Chapter 6: Waves Part 1

1
Chapter 6 Waves Part 1

• Alyssa Jean-Mary
• Source The Physical Universe by Konrad B.
  Krauskopf and Arthur Beiser

2
Waves

• A wave is a periodic disturbance (i.e. a back and
  forth change that is repeated regularly) that
  spreads out from a source and carries energy with
  it
• Water waves (produced when a stone is dropped in
  water), sound waves (produced when hands are
  clapped), and light waves (produced when a light
  is switched on) are all different types of waves
  that have some basic properties in common
• Two categories of waves
• Mechanical waves travel only though matter and
  involve the motion of particles of the matter
  they pass though. Water waves and sound waves are
  examples of mechanical waves. All mechanical
  waves transfer energy from place to place by a
  series of periodic motions of individual
  particles, but they cause no permanent shift in
  the position of the matter.
• Electromagnetic waves consist of varying electric
  and magnetic fields and can travel through matter
  AND through a vacuum. Light waves and radio waves
  are examples of electromagnetic waves.

3
Water Waves

• Water is seen as moving towards the shore. But,
  in between the breakers (i.e. the water crashing
  on the shore), water is rushing back out to the
  sea. Also, there is no pile up of water on the
  beach. Thus, the overall motion is really the
  endless movement of water back and forth.
• In another way, if seaweed is floating on water,
  it is observed that the seaweed moves very
  little. This is because as the water goes towards
  the shore, the seaweed rises and moves also
  towards the shore, but as the water moves towards
  the sea (i.e. away from the shore), it falls and
  moves also towards the sea.
• The top part of the wave is called the crest. The
  crest occurs when the wave is moving towards the
  shore. The bottom part of the wave is called the
  trough. The trough occurs when the wave is moving
  towards the sea.
• The illusion of overall movement towards the
  shore is because each molecule of water undergoes
  its circular motion after the molecule behind it
  has. At the crest, the water molecules are moving
  in the direction of the wave, and in the trough,
  the water molecules are moving backwards.
• Even though water doesnt move towards the shore,
  energy does. Waves are produced by wind. The
  energy from this wind is what is carried by the
  waves to the shore.

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Transverse Waves and Longitudinal Waves 1

• Transverse waves are waves that move
  perpendicular to the direction in which the wave
  moves
  
• In a long coil spring, if the left-hand end is
  moved back and forth, a series of compressions
  and rarefactions move along the spring. The
  compressions occur where the loops of the spring
  are pressed together, and the rarefactions occur
  where the loops of the spring are stretched
  apart. Each loop moves back and forth and thus
  transfers its motion to the loop next to it. This
  regular series of back and forth movements is
  what gives rise to the compressions and
  rarefactions. Longitudinal waves are waves that
  move along the same line as the motion of the
  individual units.

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Transverse Waves and Longitudinal Waves 2

• Water waves are a combination of transverse waves
  and longitudinal waves
  
• Transverse waves can only travel in solids, but
  longitudinal waves can travel in solids or fluids
  (i.e. gases or liquids).
  
• Transverse motion requires that each particle
  moves with the adjacent particles (i.e. its
  neighbors) to which it is tightly bound. This is
  impossible in a fluid since the molecules in a
  fluid slid past each other. Thus, transverse
  waves cannot travel in fluids.
• On the other hand, longitudinal motion only
  requires that each particle pushes on its
  neighbors, which can happen in a gas, a liquid,
  or a solid, which why longitudinal waves can
  travel through all.
• Any waves that are at the boundary between two
  fluids (i.e. surface waves on water) have both
  transverse motion and longitudinal motion
  
• Because longitudinal earthquake waves pass
  through the center of the earth, and transverse
  earthquake waves dont, it is thought that in the
  center of the earth there is a liquid.

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Wave Quantities

• As the wave moves to the right, each point on the
  curve is thought to move up and down.
  
• For longitudinal waves, the high points are the
  maximum shifts of the particles in one direction,
  and the low points are their maximum shifts in
  the other direction
• The quantities of waves
• The distance from crest to crest (or trough to
  trough) is the wavelength, ?
  
• The rate at which each crest moves is the speed,
  v
  
• The number of crests that pass a given point each
  second is the frequency, f
  
• The time needed for a complete wave (i.e. crest
  trough) to pass a given point is the period, T
  
• The height of the crest above the undisturbed
  level (or the depth of the trough below this
  level) is the amplitude, A
  
• The energy that is carried by a wave depends on
  the amplitude and the frequency of the wave, i.e.
  it depends on the violence of the wave and the
  number of waves per second. The energy is
  proportional to the square of each of these
  quantities.
• The unit of frequency is cycles per second (c/s).
  This unit is usually called a hertz (Hz), which
  was named after Heinrich Hertz, a pioneer in the
  study of electromagnetic waves.

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Basic Wave Formula

• The speed of a wave is given by the number of
  waves that pass a point per second multiplied by
  the length of each wave
  
• In equation form, the speed of a wave is given
  by
  
• v f?
• where v is the speed of the wave, f is the
  frequency of the wave, and ? is the wavelength of
  the wave.

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Example Calculation of Wave Speed

• Example What is the speed of a wave if it has a
  frequency of 34 Hz and a wavelength of 2.5 m?
  
• Answer
• Given 34 Hz, 2.5m
• Looking for speed
• Equation v f?
• Solution v (34 Hz)(2.5m) 85 m/s

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Standing Waves

• When a rope that is attached at one end and free
  at the other end is shaken on the free end, and
  one wave is sent down this rope to the attached
  end, the wave will reform itself and travel back
  along the rope.
• Now, if there is a series of waves that is sent
  down this rope, the reflected waves (i.e. those
  waves coming from the attached end) will meet the
  forward-moving waves (i.e. those waves coming
  from the free end) head on. Thus, each point on
  the rope must respond to two different impulses
  (one from each direction) on the rope at the same
  time. The two impulses will add together
• If the point on the rope is being pushed in the
  same direction by both waves, it will move in
  that direction with an amplitude equal to the sum
  of the amplitudes of the two waves.
• If the point on the rope is being pushed by the
  waves in opposite directions, it will have an
  amplitude equal to the difference of the two wave
  amplitudes.
• If the timing is right, at some points on the
  rope, the two motions will completely cancel out,
  and other points on the rope will be moving with
  twice the normal amplitude. In a situation like
  this, the waves appear not to travel at all
  because some parts of the wave only move up and
  down and other parts of the wave remain at rest.
  This is a standing wave.
• An example of standing waves are the vibrating
  strings in musical instruments.

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Sound 1

• Most sounds are produced by a vibrating object
• For example, the cone of a loudspeaker is a
  vibrating object that produces sound. When sound
  moves outward from the speaker, the cone pushes
  the air molecules that are in front of it
  together, which forms a region of high pressure
  that spreads outward. Then the cone moves
  backward, which expands the space that is
  available for nearby air molecules. Some of the
  molecules actually flow towards the cone, which
  leaves a region of low pressure that spreads
  outward behind the region of high pressure that
  was previously created. Thus, the repeated
  vibrations of the cone send out a series of
  compressions and rarefactions, which are sound
  waves.
• Sound waves are longitudinal waves because the
  molecules in their paths move back and forth in
  the same direction as the waves (i.e. not
  perpendicular like in transverse waves). The
  molecules that are in the path of a sound wave
  become alternately denser and rarer. The pressure
  change that occurs causes our eardrums to
  vibrate, which is what produces the sensation of
  sound.
• Most sounds are waves i.e. they have a series
  of compressions and rarefactions. Some sounds,
  however, have only one single compression.
  Examples of these types of sounds are the crack
  of a rifle and the first sharp sound of a
  thunderclap.

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Sound 2

• At ordinary temperatures, in the air at sea
  level, the speed of sound is about 343 m/s (767
  mi/h)
  
• Sounds travel faster in liquids and solids than
  in gases because since the molecules in liquids
  and solids are closer together than those in
  gases, they can respond faster to one anothers
  motions. Thus, in water, the speed of sound is
  about 1500 m/s and in iron, the speed of sound is
  5000 m/s.
• Our ears are the most sensitive to sounds that
  have frequencies between 3000 Hz and 4000 Hz.
  Sounds with frequencies below 20 Hz, which is
  called infrasound, and above 20,000 Hz, which is
  called ultrasound, are heard by almost no one.
  Most animals, however, have a upper limit that is
  higher than our upper limit. With age, hearing
  deteriorates, especially at the higher
  frequencies.
• Ultrasound, those sounds with frequencies above
  20,000 Hz, have many applications. They are used
  in medical imaging and in determining water
  depths. The technique to determine water depths
  is called sonar. Sonar is also used to detect
  submarines, and it is used by bats in the air to
  detect prey.

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The Decibel

• The more energy a sound wave carries, the louder
  it sounds, but our ears respond to sound waves in
  a particular way if the rate of energy flow of a
  sound is doubled, the sound doesnt sound twice
  as loud in our ears it is only slightly louder.
  This is why a single instrument can be heard in
  concerto even when an entire orchestra is playing
  at the same time. This is also why you can carry
  on a conversation at a party even when many other
  people are talking at the same time.
• The unit of sound is the decibel (dB). There is a
  special scale that uses the decibel to describe
  how powerful a sound is. If a sound can barely be
  heard by a normal person, it has 0 dB. Every 10
  dB corresponds to a 10-fold change in sound
  energy. Thus, a 50 dB sound is 10 times stronger
  than a 40 dB sound, but is 100 times stronger
  than a 30 dB sound. The sound of ordinary
  conversation is usually about 60 dB, which is 106
  (i.e. a million) times more intense than the
  faintest heard sound.
• Permanent hearing damage can happen if you are
  exposed to sounds that are 85 dB or higher. Rock
  concerts, for instance, can have sounds as high
  as 125 dB, which is why many people have
  significant hearing loss from rock concerts.
  Three-quarters of the hearing loss of a typical
  older person in the United States is due to
  exposure to such loud sounds.

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The Doppler Effect

• When vehicles are moving towards us, the sounds
  they produce seem to be higher pitched than
  normal, AND when vehicles are moving away from
  us, the sounds that they produce seem to be lower
  pitched than normal. This difference in frequency
  is known as the Doppler effect.
• The Doppler effect is due to the relative motion
  of the listener and the source of the sound.
  Either one or both need to be moving there is
  no Doppler effect when both are not moving. When
  the motion reduces the distance between the
  source of the sound and the listener (i.e. when a
  vehicle is moving towards us), the wavelength of
  the sound decreases, which makes the frequency of
  the sound higher. When the motion increases the
  distance between the source of the sound and the
  listener (i.e. when a vehicle is moving away from
  us), the wavelength of the sound increases, which
  makes the frequency of the sound lower.
• The amount of frequency difference can be seen in
  the following example if a fire engine has a 500
  Hz siren and it moving at 60 km/h (37 mi/h), when
  it is approaching you, you will hear a sound of
  526 Hz, and when it is moving away from you, you
  will hear a sound of 477 Hz. Any such frequency
  change is easy to pick up by our ears.

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Uses of the Doppler Effect

• The Doppler effect can be used to measure the
  speed of blood in an artery. When an ultrasound
  beam is directed at an artery, the moving blood
  cells with reflect waves that will exhibit a
  Doppler shift in frequency because the cells are
  acting as moving wave sources. The speed of the
  blood can be calculated from this shift in
  frequency. In the main arteries, the speed of
  blood is a few centimeters per second, and in the
  smaller ones, it is less.
• The Doppler effect occurs in light waves. Thus,
  astronomers can use the Doppler effect to detect
  and measure the motions of stars. Stars emit
  light that has only certain characteristic
  wavelengths. If a star is moving toward the
  earth, these wavelengths appear shorter than is
  characteristic, and if a star is moving away from
  the earth, these wavelengths appear longer than
  is characteristic. The amount of difference in
  the frequency from the characteristic frequency
  is used to calculate the speed at which the star
  is moving, whether it is approaching or receding.
  This is how the expansion of the universe was
  discovered.
• There are many other uses for the Doppler effect.

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Musical Sounds

• Just like other sounds, musical sounds are
  produced by vibrating objects
  
• The vibrating objects that produce musical sounds
  vary. Some examples are stretched wire in
  stringed instruments, vocal cords in the throat,
  membranes in drums, and air columns in wind
  instruments.
• The simplest vibration of a stretched string is
  when a single standing wave takes up the entire
  length of the string.
  
• The frequency of a sound may be changed by
  changing the tension in the string. The more
  tension the string has (i.e. the tighter it is),
  the higher the frequency is. The frequency can
  also be changed for a given amount of tension by
  changing the length of the string.
• Depending on where the string is plucked, bowed,
  or struck, more complex vibrations can occur.
  Thus, instead of a standing wave with a single
  crest, the standing wave might have two, three,
  or even more crests. A sound wave that has more
  crests because it has a shorter standing wave has
  a higher frequency. The frequencies are related
  to the frequency of the longest wave (i.e. the
  single standing wave) by simple ratios 21, 31,
  etc. The tone that is produced by a single
  standing wave is called the fundamental tone. The
  tones that occur at higher frequencies when the
  spring vibrates in segments are called overtones.

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Resonance

• The motion of the string and thus the form of the
  sound wave may be very complex
  
• The strings of a musical instrument never just
  give a fundamental tone or a single overtone
  they give a combination of the fundamental tone
  and several overtones
• The fundamental tone sounds flat and
  uninteresting to the ear, but as overtones are
  added, the sound becomes richer. The quality, or
  timbre, of the tone depends on which overtones
  are emphasized in the sound. Which overtones are
  emphasized depends mostly on the shape of the
  instrument, which enables it to resonate at
  particular frequencies. The sound part of the
  instrument (i.e. the belly of the violin or the
  soundboard of the piano) has certain natural
  frequencies of vibration. The instrument is more
  readily set to vibrate at these natural
  frequencies than at any other frequencies. A
  resulting sound may have a large number of
  overtones, but the musical quality characteristic
  of an instrument is due to which overtones get
  greater emphasis.

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Other Musical Sounds

• Wind instruments produce sounds from vibrating
  air columns.
  
• An organ has a separate pipe for each note. The
  shorter the pipe, the higher the pitch.
  
• Woodwinds (i.e. flutes and clarinets) have a
  single tube with holes. The length of the air
  column is controlled by the opening and closing
  of these holes.
• Most brass instruments have valves that are
  connected to loops of tubing. If a valve is
  opened, the length of the air column is
  increased, and thus, a note of lower pitch is
  produced.
• A slide trombone varies the length of its air
  column by sliding in or out its telescoping U
  tube.
  
• Since a bugle has neither holes nor valves,
  different notes are obtained using the lips of
  the bugler.
  
• The fundamental frequencies of the speaking voice
  on an average 145 Hz for men and 230 Hz for
  women. Even when the overtones that are present
  are considered, the frequencies in ordinary speed
  are for the most part below 1000 Hz. In singing,
  the first and second overtones may be louder than
  their fundamental tones, and even higher
  overtones add to the beauty of the sound.