an image onto light-sensitive paper,
thus creating the photographic camera!
10
Figure 11. Pinhole camera
3.1.2
Polarization
Do you still have the rope we used in the previous chapter lying around? Once
again grab the free end of the rope and shake it upside down. Note that the
wave that travels away from you will do so only in the vertical direction, or in
other words, it will travel in one plane. If you now shake it in the horizontal
direction, the wave will travel away in the horizontal plane. Waves traveling in
just one plane are called plane polarized or linearly polarized. If you were to
shake the rope in random directions the wave that moves away from you won’t
be polarized anymore.
Light can be thought of as a wave with two components perpendicular to
each other. Most light is not polarized because the two components oscillate
rapidly and randomly.
Polarizers There are some substances capable of absorbing light waves that
oscillate in a particular direction while allowing those vibrating in a perpendicular direction to pass. this is shown schematically in Figure 12.
(a)
(b)
Figure 12. Two plane polarized waves. (a) The wave will go through the slit if
the plane of polarization coincides with the lenght of the slit. (b) If the plane
of polarization is perpendicular to the lenght of the slit the wave is stopped.
11
Example 11 Cellophane sandwich. First set the axes of the two polarizers
perpendicular to each other without the cellophane in between, no light should
be transmitted through the second polarizer. Introduce the cellophane in between
and rotate it about the beam axis. You’ll see that the cellophane makes it possible
for light to go through the second polarizer. You’ll also notice that there are two
positions of the cellophane, at right angles to each other that permit no light
to pass through. Ordinary cellophane has the right half-wave thickness for a
particular component of white light, and the transmitted beam will have the color
of this component.
Example 12 Corn syrup sandwich. Get a bottle of corn syrup and a pair of
polarizers. These could be a couple of lenses from an old pair of sunglasses.
Position the bottle of corn syrup between the two polarizers and rotate one of
them. What do you see?
3.1.3
Light we cannot see.
Sunlight warms up everything it touches. Back in 1800 Herschel (an astronomer)
placed a thermometer in the colors of the spectrum and recorded temperatures
at each di¤erent wavelength. He was surprised when he realized that the temperature raised the most when the thermometer was o¤ to the side of the red
band, where there was no visible light. He was in the infrared region whose
wavelength is longer than visible light.
In 1801 Johann Wilhelm Ritter placed “photographic paper”(paper soaked
in silver chloride) in the spectrum and discovered that the greatest blackening
of the paper occurred just beyond the violet. He was in the ultraviolet region
where wavelengths are shorter than visible light. This is why overexposure
causes sunburn.
4
Mechanical waves. Sound.
When we hit a tuning fork, it moves against the air moving particles on its way.
When the molecules are pushed together they create a region of compression
that travels away from the fork.
When the fork recedes it leaves a partial vacuum known as rarefaction or
expansion. Both the compression and rarefaction regions move away from the
fork. This traveling disturbance is a sound wave and is shown in Fig. 13.
12
Figure 13. Sound wave
When the waves enter your ear, the hearing system senses the vibrations
and triggers nerve signals to your brain. Therefore, a sound is your brain’s
interpretation of how the pressure changes in the air as the wave passes by.
The above discussion should help you to realize that sound waves belong to
the category of longitudinal waves.
We can hear waves that oscillate from 20 cycles=s = 20 Hz up to 20000
cycles=s = 20 kHz:
One thing to note is that sound waves of all frequencies travel at the same
speed in air, namely v = 331 m= s: This means that the speed of sound is an
average of the speeds of the air molecules and it doesn’t depend on the rate
at which the pressure ‡uctuates. Otherwise it would be impossible to attend a
concert or to carry on a conversation across a room.
4.1
Re‡ection and absorption
Flat walls, bare ‡oors and ceilings are good re‡ectors of sound. The waves
bounce o¤ them with little distortion, i.e. echoes resemble the original sound.
On the other hand, curtains, carpets, upholstered furniture scatter sound
waves in may directions. In fact, they absorb a large percentage of energy and
so they soften noises.
Sound re‡ections can be used to detect things, and this is the principle of
radar, sonar and ultrasound imaging.
4.2
Di¤raction
Sound waves di¤ ract or spread when they pass through an opening. It is partially because of di¤raction that you can hear a person talking around a corner.
Di¤raction is in fact a limiting factor in the resolution of imaging systems.
It gets a¤ected by aperture size, frequency, and focusing as is shown in Figure
14.
13
Aperture size
d
d /2
Frequency
f
f/2
fo c u sin g
Figure 14. Factors a¤ecting di¤raction. From C.J.Daly, N.A.K. Rao “Scalar
Di¤ raction from a Circular Aperture”
4.3
Refraction
We have discussed already that waves bend and change direction of propagation
when they impinge at an angle that is not perpendicular, di¤erent media with
di¤erent speeds of propagation. Sound waves do refract as well. They will turn
unto a cooler region of air, where they travel more slowly, and they will turn
away from a warmer area, where they travel faster.
Example 13 Refraction is also encountered in ultrasound medical imaging.
Suppose that a doctor is imaging a fetus holding the ultrasound transducer in
such a way that the ultrasound beam makes an angle of 30 when it encounters
an interface between bone and soft tissue. Assume that the speed of sound in
bone is 4080 m= s; and the speed of sound in soft tissue is 1540 m= s: According
to Eq. 4 we have
sin t
sin 30
sin t
sin 30
sin t
sin t
t
=
1540
4080
= 0:38
= 0:38 sin 30
= 0:38 0:5 = 0:189
' 11
This means that there has been a 30
11 = 19 shift in the trajectory of
the beam! As you can well imagine, refraction may introduce artifacts on an
ultrasound image. They manifest themselves in the image as lateral displacement
of anatomical structures.
14
4.4
Doppler shift
How many times have you been standing in the street while an ambulance is
rushing towards you? You may have noticed that as the ambulance approaches
the siren has a higher frequency than when it recedes in the distance after
passing you. Don’t confuse loudness with frequency. The sound is louder as the
ambulance approaches you, in other words, loudness depends on distance. On
the other hand, the change in frequency is due to the motion of the vehicle. In
this case the source (the ambulance) is moving towards you (the receiver) and
then away from you.
As a matter of fact, you’ll notice the same e¤ect if you walk towards a parked
car blowing its horn. In this instance the source (the car) is standing while you
(the receiver) move towards it.
Any change in frequency due to motion is called Doppler shift. Figure 15
will help you understand why this happens:
Figure 15. Doppler E¤ect. If both the source (S) and the receiver (R) are
stationary the sound waves propagate at the same frequency. If the source (S)
moves towards the receiver (R) the wavefront is squeezed and the frequency is
higher.
In medical ultrasound the Doppler e¤ect is used to quantify and to image
blood ‡ow and to detect fetal movements.
15
References
1. Boleman, J. Physics. An Introduction. Prentice Hall, 1989
2. Falk, D., Brill, D., Stork, D. Seeing the Light. John Wiley & Sons, 1986
3. Pierce, J.R. Almost all About Waves. The MIT Press, 1974
4. Daly, C.J., Rao, N.A.H.K. Scalar Di¤raction from a Circular Aperture.
Kluwer Academic Publishers, 2000.
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