# waves

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```					                      Topic List

Types of Waves       Wave speed
Longitudinal Waves   Transmission of Waves
Transverse Waves     Reflection
Surface Waves        Refraction
Frequency            Superposition Principle
Wavelength           Interference
Period               Diffraction
Amplitude            Standing Waves & Resonance
Waves & Information
Marvin the Martian would like to send a message from Mars to
Earth. There are two ways of sending a message. He could enclose
the message in a rocket and physically send it to Earth. Or, he could
send some type of signal, maybe in the form of radio waves.
Information can be sent via matter or waves. If sent via waves,
nothing material is actually transmitted from sender to receiver. If
you talk to a friend, be it in
person or on the phone, you are
transmitting information via
waves. Nothing is physically
transported from you to your
friend. This would not be the
case, however, if you sent him
a letter.
Waves & Energy
Suppose Charlie Brown wants to wake up Snoopy. Some energy is
required to rouse Snoopy from his slumber. Like information, energy
can also be transmitted via physical objects or waves. Charlie Brown
can transmit energy from himself to Snoopy via Woodstock:
Woodstock flies over; his kinetic energy is physically transported in
the form a little, yellow bird. Alternatively, Charlie Brown could
send a pulse down a rope that’s attached to Snoopy’s dog house. The
rope itself is not transported, but the pulse and its energy are!

pulse
Types of Waves
A mechanical wave is just a disturbance that propagate through a
medium. The medium could be air, water, a spring, the Earth, or even
people. A medium is any material through which a wave travels.
Mechanical wave examples: sound; water waves; a pulse traveling on
a spring; earthquakes; a “people wave” in a football stadium.
An electromagnetic wave is simply light of a visible or invisible
wavelength. Oscillating intertwined electric and magnetic fields
comprise light. Light can travel without medium—super, duper fast.
A matter wave is a term used to describe particles like electrons that
display wavelike properties. It is an important concept in quantum
mechanics.
A gravity wave is a ripple in the “fabric of spacetime” itself. They are
predicted by Einstein’s theroy of relativity, but they’re very difficult to
detect.
Mechanical Waves: Three Types
Mechanical waves require a physical medium. The particles in the
medium can move in two different ways: either perpendicual or
parallel to direction of the wave itself.
In a longitudinal wave, the particles in the medium move parallel to
the direction of the wave.
In a transverse wave, the particles in the medium move perpen-
dicular to the direction of the wave.
A surface wave is often a combination of the two. Particles typically
move in circular or elliptical paths at the surface of a medium.

Longitudinal  Parallel
Transverse  Perpendicular
Surface  Combo
Longitudinal Waves
A whole bunch of kids are waiting in line to get their picture taken
with Godzilla. The bully in back pushes the kid in front of him, who
bumps into the next kid, and so on down the line. A longitudinal
pulse is sent through the line of kids. It’s longitudinal because as
each kid gets bumped, he moves forwards, then backwards (red
arrow), parallel to the direction of the pulse. The location of the
pulse is the point where two kids are being compressed together. The
next slide shows how the pulse progresses
through the line.

pulse direction
Ouch                   Longitinal
!
Waves (cont.)
C = Compression (high
C                     kid density)
Ouch
!            R = Rarefaction (low
kid density)
The compression (the
pulse) moves up the
R C            line, but each kid keeps
Ouch   his place in line.
!
I hope
Godzilla eats
that bully!
R C
Sound is a Longitudinal Wave
As sound travels through air, water, a solid, etc., the molecules
of the medium move back and forth in the direction of the
wave, just like the kids in the last example, except the
molecules continually move back and forth for as long as the
sound persists. If the bully kept shoving the kid in front of
him, a series of pulses would be generated. If
he shoved with equal force each time and did
this at a regular rate, we would call these
pulses a wave. Similarly, when a speaker or a
tuning fork vibrates, it repeatedly shoves the
air in front of it, and a longitudinal wave
propagates through to the air. The speaker
shoves air molecules; the bully shoves people.
In either case, the components of the medium
must bump into their neighbors.
Transverse Waves
After a great performance at a drum and bugle corps contest, the
audience decides to start a wave in the stands. Each person rises and
sits at just the right time so the effect is similar to the pulse in Charlie
Brown’s rope. Like the Godzilla example, people make up the wave
medium here. But this is a transverse wave because, as the wave
moves across the stands, folks are moving up and down.
wave direction
Transverse Waves (cont.)
In a transverse wave, molecules aren’t being compressed and spread
out as they are in a longitudinal wave. The reason a transverse wave
can propagate is because of the attraction between adjacent molecules.
Imagine if each person in the stands on the last slide were connected to
the person on his left and right with giant rubber bands. As soon the
person on one end stood up, the band stretches. The tension in the
band pulls his neighbor up, who, in turn, lifts the next guy.
The tension in the rubber bands is analogous to the forces connecting
particles of the medium to their neighbors. The colored sections of
rope tug on each other as the waves travels through them. If they
didn’t, it would be as if the rope were cut, and no wave could travel
through it.
Surface Waves
Below the surface fluids can typically only transmit longitudinal
waves, since the attraction between neighboring molecules is not
as strong as in a fluid. At the surface of a lake, water molecules
(white dots) move in circular paths, which are partly longitudinal
and partly transverse. The molecules are offset, though: when
one is at the top of the circle, the one in front of it is near the top.
As in any wave, the particles of the medium do not move along
with the wave. The water molecules complete a circle each time
a crest passes by. Animation
wave direction
Breaking Waves
Waves break near the
shore because the
water becomes
shallow. Close to the
shore the ground
beneath the water
interferes with the
circular motions of the
water molecules as
they participate in a
passing wave.
Sandbars further off
shore can have the
same effect, much to
the delight of surfing
enthusiasts like Bart.
Seismic Waves
Seismic waves use Earth itself as their medium. Earthquakes produce
them and so does a nation when it carries out an underground nuclear
test. (Other countries can detect them.) Seismic waves can be
longitudinal, transverse, or surface waves. P and S type waves are
called body waves, since they are not confined to the surface. Rayleigh
waves do most of the shaking during a quake.
Name            Type                          Info

P Wave       Longitudinal    Also known as primary, compressional,
or acoustic waves; fastest seismic wave
S wave        Transverse       Also known as secondary, or shear
waves; do not travel through fluids;
Rayleigh       Surface        Rolls along surface like a water wave;
Wave                                    large amplitude
Love Wave       Surface         Ground moves side to side as wave
moves forward
“Mini Seismic” Waves
Though we might not refer to them as
seismic, anything moving on the ground can
transmit waves through the ground. If you
stand near a moving locomotive or a heard
of charging elephants, you would feel these
vibrations. Even something as small as a
beetle generates pulses when it moves. These pulses can be detected
by a nocturnal sand scorpion. Sensors on its eight legs can detect both
longitudinal and surface waves. The scorpion can determine the
direction of the waves based on which legs feel the waves first. It can
determine the distance of the prey based on the time delay between the
fast moving longitudinal waves and the slower moving surface waves.
The greater the time delay, the farther away the beetle. This is the
same way seismologists determine the distance of a quake’s epicenter.
Sand is not the best conductor of waves, so the scorpion will only be
able to detect beetles within about a half meter.
Wave Characteristics
Amplitude (A) – Maximum displacement of particle of the medium
from its equilibrium point. The bigger the amplitude, the more energy
the wave carries.
Wavelength () – Distance from crest (max positive displacement) to
crest; same as distance from trough (max negative displacement) to
trough.
Period (T) – Time it takes consecutive crests (or troughs) to pass a
given point, i.e., the time required for one full cycle of the wave to pass
by. Period is the reciprocal of frequency: T = 1 / f.
Frequency (f ) – The number of cycles passing by in a given time. The
SI unit for frequency is the Hertz (Hz), which is one cycle per second.
Wave speed (v) – How fast the wave is moving (the disturbance itself,
not how fast the individual particles are moving, which constantly
varies). Speed depends on the medium. We’ll prove that v =  f.
Amplitude & Wavelength
The red transverse wave has the same wavelength as the longitudinal
wave in the spring. (P to Q is one full cycle.) Note that where the spring
is most compressed, the red wave is at a crest, and where the spring is
most stretched (rarified), the red wave is at a trough. The amplitude in
the red wave is easy to see. In the longitudinal wave, the amplitude
refers to how far a particle on the spring moves to the left or right of its
equilibrium point. Often a graph like the red wave is used to represent
a longitudinal wave. For sound, the y-axis might be pressure deviation
from normal air pressure, and the x-axis might be time or position.

P                         Q
A


Frequency & Period
Why is the
Period = seconds per cycle.
frequency of a wave
the reciprocal of its    Frequency = cycles per second.
period?
They’re reciprocals no matter
what unit we use for time. A
sound wave that has a frequency
of 1,000 Hz has a period of
1 / 1,000 of a second. This means
that 1,000 high pressure fronts
are moving through the air and
Speed, Wavelength, & Frequency
Barney Rubble, a.k.a. “Barney the Wave Watcher,” is excited because
he just made a discovery: v =  f. With some high tech, prehistoric
equipment, Barney measures the wavelength of the incoming waves to
be 18 ft. He counts 10 crests hitting the shoreline every minute. So,
10 crests pass any given point
in a time of one minute. But
18 ft               10 crests corresponds to a
distance of 180 ft, which
means the wave is traveling
at 180 ft / min. This result is
the product of wavelength
and frequency, yielding the
result:

v = f
Harmonic Waves
Imagine a whole bunch of equal masses hanging from identical springs.
If the masses are set to bobbing at staggered time intervals, a snapshot
of the masses forms a transverse wave. Each mass undergoes simple
harmonic motion, and the period of each is the same. If the release of
the masses is timed so that the masses form a sinusoid at each point in
time, the wave is called harmonic. Right now, m4 is peaking. A little
later m4 will be lower and m3 will be peaking. The masses (the
particles of the medium) bob up and down but do not move
horizontally, but the wave does move horizontally.

m4                              m9
m3           m5                 m8          m10
m2                              m7
m1                              m6          wave direction
Making a Harmonic Wave
A generator attached to a rope moves up and down in simple harmonic
motion. This generates a harmonic wave in the rope. Each little piece
of rope moves vertically just like the masses on the last slide. Only the
wave itself moves horizontally. The time it takes the wave to move
from P to Q is the period of the wave, T. The distance from P to Q is
the wavelength, . So, the wave speed is given by: v =  / T =  f
(since frequency and period are reciprocals).
Since the generator moves vertically in SHM, the vertical position of
the black doo-jobber is given by: y(t) = A cos  t. The doo-jobber’s
period is given by T = 2 /  . This is also the period of the wave.

P                                     Q

wave direction
Making a Non-harmonic Wave
If the black doo-jobber does not move in SHM, the wave it generates
will not be harmonic. As long as the generator has some sort of
periodic motion, the wave generated will have a well defined period
and wavelength. Here the generator pauses at the high and low points,
causing the wave to flatten.
If the wave had moved at a constant speed and changed direction
instantly, a saw-tooth wave would have been the result. Sound is not
a transverse wave, but a graph of pressure vs. time as a sound waves
pass by would look like a very few simple sinusoid in the case of a
pure tone. It would be a very complicated wave if the sound is a
musical instrument of someone’s voice.
P                                Q

wave direction
Wave Speed on a Rope
If a pulse is traveling along a rope to the right at a speed v, from its
point of view it’s still and the rope is moving to the left at a speed v.
As the red segment of rope of length s rounds the turn in the pulse, a
centripetal force must act on it. The tension in the rope is F, and the
downward components of the tension vectors add to make the
centripetal force.
FC = m v 2 / r        2 F sin ( / 2) = m v 2 / r     2 F ( / 2) = m v 2 / r
(since the sine of an angle  the angle itself in radians)

F = mv2 / r          F r/ m = v2        F s / m = v2
(since s = r  )
m
s                                          (continued)
/2               /2     v (rope)
F      r          F
Wave Speed on a Rope (cont.)
If the rope is uniform density, then the mass per unit length is a
constant. We’ll call this constant µ. Thus, µ = m / s. From the last
slide we have:
v2 = F s / m = F / µ
F
v =
µ
This shows that waves travel faster in materials that are stiff
(high tension) and light weight. Unit check: [N / (kg / m)] ½
= [N m / kg ] ½ = [(kg m / s 2) m / kg ] ½ = [m 2 / s 2] ½ = m / s.
m
s
/2                 /2    v (rope)
F      r            F
Reflection of Waves
Whenever a wave encounters different medium, some
of the wave may be reflected back, and some of the
wave penetrate and be absorbed or transmitted through
the new medium. Light waves reflects off of objects.
If it didn’t, we would only be able to see objects that
emitted their own light. We see the moon because it’s
reflecting sunlight. Sound waves also reflect off of
objects, creating echoes. Water waves, seismic waves,
and waves traveling on a rope all can reflect.
Transmission & Reflection
Let’s look at 4 different scenarios of a waves traveling along a rope.
The link below has an animation of each.
1. Hard boundary (fixed end): Reflected wave is inverted.
2. Soft boundary (free end): Reflected wave is upright.
3. Light rope to heavy rope: Reflected wave is faster and wider than
transmitted wave. Transmitted wave is upright, but reflected
wave is inverted (since to the thin rope, the thick rope is like a
hard boundary).
4. Heavy rope to light rope: Transmitted wave is faster, wider, and
has a greater amplitude than reflected wave. Both waves are
upright. (The transmitted wave is upright this time since, to the
thick rope, the thin rope is like a soft boundary).
Animation
Frequency of Transmitted Waves
The frequency of a transmitted wave is always unchanged. Say a
wave with a frequency of 5 Hz is traveling along a rope that changes
thickness at some point. Since 5 pulses hit this point every second, 5
pulses will be transmitted every second. Since the speed will vary
depending on the thickness of the rope, the wavelength must vary too.
Here a wave travels from a thin rope to a thick one. Because µ is
larger in the thick rope, the wave is slower there. This causes the
waves to “bunch up,”which means a decrease in wavelength. (For
clarity the reflected waves are not shown here.)

v                           v
Amplitude & Energy
The energy carried by a wave is proportional to the square

EA
of its amplitude:                  2
Consider our masses on a string again. The amount of potential
energy stored in a spring is given by: U = ½ k x 2, where k is the
spring constant and x is the distance from equilibrium. For m1 or
m4, U = ½ k A2. The other masses have kinetic energy but less
potential. Since energy is conserved, the total energy any mass has
is ½ k A2. This shows that
energy varies as the square
of the amplitude. The
m4                   constant of proportionality
m3            m            depends on the medium.
5

m2
m1
Amplitude of Reflected & Transmitted
Waves: Light to Heavy
When a pulse on the light rope reaches the interface, the heavy rope
offers a lot of resistance. The heavy rope is not affected much by the
light rope, so the transmitted pulse has a smaller amplitude. The
reflected pulse’s amplitude diminishes since some of the light rope’s
energy it transmitted to the heavy rope.

before
incident pulse

inverted reflected pulse
after                                   transmitted pulse

Back to Animation
Amplitude of Reflected & Transmitted
Waves: Heavy to Light
When a pulse on the heavy rope reaches the interface, the light rope
offers little resistance. The light rope is greatly affected by the heavy
rope, so the transmitted pulse has a greater amplitude. The upright
reflected pulse’s amplitude diminishes since some of the heavy rope’s
energy it transmitted to the light rope.

before
incident pulse

upright reflected pulse

after                                    transmitted pulse

Back to Animation
Refraction
We’ve seen that when a wave reaches an interface (a change from
one medium to another), part of the wave can be transmitted, and
part can be reflected back. A rope is a 1-dimensional medium; in a
2-dimensional medium a transmitted wave can change direction.
This is refraction—the bending of a wave as it passes from one
medium to another. The most well know type of refraction is that
of light bending as it passes from air to glass or water, which we’ll
study in detail in a unit on light.
As ocean waves approach the shore at an angle, the part of the
wave closer to shore begins to slow down because the water is
shallower. This causes refraction, and the waves bend so that it the
wave fronts (crests) come in nearly parallel with the shore. See
pic on next slide. Even though the medium (water) doesn’t
change, one of its properties does—the speed of the wave.
Refraction of Ocean Waves
Wave fronts are shown in white heading toward the beach. The water
gets shallow at the bottom first, which causes the waves to slow down
and bend, and the wavelength to decrease. By the time the waves
reach shore, they’re nearly parallel to the shoreline. The effect can
even be seen on islands, where winds nearly wrap around it and come
toward the island from all sides.
Superposition
Check out this animation to see what happens when two pulses
approach each other from opposite ends on a rope.

Superposition Animation

Note the following:
1. The waves pass through each other unaffected by their meeting.
2. As they’re passing through each other the waves combine to
create a changing waveform.
3. The displacement of the rope at any point in this “combo wave”
is the sum of the displacements of the displacements of the
original waves. In other words, we add amplitudes. This is
called superposition.
Constructive & Destructive Interference

Destructive Interference
Constructive Interference
Waves are “out of phase.” By
Waves are “in phase.” By super-        superposition, red and blue
position, red + blue = green. If red   completely cancel each other
and blue each have amplitude A,        out, if their amplitudes and
then green has amplitude 2A.           frequencies are the same.
Interference
Like force vectors, waves can work together or opposition.
Sometimes they can even do some of both at the same time.
Superposition applies even when the waves are not identical.
Interference Animation
Constructive interference occurs at a point when two waves
have displacements in the same direction. The amplitude of the
combo wave is larger either individual wave.
Destructive interference occurs at a point when two waves have
displacements in opposite directions. The amplitude of the combo
wave is smaller than that of the wave biggest wave.
Superposition can involve both constructive and destructive
interference at the same time (but at different points in the
medium).
Wave Interference
Diffraction
When waves bounce off a barrier, this is reflection. When waves
bend due to a change in the medium, this is refraction. When waves
change direction as they pass around a barrier or through a small
opening, this is diffraction. Refraction involves a change in wave
speed and wavelength; diffraction doesn’t.
Diffraction of water happens as waves bend around a boat in a
harbor. This is different than the refraction of waves near shore
because the depth of water does not decrease around the boat like it
does near shore. Diffraction is most noticeable when the wavelength
is large compared to the obstacle or opening. Thus, no noticeable
diffraction may occur if the boat in the harbor is very big.
The sound waves from an owl’s hoot travel a greater distance in the
forest than a song bird’s call, because a low pitch owl hoot has a
longer wavelength than a high pitch songbird call, and the owl’s
waves are able to diffract around trees.             Pics on next slide
Diffraction Pics
When waves pass a barrier they curve around it slightly. When they
pass through a small opening, they spread out almost as if they had
come from a point source. These effects happen for any type of wave:
water; sound; light; seismic waves, etc.
Diffraction & Bats
Bats use ultrasonic sound waves (a frequency too high for
humans to hear) to hunt moths. The reason they use
ultrasound is because at lower frequencies much of the sound
waves would have a wavelength close to the size of a moth,
which means much of the sound would diffract around it.
Bats hunt by echolocation—bouncing sound waves off of
prey and listening for the echoes, so they need to emit sound
with a wavelength smaller that the typical moth, which means
a high frequency is required. High frequency sound waves
reflect off the moths rather than diffracting around them. If
bats hunted bigger prey, we might have emitted sounds that
we could hear.
Standing Waves
Animations:
When waves on a rope hits a fixed end, it
reflects and is inverted. This reflected
1st Harmonic          waves then combine with oncoming
( The Fundamental )   incident waves. At certain frequencies the
resulting superposition yields a standing
wave, in which some points on the rope
2nd Harmonic          called nodes never move at all, and other
points called antinodes have an amplitude
twice as big as the original wave.
3rd Harmonic          A rope of given length can support
standing waves of many different
frequencies, called harmonics, which are
named based on the number of antinodes.
4th Harmonic
continued
Standing Waves (cont.)
Animations:          It is important to understand that a standing
is the result of the a wave interfering
constructively and destructively with its
Standing Wave with     reflection. Only certain wavelengths will
Superposition Shown    interfere with themselves and produce a
(scroll down)          standing wave. The wavelengths that work
depend on the length of the rope, and we’ll
learn how to calculate them in the sound
unit. (Standing waves are very important in
Standing Wave with     music.)
Incident & Reflected   Wavelengths that don’t work result in
Waves Shown            irregular patterns. A standing wave could be
Separately             simulated with a series of masses on springs,
(scroll down)          as long as their amplitudes varied
sinusoidally.
Resonance
Objects that oscillate or vibrate tend to do so at a particular frequency
called the natural frequency. For example, a pendulum will swing
back and forth at a certain frequency that only depends on its length,
and a mass on a spring will bob up in down at a frequency that
depends on the mass and the spring constant. It is possible physically
to grab hold of the pendulum or mass and force it to swing or bob at
any frequency, but if no one forces them, each will swing of bob at its
own natural frequency. If left alone,
friction will rob the masses of their
energy, and their amplitudes will
decay. If a periodic force, like an
occasional push, matches the period
of one of the masses, this is called
resonance, and the mass’s amplitude
M                 will grow.               (continued)
m                                      Resonance Animation
Resonance           (cont.)
Jane does
Tarzan is swinging through
positive
the jungle, but he can’t quite
work
make it across the river to the
next tree. So, he asks Jane
x
for a little help. She obliges by giving him a push
F       every time he’s just about to swing away from her.
In order to maximize his amplitude to get him
across the river, her pushing frequency must match
Jane does       his natural frequency. This is resonance. When
negative        resonance occurs her applied force does the
work            maximum amount of positive work. If she mis-
times the push, she might do negative work, which
would diminish his amplitude. The moral of the
x           story is: Resonance involves timing and matching
the natural frequency of an oscillator. When it
F       happens, the oscillator’s amplitude increases.
Resonance Question
Explain how you could get a 700 lb
wrecking ball swing with a large
amplitude only by pulling on it with a
scrawny piece of dental floss.
Give the ball a little tug, as much as you
can without breaking the floss. The ball
with barely budge. Continue giving it
tugs every time the ball is at its closest to
you. If you match the natural frequency
Wrecking   of the ball, its amplitude will slowly
Ball     increase to the desired amount. In this
way you are adding energy to the ball
very slowly.
Tacoma Narrows Bridge
Even bridges have resonant (natural) frequencies. The Tacoma
Narrows bridge in Washington state collapsed due to the
complicated effects of wind. One day in 1940 the wind blew at
just the right speed. The wind was like Jane pushing Tarzan, and
the bridge was like Tarzan. The bridge twisted and shook
Eventually, the vibrations
caused the by wind grew in
amplitude until the bridge was
destroyed.

Click the pic to see the
MPEG video clip.
Credits
The following images were obtained for these websites:
Marvin the Martian         http://store.yahoo.com/rnrdist/warnerbrothers.html

Charlie Brown & Snoopy http://www.snoopy.com/
Godzilla http://www.cinescape.com/godzilla/
Drum & Bugle Corps (Cavaliers of Rosemont, IL)
http://www.cavaliers.org/

Sand Scorpion      http://www.aps.org/meet/MAR00/baps/vpr/layy3-03-04.html

Beach pic http://www.ssdsupply.com/hawaii.htm
Diffraction http://www.glenbrook.k12.il.us/gbssci/phys/Class/sound/u11l3d.html
http://hea-www.harvard.edu/ECT/the_book/Chap1/Chapter1.html
Credits
Wave movies: Dr. Ken Russel, Kettering University
http://www.kettering.edu/~drussell/Demos.html
Standing wave animated gifs: Tom Henderson, Glenbrook South High School
http://www.physicsclassroom.com/Class/waves/U10L4b.html
Tacoma Narrows Bridge:
http://www.civeng.carleton.ca/Exhibits/Tacoma_Narrows/DSmith/fig06.gif

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