Chapter Twenty Five Notes:
Vibrations and Waves
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A repeating back-and-forth motion about an equilibrium
position in a VIBRATION. A vibration cannot exist in one
instant. It needs time to move back and forth.
A disturbance that is transmitted progressively from one
place to the next with no actual transport of matter is a
WAVE. A wave cannot exist in one place but must extend
from one place to another.
Pendulums swing to and fro with regularity.
A complete to-and-fro oscillation is one vibration.
The time of a to-and-fro vibration (or swing) is called the
period of the pendulum. The period depends only on the
length of a pendulum and the acceleration of gravity.
A long pendulum has a longer period than a short pendulum;
that is, it swings to and fro less frequently than a short
pendulum.
Vibration of a pendulum. The to-and-fro
vibratory motion is also called oscillatory
motion (or oscillation).
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The back and forth vibratory motion (often called oscillatory motion)
of a swinging pendulum is called simple harmonic motion (SHM). If
you fill the pendulum with sand, and allow a conveyor belt to move
at constant speed beneath the pendulum, the sand traces out a
special curve known as a sine curve. A sine curve is a pictorial
representation of a wave. The source of all waves is something that
vibrates.
 Wave
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description (parts of wave)
Crests and troughs: the high points of a wave are called crests, and
the low points are called troughs.
Midpoint of vibration: the straight dashed line represents the “home”
position, or midpoint of the vibration. The midpoint position is also
the equilibrium position.
Amplitude: the distance from the midpoint to the crest (or trough).
The amplitude equals the maximum displacement from equilibrium.
Wavelength: the distance from the top of one crest to the top of the
next one. Or equivalently, the wavelength is the distance between
any successive identical parts of the wave.
Frequency: how frequently a vibration occurs is described by its
frequency. For example, the frequency of a vibrating pendulum
specifies the number of complete to-and-fro vibrations it makes in a
given time (usually one second).
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More about frequency
• The unit of frequency is called the hertz (Hz).
For example, if a pendulum makes two vibrations in one second, its
frequency is 2 Hz.
• The source of all waves is something that vibrates.
• The frequency of the vibrating source and the frequency of the
wave it produces are the same.
• Relationship between frequency and period:
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Frequency = 1/ Period or Period = 1/ Frequency.
– For example, suppose that a pendulum makes two vibrations in
one second. Its frequency is 2 Hz. Its period, that is, the time
needed to complete one vibration is 1/2 second.
Electrons in the transmitting antenna
vibrate 940,000 times each second
and produce 940-kHz radio waves.
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Most information about our surroundings comes to us in some form
of waves. It is through wave motion that sounds come to ears, light
to our eyes, and electromagnetic signals to our radios and television
sets.
• Through wave motion, energy can be transferred from a source to
a receiver without the transfer of matter between the two points.
• In wave motion, what is transported from one place to another is a
disturbance (vibration) in a medium, not the medium itself.
If one end of a rope is shaken up and down, a rhythmic disturbance travels along the rope. Each
particle of the rope moves up and down, while at the same time the disturbance moves along the
length of the rope. The medium, rope or whatever, returns to its initial condition after the
disturbance has passed. What is propagated is the disturbance, not the medium itself.
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A wave is a disturbance which moves along a medium from one end
to the other. If one watches an ocean wave moving along the
medium (the ocean water), one can observe that the crest of the
wave is moving from one location to another over a given interval of
time. The crest is observed to cover distance. The speed of an object
refers to how fast an object is moving and is usually expressed as
the distance traveled per time of travel. In the case of a wave, the
speed is the distance traveled by a given point on the wave (such as
a crest) in a given interval of time. In equation form,
If the crest of an ocean wave moves a distance of 20 meters in 10
seconds, then the speed of the ocean wave is 2 m/s. On the other
hand, if the crest of an ocean wave moves a distance of 25 meters in
10 seconds (the same amount of time), then the speed of this ocean
wave is 2.5 m/s. The faster wave travels a greater distance in the
same amount of time.
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Sometimes a wave encounters the end of a medium and the
presence of a different medium. For example, a wave introduced by
a person into one end of a slinky will travel through the slinky and
eventually reach the end of the slinky and the presence of the hand
of a second person. One behavior which waves undergo at the end
of a medium is reflection. The wave will reflect or bounce off the
person's hand. When a wave undergoes reflection, it remains within
the medium and merely reverses its direction of travel. In the case of
a slinky wave, the disturbance can be seen traveling back to the
original end. A slinky wave which travels to the end of a slinky and
back has doubled its distance. That is, by reflecting back to the
original location, the wave has traveled a distance which is equal to
twice the length of the slinky.
Reflection phenomenon are commonly observed with sound waves.
When you let out a holler within a canyon, you often hear the echo of
the holler. The sound wave travels through the medium (air in this
case), reflects off the canyon wall and returns to its origin (you). The
result is that you hear the echo (the reflected sound wave) of your
holler. A classic physics problem goes like this:
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In this instance, the sound wave travels
340 meters in 1 second, so the speed of
the wave is 340 m/s. Remember, when
there is a reflection, the wave doubles its
distance. In other words, the distance
traveled by the sound wave in 1 second
is equivalent to the 170 meters down to
the canyon wall plus the 170 meters back
from the canyon wall.
Since the distance a single wave moves is
measured in Wavelength, and the time it
takes to move this distance is measured
as period, it can be said that the speed of
a wave is also the wavelength/period.
Noah stands 170 meters
away from a steep canyon
wall. He shouts and hears
the echo of his voice one
second later. What is the
speed of the wave?
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Since the period is the reciprocal of the frequency, the expression
1/f can be substituted into the above equation for period.
Rearranging the equation yields a new equation of the form:
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Speed = Wavelength • Frequency
The above equation is known as the wave equation. It states the
mathematical relationship between the speed (v) of a wave and its
wavelength (λ) and frequency (f). Using the symbols v, λ, and f, the
equation can be rewritten as
 v = λ • f
• Shake the rope with a regular continuing up-and-down motion, the
series of pulses will produce a wave.
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Since the motion of the medium (up and down arrows in the rope
in this case) is at right angles to the direction the wave travels, this
type of wave is called a transverse wave.
• Examples of transverse waves:
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Waves in the stretched strings of musical instruments.
Waves upon the surfaces of liquids.
Electromagnetic waves, which make up radio waves and light.
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Not all waves are transverse.
Sometimes parts that make up a medium move to and fro in the
same direction in which the wave travels.
In this case, motion is along the direction of the wave rather than at
right angles to it. This produces a longitudinal wave.
Sound waves are longitudinal waves.
What happens when two waves meet while they travel through the same
medium? What affect will the meeting of the waves have upon the
appearance of the medium? Will the two waves bounce off each other
upon meeting (much like two billiard balls would) or will the two waves
pass through each other? These questions involving the meeting of two
or more waves along the same medium pertain to the topic of wave
interference.
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Wave interference is the phenomenon which occurs when two
waves meet while traveling along the same medium. The
interference of waves causes the medium to take on a shape
which results from the net effect of the two individual waves
upon the particles of the medium. To begin our exploration
of wave interference, consider two pulses of the same
amplitude traveling in different directions along the same
medium. Let's suppose that each displaced upward 1 unit at
its crest and has the shape of a sine wave. As the sine pulses
move towards each other, there will eventually be a moment
in time when they are completely overlapped. At that
moment, the resulting shape of the medium would be an
upward displaced sine pulse with an amplitude of 2 units. The
diagrams below depict the before and during interference
snapshots of the medium for two such pulses. The individual
sine pulses are drawn in red and blue and the resulting
displacement of the medium is drawn in green.
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This type of interference is sometimes called constructive
interference. Constructive interference is a type of interference
which occurs at any location along the medium where the two
interfering waves have a displacement in the same direction. In this
case, both waves have an upward displacement; consequently, the
medium has an upward displacement which is greater than the
displacement of the two interfering pulses. .
Constructive interference is observed at any location where the two
interfering waves are displaced upward. But it is also observed when
both interfering waves are displaced downward. This is shown in the
diagram below for two downward displaced pulses.
In this case, a sine pulse with a maximum displacement of -1 unit
(negative means a downward displacement) interferes with a sine
pulse with a maximum displacement of -1 unit. These two pulses
are drawn in red and blue. The resulting shape of the medium is a
sine pulse with a maximum displacement of -2 units.
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Destructive interference is a type of interference which occurs at any
location along the medium where the two interfering waves have a
displacement in the opposite direction. For instance, when a sine
pulse with a maximum displacement of +1 unit meets a sine pulse
with a maximum displacement of -1 unit, destructive interference
occurs. This is depicted in the diagram below.
In the diagram above, the interfering pulses have the same
maximum displacement but in opposite directions. The result is that
the two pulses completely destroy each other when they are
completely overlapped. At the instant of complete overlap, there is
no resulting displacement of the particles of the medium. This
"destruction" is not a permanent condition. In fact, to say that the
two waves destroy each other can be partially misleading. When it is
said that the two pulses destroy each other, what is meant is that
when overlapped, the affect of one of the pulses on the
displacement of a given particle of the medium is destroyed or
canceled by the affect of the other pulse.
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Recall from an earlier lesson that waves transport energy through a
medium by means of each individual particle pulling upon its
nearest neighbor. When two pulses with opposite displacements
(i.e., one pulse displaced up and the other down) meet at a given
location, the upward pull of one pulse is balanced (canceled or
destroyed) by the downward pull of the other pulse. Once the two
pulses pass through each other, there is still an upward displaced
pulse and a downward displaced pulse heading in the same direction
which they were heading before the interference. Destructive
interference leads to only a momentary condition in which the
medium's displacement is less than the displacement of the largestamplitude wave.
The two interfering waves do not need to have equal amplitudes in
opposite directions for destructive interference to occur. For
example, a pulse with a maximum displacement of +1 unit could
meet a pulse with a maximum displacement of -2 units. The
resulting displacement of the medium during complete overlap is -1
unit.
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This is still destructive interference since the two interfering pulses
have opposite displacements. In this case, the destructive nature of
the interference does not lead to complete cancellation.
Interestingly, the meeting of two waves along a medium does not
alter the individual waves or even deviate them from their path. This
only becomes an astounding behavior when it is compared to what
happens when two billiard balls meet or two football players meet.
Billiard balls might crash and bounce off each other and football
players might crash and come to a stop. Yet two waves will meet,
produce a net resulting shape of the medium, and then continue on
doing what they were doing before the interference.
The task of determining the shape of the resultant demands that the
principle of superposition is applied. The principle of superposition
is sometimes stated as follows:
When two waves interfere, the resulting displacement of the
medium at any location is the algebraic sum of the
displacements of the individual waves at that same location.
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The animation below depicts two waves moving through a medium
in opposite directions. The blue wave is moving to the right and the
green wave is moving to the left. As is the case in any situation in
which two waves meet while moving along the same medium,
interference occurs. The blue wave and the green wave interfere to
form a new wave pattern known as the resultant. The resultant in the
animation below is shown in black. The resultant is merely the result
of the two individual waves - the blue wave and the green wave added together in accordance with the principle of superposition.
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The result of the interference of the two waves above is a new
wave pattern known as a standing wave pattern. Standing
waves are produced whenever two waves of identical
frequency interfere with one another while traveling opposite
directions along the same medium. Standing wave patterns
are characterized by certain fixed points along the medium
which undergo no displacement. These points of no
displacement are called nodes (nodes can be remembered as
points of no displacement). The nodal positions are labeled
by an N in the animation above. The nodes are always located
at the same location along the medium, giving the entire
pattern an appearance of standing still (thus the name
"standing waves"). A careful inspection of the above
animation will reveal that the nodes are the result of the
destructive interference of the two interfering waves. At all
times and at all nodal points, the blue wave and the green
wave interfere to completely destroy each other, thus
producing a node.
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Midway between every consecutive nodal point are points
which undergo maximum displacement. These points are
called antinodes; the anti-nodal nodal positions are labeled
by an AN. Antinodes are points along the medium which
oscillate back and forth between a large positive
displacement and a large negative displacement. A careful
inspection of the above animation will reveal that the
antinodes are the result of the constructive interference of the
two interfering waves.
In conclusion, standing wave patterns are produced as the
result of the repeated interference of two waves of identical
frequency while moving in opposite directions along the same
medium. All standing wave patterns consist of nodes and
antinodes. The nodes are points of no displacement caused
by the destructive interference of the two waves. The
antinodes result from the constructive interference of the two
waves and thus undergo maximum displacement from the
rest position.
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Suppose that there is a happy bug in the center of a
circular water puddle. The bug is periodically shaking its
legs in order to produce disturbances that travel through
the water. If these disturbances originate at a point, then
they would travel outward from that point in all directions.
Since each disturbance is traveling in the same medium,
they would all travel in every direction at the same speed.
The pattern produced by the bug's shaking would be a series of
concentric circles as shown in the diagram at the right. These circles
would reach the edges of the water puddle at the same frequency. An
observer at point A (the left edge of the puddle) would observe the
disturbances to strike the puddle's edge at the same frequency that
would be observed by an observer at point B (at the right edge of the
puddle). In fact, the frequency at which disturbances reach the edge of
the puddle would be the same as the frequency at which the bug
produces the disturbances. If the bug produces disturbances at a
frequency of 2 per second, then each observer would observe them
approaching at a frequency of 2 per second.
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Now suppose that our bug is moving to the right
across the puddle of water and producing
disturbances at the same frequency of 2 disturbances
per second. Since the bug is moving towards the
right, each consecutive disturbance originates from a
position which is closer to observer B and farther
from observer A. Subsequently, each consecutive
disturbance has a shorter distance to travel before
reaching observer B and thus takes less time to reach
observer B. Thus, observer B observes that the frequency of arrival of
the disturbances is higher than the frequency at which disturbances
are produced. On the other hand, each consecutive disturbance has a
further distance to travel before reaching observer A. For this reason,
observer A observes a frequency of arrival which is less than the
frequency at which the disturbances are produced. The net effect of
the motion of the bug (the source of waves) is that the observer
towards whom the bug is moving observes a frequency which is
higher than 2 disturbances/second; and the observer away from
whom the bug is moving observes a frequency which is less than 2
disturbances/second. This effect is known as the Doppler effect.
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The Doppler effect is observed whenever the source of waves is
moving with respect to an observer. The Doppler effect can be
described as the effect produced by a moving source of waves in
which there is an apparent upward shift in frequency for observers
towards whom the source is approaching and an apparent
downward shift in frequency for observers from whom the source is
receding. It is important to note that the effect does not result
because of an actual change in the frequency of the source. Using
the example above, the bug is still producing disturbances at a rate
of 2 disturbances per second; it just appears to the observer whom
the bug is approaching that the disturbances are being produced at
a frequency greater than 2 disturbances/second. The effect is only
observed because the distance between observer B and the bug is
decreasing and the distance between observer A and the bug is
increasing.
The Doppler effect can be observed for any type of wave - water
wave, sound wave, light wave, etc. We are most familiar with the
Doppler effect because of our experiences with sound waves.
Perhaps you recall an instance in which a police car or emergency
vehicle was traveling towards you on the highway. As the car
approached with its siren blasting, the pitch of the siren sound
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(a measure of the siren's frequency) was high; and then suddenly
after the car passed by, the pitch of the siren sound was low. That
was the Doppler effect - an apparent shift in frequency for a sound
wave produced by a moving source.
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Light:
The Doppler effect is of intense interest to
astronomers who use the information about the shift in
frequency of electromagnetic waves produced by moving
stars in our galaxy and beyond in order to derive information
about those stars and galaxies. The belief that the universe is
expanding is based in part upon observations of
electromagnetic waves emitted by stars in distant galaxies.
Furthermore, specific information about stars within galaxies
can be determined by application of the Doppler effect.
Galaxies are clusters of stars which typically rotate about
some center of mass point. Electromagnetic radiation emitted
by such stars in a distant galaxy would appear to be shifted
downward in frequency (a red shift) if the star is rotating in
its cluster in a direction which is away from the Earth. On the
other hand, there is an upward shift in frequency (a blue shift)
of such observed radiation if the star is rotating in a direction
that is towards the Earth.
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When the source moves at a speed equal to the speed of the
wave, a barrier wave is produced in front of the source as
each successive wave front piles on top of the previous
one. This regions of constructive interference pattern is a
physical reality that must be overcome if the source is to
move any faster.
Boats, after they break through the barrier wave that is
produced when their speeds equal the speed of the
water waves in that region, start trailing a two-dimensional
bow wave. Down the center of the bow wave is a region of
destructive interference while the edges, or wake, are regions
of high amplitude constructive interference. As a general
rule, the speed of a water wave is principally determined by
the water's depth and its temperature.
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When planes fly through the "sound barrier," they are doing more
than just traveling in excess of 340 m/sec. They have traveled also
through a region of high amplitude constructive interference which,
for sound waves, is a region of high compression followed by a
region of low rarefaction. The pressure differential is
tremendous. After breaking through the barrier wave, the plane then
trails a three-dimensional bow wave, or a shock cone, and is said to
be traveling supersonic. The width of the bow wave or shock cone
depends on the speed of the source; the faster the source travels,
the narrower the wave becomes.
When a shock wave, trailing a supersonic plane, passes through your
position, you heard a sonic boom. It is actually a "double
boom." When the leading edge of the cone passes, it raises
equilibrium pressure rapidly upwards (compression), followed by a
rapid drop in pressure (rarefaction), followed by a return to
equilibrium. We hear a sonic boom whenever the Shuttle's glide path
passes over Daytona for its landing at the Cape.
To determine the "Mach" number for a supersonic plane you
compare the ratio of the plane's radius to the ratio of the sound's
radius.
Doppler Effect
a stationary wave
source
a wave source moving to
the right at a speed less
than the wave speed
Barrier Wave
Bow Wave (2-D)
Shock Wave (3-D)
a wave source moving to wave source moving to the right at a
the right at a speed equal speed in excess of the wave speed
to the wave speed
Use this physlet animation by Wolfgang
Christian at Davidson College to watch the
wave patterns as the red particle travels
faster and faster.
The Doppler Effect