Vibrations and Waves
Section 3
What do you think?
• Consider different types of waves, such as water
waves, sound waves, and light waves. What
could be done to increase the speed of any one
of these waves? Consider the choices below.
• Change the size of the wave? If so, in what way?
• Change the frequency of the waves? If so, in what
way?
• Change the material through which the wave is
traveling? If so, in what way?
Vibrations and Waves
Section 3
Wave Motion
• A wave is a disturbance that propagates through
a medium.
– What is the meaning of the three italicized terms?
– Apply each word to a wave created when a child
jumps into a swimming pool.
• Mechanical waves require a medium.
• Electromagnetic waves (light, X rays, etc.) can
travel through a vacuum.
Vibrations and Waves
Section 3
Wave Types
• The wave shown is a pulse wave.
– Starts with a single disturbance
• Repeated disturbances produce periodic waves.
Vibrations and Waves
Section 3
Wave Types
• If a wave begins with a disturbance that is SHM, the
wave will be a sine wave.
• If the wave in the diagram is moving to the right, in which
direction is the red dot moving in each case?
Vibrations and Waves
Section 3
Transverse Waves
• A wave in which the particles move perpendicular to the
direction the wave is traveling
• The displacement-position graph below shows the
wavelength () and amplitude (A).
Vibrations and Waves
Section 3
Click below to watch the Visual Concept.
Visual Concept
Transverse Wave
Vibrations and Waves
Section 3
Longitudinal Wave
• A wave in which the particles move parallel to the
direction the wave is traveling.
– Sometime called a pressure wave
• Try sketching a graph of density vs. position for the
spring shown below.
Vibrations and Waves
Section 3
Click below to watch the Visual Concept.
Visual Concept
Longitudinal Wave
Vibrations and Waves
Section 3
Wave Speed
xv
t
• Use the definition of speed to determine the speed of a
wave in terms of frequency and wavelength.
– A wave travels a distance of one wavelength () in
the time of one period (T), so
– Because frequency is inversely related to period:
x
t T
v fT
Vibrations and Waves
Section 3
Wave Speed
• SI unit: s-1 m = m/s
• The speed is constant for any given medium.
– If f increases, decreases proportionally.
– Wavelength () is determined by frequency and speed.
• Speed only changes if the medium changes.
– Hot air compared to cold air
– Deep water compared to shallow water
Vibrations and Waves
Section 3
Click below to watch the Visual Concept.
Visual Concept
Characteristics of a Wave
Vibrations and Waves
Section 3
Waves Transfer Energy
• Waves transfer energy from one point to another
while the medium remains in place.
– A diver loses his KE when striking the water but the
wave carries the energy to the sides of the pool.
• Wave energy depends on the amplitude of the
wave.
– Energy is proportional to the square of the amplitude.
• If the amplitude is doubled, by what factor does the energy
increase?
• Answer: by a factor of four
Vibrations and Waves
Section 3
Now what do you think?
• Consider different types of waves, such as water
waves, sound waves, and light waves. What
could be done to increase the speed of any one
of these waves? Consider the choices below.
– Change the size of the wave? If so, in what way?
– Change the frequency of the waves? If so, in what
way?
– Change the material through which the wave is
traveling? If so, in what way?
Vibrations and Waves
Section 4
What do you think?
• Imagine two water waves traveling toward each other in a swimming pool. Describe the behavior of the two waves when they meet and afterward by considering the following questions. • Do they reflect off each other and reverse direction?
• Do they travel through each other and continue?
• At the point where they meet, does it appear that only one wave is present, or can both waves be seen?
• How would your answers change for a crest meeting a trough?
Vibrations and Waves
Section 4
Wave Interference
• Superposition is the combination of two
overlapping waves.
– Waves can occupy the same space at the same time.
– The observed wave is the combination of the two
waves.
– Waves pass through each other after forming the
composite wave.
Vibrations and Waves
Section 4
Constructive Interference
• Superposition of waves that produces a resultant wave
greater than the components
– Both waves have displacements in the same direction.
Vibrations and Waves
Section 4
Destructive Interference
• Superposition of waves that produces a resultant wave
smaller than the components
– The component waves have displacements in opposite
directions.
Vibrations and Waves
Section 4
Click below to watch the Visual Concept.
Visual Concept
Comparing Constructive and Destructive
Interference
Vibrations and Waves
Section 4
Reflection: Free End
• The diagram shows a
wave reflecting from an
end that is free to move
up and down.
• The reflected pulse is
upright.
– It is produced in the same
way as the original pulse.
Vibrations and Waves
Section 4
Reflection: Fixed End
• This pulse is reflected
from a fixed boundary.
• The pulse is inverted
upon reflection.
– The fixed end pulls
downward on the rope.
Vibrations and Waves
Section 4
Standing Waves
• Standing waves are produced when two
identical waves travel in opposite directions and
interfere.
– Interference alternates between constructive and
destructive.
• Nodes are points where interference is always
destructive.
• Antinodes are points between the nodes with
maximum displacement.
Vibrations and Waves
Section 4
Standing Waves
• A string with both ends fixed
produces standing waves.
– Only certain frequencies are
possible.
• The one-loop wave (b) has
a wavelength of 2L.
• The two-loop wave (c) has
a wavelength of L.
• What is the wavelength of
the three-loop wave (d)?
– 2/3L
Vibrations and Waves
Section 4
Click below to watch the Visual Concept.
Visual Concept
Standing Wave
Vibrations and Waves
Section 4
What do you think?
• Imagine two water waves traveling toward each other in a swimming pool. Describe the behavior of the two waves when they meet and afterward by considering the following questions. • Do they reflect off each other and reverse direction?
• Do they travel through each other and continue?
• At the point where they meet, does it appear that only one wave is present or can both waves be seen?
• How would your answers change if it was a crest and a trough?
Sound
Section 1
What do you think?
• What is sound? • What do all of the sounds that you hear have in
common?
• How do they differ?
• Can sounds travel through solids? Liquids? Gases? • Is one type of material better for transmitting sound
waves?
• When race cars or emergency vehicles pass you, the sound changes. In what way, and why?
Sound
Section 1
What is Sound?
• Sound is a longitudinal wave.
• All sound waves are produced
by vibrating objects.
– Tuning forks, guitar strings, vocal
cords, speakers
• The vibrating object pushes the
air molecules together, forming
a compression.
• It then spreads them apart,
forming a rarefaction.
Sound
Section 1
Graphing Sound Waves
• The diagram shows
compressions (dark) and
rarefactions(white). If you
measured the pressure or
density of the air and
plotted these against
position, how would the
graph appear?
Sound
Section 1
Click below to watch the Visual Concept.
Visual Concept
Sound Waves
Sound
Section 1
Characteristics of Sound
• Frequency is the number of waves per second.
• You have heard of ultrasound. What is it?
• Frequencies audible to humans are between
20 Hz and 20 000 Hz.
– Middle C on a piano is 262 Hz.
– The emergency broadcast signal is 1 000 Hz.
• Infrasound frequencies are lower than 20 Hz.
• Ultrasound frequencies are greater than
20 000 Hz.
Sound
Section 1
Click below to watch the Visual Concept.
Visual Concept
Comparing Infrasonic and Ultrasonic Sounds
Sound
Section 1
Pitch
• Pitch is the human perception of how high or low
a sound appears to be.
– Pitch is primarily determined by frequency.
– Pitch also depends slightly on other factors.
• Higher frequencies appear to have a higher pitch when
played loudly, even though the frequency does not change.
Sound
Section 1
Speed of Sound
• Sound waves travel through solids, liquids and
gases.
– In which would the speed generally be greatest?
Why?
• Solids. Because the molecules are more closely packed, the
particles respond more rapidly to compressions.
– How might the temperature of air affect the speed of
sound waves? Why?
• Higher temperature increases the speed of the waves
because the particles are moving faster and colliding more
often.
Sound
Section 1
Speed of Sound
Sound
Section 1
Spherical Waves
• Sound propagates in three dimensions.
• The diagram shows: – Crests or wave fronts (blue
circles)
– Wavelength ()
– Rays (red arrows)
• Rays indicate the direction of propagation.
• How would these wave fronts appear different if they were much farther from the source?
Sound
Section 1
Spherical Waves
• Wave fronts and rays become
more nearly parallel at great
distances.
• Plane waves are simply very
small segments of a spherical
wave a long distance from the
source.
Sound
Section 1
Doppler Effect
• Why are the waves closer together on the left?
– Waves are closer because the vehicle moves to the left
along with the previous wave.
• How will the
sound be different
for observer A and
observer B?
– Higher frequency
(pitch) for
observer A
• Continued on the next
slide….
Sound
Section 1
Doppler Effect
• How would the wave
pattern change if the
vehicle moved at a faster
speed? How would it
sound different?
– At a higher speed, waves
would be even closer
together and the pitch
difference would be even
greater.
• The Doppler effect is the observed change in
frequency due to the motion of the source or observer.
Sound
Section 1
Click below to watch the Visual Concept.
Visual Concept
Doppler Effect and Sound
Sound
Section 1
Now what do you think?
• What is sound? – What do all of the sounds that you hear have in
common?
– How do they differ?
• Can sounds travel through solids? Liquids? Gases? – Is one type of material better for transmitting sound
waves?
• When race cars or emergency vehicles pass you, the sound changes. In what way, and why?
Sound
Section 2
What do you think?
• Members of rock bands generally protect their
ears from the loud sounds to prevent damage to
their hearing.
• How do we determine the loudness of a sound?
• What quantity is loudness measuring?
• What units are used?
• Name three ways you can reduce the loudness of the
music heard by a person in the audience.
Sound
Section 2
Sound Intensity
• Vibrating objects do work on the air as they push
against the molecules.
• Intensity is the rate of energy flow through an
area.
– What is “rate of energy flow” called?
• E/t is called power (P).
– Since the waves spread out spherically, you must
calculate the area of a sphere. How?
• A = 4r2
– So, what is the equation for intensity?
Sound
Section 2
Sound Intensity
• SI unit: W/m2
• This is an inverse square relationship.
– Doubling r reduces intensity by ¼.
– What happens if r is halved?
• Intensity increases by a factor of 4.
Sound
Section 2
Intensity and Decibels
• An intensity scale based on human perception of
loudness is often used.
• The base unit of this scale is the bel. More
commonly, the decibel (dB) is used.
– 0.1 bel = 1 dB,1 bel = 10 dB, 5 bels = 50 dB, etc.
– The lowest intensity humans hear is assigned a value
of zero.
• The scale is logarithmic, so each increase of 1
bel is 10 times louder.
– An increase in intensity of 3 bels is 1 000 times louder.
Sound
Section 2
Sound
Section 2
Classroom Practice Problems
• The intensity of the sound from an explosion is
0.10 W/m2 at a distance of 1.0 × 103 m. Find the
intensity of the sound at a distance of 5.0 × 102
m, 1.0 × 102 m and 10.0 m.
– Answers: 0.41 W/m2, 1.0 × 101 W/m2, 1.0 × 103 W/m2
• Find the approximate decibel equivalents of
these sound intensities using Table 2.
– Answers: 110 dB, 130 dB, 150 dB
Sound
Section 2
Audible Sounds
• The softest sound humans can hear is called the
threshold of hearing.
– Intensity = 1 10-12 W/m2 or zero dB
• The loudest sound humans can tolerate is called
the threshold of pain.
– Intensity = 1.0 W/m2 or 120 dB
• Human hearing depends on both the frequency
and the intensity.
Sound
Section 2
Sound
Section 2
Forced Vibrations
• Sympathetic vibrations occur when a vibrating
object forces another to vibrate as well.
– A piano string vibrates the sound board.
– A guitar string vibrates the bridge.
• This makes the sound louder and the vibrations
die out faster.
– Energy is transferred from the string to the sound
board or bridge.
Sound
Section 2
Resonance
• The red rubber band links the 4
pendulums.
• If a blue pendulum is set in
motion, only the other blue
pendulum will have large-
amplitude vibrations.
– The others will just move a small
amount.
• Since the vibrating frequencies
of the blue pendulums match,
they are resonant.
Sound
Section 2
Resonance
• Large amplitude vibrations produced when the
frequency of the applied force matches the
natural frequency of receiver
– One blue pendulum was the driving force and the
other was the receiver.
• Bridges have collapsed as a result of structural
resonance.
– Tacoma Narrows in the wind
– A freeway overpass during an earthquake
Sound
Section 2
Click below to watch the Visual Concept.
Visual Concept
Resonance (Frequency)
Sound
Section 2
Now what do you think?
• Members of rock bands generally protect their
ears from the loud sounds to prevent damage to
their hearing.
– How do we determine the loudness of a sound?
• What quantity is loudness measuring?
• What units are used?
– Name three ways you can reduce the loudness of the
music heard by a person in the audience.
Sound
Section 3
What do you think?
• A violin, a trumpet, and a clarinet all play the
same note, a concert A. However, they all sound
different.
• What is the same about the sound?
• Are the frequencies produced the same?
• Are the wave patterns the same?
• Why do the instruments sound different?
Sound
Section 3
Standing Waves on a String
• There is a node at each end because
the string is fixed at the ends.
• The diagram shows three possible
standing wave patterns.
• Standing waves are produced by
interference as waves travel in
opposite directions after plucking or
bowing the string.
• The lowest frequency (one loop) is
called the fundamental frequency (f1).
Sound
Section 3
Standing Waves on a String
• To the left is a snapshot of a single loop standing wave on a string of length, L.
• What is the wavelength for this wave? – Answer: = 2L
• What is the frequency? – Answer:
12
v vf
L
Sound
Section 3
Harmonics
• n is the number of loops or harmonic number.
• v is the speed of the wave on the string. – Depends on tension and density of the string
• L is the length of the vibrating portion of the string.
• How could you change the frequency (pitch) of a string?
Sound
Section 3
Click below to watch the Visual Concept.
Visual Concept
Fundamental Frequency
Sound
Section 3
Standing Waves in an Air Column
• Wind instruments also use standing waves.
– Flutes, trumpets, pipe organs, trombones, etc.
• Some instruments have pipes open at both ends
while others have one end closed.
– Air is free to move at open ends so antinodes occur.
– Closed ends are nodes.
• The velocity of the wave is now the velocity of
sound in air (346 m/s at 25°C).
Sound
Section 3
Wind Instruments
• Wind instruments are not as simple as organ
pipes.
– The shape is not always cylindrical.
– The holes change the wave patterns as well.
– The size of the “pipe” varies along the length.
Sound
Section 3
Classroom Practice Problems
• One string on a toy guitar is 34.5 cm long.
– What is the wavelength of the first harmonic or the
fundamental wavelength?
• Answer: 69.0 cm or 0.690 m
– The string is plucked and the speed of the waves on
the string is 410 m/s. What are the frequencies of the
first three harmonics?
• 590 Hz, 1200 Hz, 1800 Hz
• Note: The use of significant figures causes the multiples of
590 to be 1200 and 1800 because only two significant figures
are present in the answer.
Sound
Section 3
Classroom Practice Problems
• An organ pipe open at both ends is 34.5 cm long. – What is the wavelength of the first harmonic or the
fundamental wavelength? • Answer: 69.0 cm or 0.690 m
– What are the frequencies of the first three harmonics if the air temperature is 25.0°C?
• Answers: 501 Hz, 1000 Hz, 1500 Hz
– Answer the same questions if the pipe is closed at one end.
• Answers: 251 Hz, 753 Hz, 1250 Hz
Sound
Section 3
Timbre or Quality of Sound
• Instruments do not vibrate in a single mode.
– Several harmonics are produced at the same time.
– The particular harmonics and intensity of each vary
with different instruments.
• Timbre is the quality of the tone resulting from
the combination of harmonics.
• The fundamental frequency (1st harmonic)
determines the pitch.
– Adding other harmonics changes the timbre.
Sound
Section 3
Click below to watch the Visual Concept.
Visual Concept
Timbre
Sound
Section 3
Beats
• The diagram
shows two waves
of different
frequencies.
Sketch the
superposition or
sum of these
waves.
• How would the
combined wave
sound?
Sound
Section 3
Beats
• Produced by two waves with the same intensity and different frequencies – Generally the frequencies are nearly the same.
• The sound pulses or changes from loud to soft and back.
• Beats are used to tune instruments. – If a tuner hears beats, the instruments are slightly out
of tune.
• The number of beats heard per second is the difference in the two frequencies.
Sound
Section 3
Click below to watch the Visual Concept.
Visual Concept
Beat
Sound
Section 3
Now what do you think?
• A violin, a trumpet, and a clarinet all play the
same note, a concert A. However, they all sound
different.
– What is the same about the sound?
• Are the frequencies produced the same?
• Are the wave patterns the same?
– Why do the instruments sound different?
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