Conceptual Physics

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Conceptual Physics

• Chapter Thirty Two Notes
• Electrostatics

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32.1 Electrical Force Charge

• Electrostatics
• Electrostatics, as the name implies, is the study
  of stationary electric charges. A rod of plastic
  rubbed with fur or a rod of glass rubbed with
  silk will attract small pieces of paper and is
  said to be electrically charged. The charge on
  plastic rubbed with fur is defined as negative,
  and the charge on glass rubbed with silk is
  defined as positive.
• 32.1 Electrical Forces and Charge

The enormous attractive and repulsive electrical
forces between the charges in Earth and the
charges in your body balance out, leaving the
relatively weaker force of gravity, which only
attracts. Hence your weight is due only to
gravity.
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• The Atom Electrical forces arise from particles
  in atoms. In the simple Bohr model, the
  positively charged nucleus is surrounded by
  electrons. The protons in the nucleus attract
  the electrons and hold them in orbit. Electrons
  are attracted to protons, but repel other
  electrons. This fundamental electrical property
  to which attraction and repulsion is attributed
  is called charge.
• By agreement, electrons are negatively charged
  and protons are positively charged. Neutrons
  have no charge.
• Attraction and Repulsion The fundamental rule
  at the base of all electrical phenomena is that
  like charges repel and opposite charges attract.

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32.2 Conservation of Charge

• Conservation of Charge is the principle that
  electrons (and protons) are neither created or
  destroyed, but simply transferred from one
  material to another.
• In a neutral atom, there are as many electrons as
  protons. If an electron is added or removed, we
  have an ion which is a charged atom. Negatively
  charged if an electron is added, and positively
  charged if an electron is removed.

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• the sound box, being connected to the air inside
  of it, sets the air inside of the sound box into
  vibrational motion. As the tines of the tuning
  fork, the structure of the sound box, and the air
  inside of the sound box begin vibrating at the
  same frequency, a louder sound is produced. In
  fact, the more particles which can be made to
  vibrate, the louder or more amplified the sound.
  This concept is often demonstrated by the
  placement of a vibrating tuning fork against the
  glass panel of an overhead projector or on the
  wooden door of a cabinet. The vibrating tuning
  fork sets the glass panel or wood door into
  vibrational motion and results in an amplified
  sound.

We know that a tuning fork is vibrating because
we hear the sound which is produced by its
vibration. Nonetheless, we do not actually
visibly detect any vibrations of the tines. This
is because the tines are vibrating at a very high
frequency. If the tuning fork which is being used
corresponds to middle C on the piano keyboard,
then the tines are vibrating at a frequency of
256 Hertz that is, 256 vibrations per second. We
are unable to visibly detect vibrations of such
high frequency.
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• A common physics demonstration involves slowing
  down the vibrations by through the use of a
  strobe light. If the strobe light puts out a
  flash of light at a frequency of 512 Hz (two
  times the frequency of the tuning fork), then the
  tuning fork can be observed to be moving in a
  back and forth motion. With the room darkened,
  the strobe would allow us to view the position of
  the tines two times during their vibrational
  cycle. Thus we would see the tines when they are
  displaced far to the left and again when they are
  displaced far to the right. This would be
  convincing proof that the tines of the tuning
  fork are indeed vibrating to produce sound.
• As discussed in an earlier unit, the frequency is
  simply the reciprocal of the period. For this
  reason, a sound wave with a high frequency would
  correspond to a pressure time plot with a small
  period - that is, a plot corresponding to a small
  amount of time between successive high pressure
  points. Conversely, a sound wave with a low
  frequency would correspond to a pressure time
  plot with a large period - that is, a plot
  corresponding to a large amount of time between
  successive high pressure points. The diagram
  below shows two pressure-time plots, one
  corresponding to a high frequency and the other
  to a low frequency.

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• The ears of a human (and other animals) are
  sensitive detectors capable of detecting the
  fluctuations in air pressure which impinge upon
  the eardrum. The mechanics of the ear's detection
  ability will be discussed later in this lesson.
  For now, it is sufficient to say that the human
  ear is capable of detecting sound waves with a
  wide range of frequencies, ranging between
  approximately 20 Hz to 20 000 Hz. Any sound with
  a frequency below the audible range of hearing
  (i.e., less than 20 Hz) is known as an infrasonic
  and any sound with a frequency above the audible
  range of hearing (i.e., more than 20 000 Hz) is
  known as an ultrasound. Humans are not alone in
  their ability to detect a wide range of
  frequencies. Dogs can detect frequencies as low
  as approximately 50 Hz and as high as 45 000 Hz.
  Cats can detect frequencies as low as
  approximately 45 Hz and as high as 85 000 Hz.
  Bats, being nocturnal creature, must rely on
  sound echolocation for navigation and hunting.
  Bats can detect frequencies as high as 120 000
  Hz. Dolphins can detect frequencies as high as
  200 000 Hz. While dogs, cats, bats, and dolphins
  have an unusual ability to detect ultrasound, an
  elephant possesses the unusual ability to detect
  infrasound, having an audible range from
  approximately 5 Hz to approximately 10 000 Hz.

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• The sensation of a frequencies is commonly
• referred to as the pitch of a sound. A high
• pitch sound corresponds to a high frequency
• sound wave and a low pitch sound
• corresponds to a low frequency sound wave.
• Amazingly, many people, especially those
• who have been musically trained, are capable
  of detecting a difference in frequency between
  two separate sounds which is as little as 2 Hz.
  When two sounds with a frequency difference of
  greater than 7 Hz are played simultaneously, most
  people are capable of detecting the presence of a
  complex wave pattern resulting from the
  interference and superposition of the two sound
  waves. Certain sound waves when played (and
  heard) simultaneously will produce a particularly
  pleasant sensation when heard, are are said to be
  consonant. Such sound waves form the basis of
  intervals in music. For example, any two sounds
  whose frequencies make a 21 ratio are said to be
  separated by an octave and result in a
  particularly pleasing sensation when heard. That
  is, two sound waves sound good when played
  together if one sound has twice the frequency of
  the other. Similarly two sounds with a frequency
  ratio of 54 are said to be separated by an
  interval of a third
• such sound waves also sound good when played
• together.

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26.2 Sound in Air

• A sound wave, like any other wave, is introduced
  into a medium by a vibrating object. The
  vibrating object is the source of the disturbance
  which moves through the medium. The vibrating
  object which creates the disturbance could be the
  vocal chords of a person, the vibrating string
  and sound board of a guitar or violin, the
  vibrating tines of a tuning fork, or the
  vibrating diaphragm of a radio speaker.
  Regardless of what vibrating object is creating
  the sound wave, the particles of the medium
  through which the sound moves is vibrating in a
  back and forth motion at a given frequency. The
  frequency of a wave refers to how often the
  particles of the medium vibrate when a wave
  passes through the medium. The frequency of a
  wave is measured as the number of complete
  back-and-forth vibrations of a particle of the
  medium per unit of time. If a particle of air
  undergoes 1000 longitudinal vibrations in 2
  seconds, then the frequency of the wave would be
  500 vibrations per second. A commonly used unit
  for frequency is the Hertz (abbreviated Hz),
  where 1 Hertz 1
  vibration/second

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• As a sound wave moves through a medium, each
  particle of the medium vibrates at the same
  frequency. This is sensible since each particle
  vibrates due to the motion of its nearest
  neighbor. The first particle of the medium begins
  vibrating, at say 500 Hz, and begins to set the
  second particle into vibrational motion at the
  same frequency of 500 Hz. The second particle
  begins vibrating at 500 Hz and thus sets the
  third particle of the medium into vibrational
  motion at 500 Hz. The process continues
  throughout the medium each particle vibrates at
  the same frequency. And of course the frequency
  at which each particle vibrates is the same as
  the frequency of the original source of the sound
  wave. Subsequently, a guitar string vibrating at
  500 Hz will set the air particles in the room
  vibrating at the same frequency of 500 Hz which
  carries a sound signal to the ear of a listener
  which is detected as a 500 Hz sound wave.
• The back-and-forth vibrational motion of the
  particles of the medium would not be the only
  observable phenomenon occurring at a given
  frequency. Since a sound wave is a pressure wave,
  a detector could be used to detect oscillations
  in pressure from a high pressure to a low
  pressure and back to a high pressure. As the
  compressions

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• (high pressure) and rarefactions (low pressure)
  move through the medium, they would reach the
  detector at a given frequency. For example, a
  compression would reach the detector 500 times
  per second if the frequency of the wave were 500
  Hz. Similarly, a rarefaction would reach the
  detector 500 times per second if the frequency of
  the wave were 500 Hz. The frequency of a sound
  wave not only refers to the number of
  back-and-forth vibrations of the particles per
  unit of time, but also refers to the number of
  compressions or rarefactions which pass a given
  point per unit of time. A detector could be used
  to detect the frequency of these pressure
  oscillations over a given period of time. The
  typical output provided by such a detector is a
  pressure-time plot as shown below.

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• Sound as a Longitudinal Wave
• In the first part of Lesson 1, it was mentioned
  that sound is a mechanical wave which is created
  by a vibrating object. The vibrations of the
  object set particles in the surrounding medium in
  vibrational motion, thus transporting energy
  through the medium. For a sound wave traveling
  through air, the vibrations of the particles are
  best described as longitudinal. Longitudinal
  waves are waves in which the motion of the
  individual particles of the medium is in a
  direction which is parallel to the direction of
  energy transport. A longitudinal wave can be
  created in a slinky if the slinky is stretched
  out in a horizontal direction and the first coils
  of the slinky are vibrated horizontally. In such
  a case, each individual coil of the medium is set
  into vibrational motion in directions parallel to
  the direction which the energy is transported.

Sound waves in air (and any fluid medium) are
longitudinal waves because particles of the
medium through which the sound is transported
vibrate parallel to the direction which the sound
wave moves.
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• A vibrating string can create longitudinal waves
  as depicted in the animation below. As the
  vibrating string moves in the forward direction,
  it begins to push upon surrounding air molecules,
  moving them to the right towards their nearest
  neighbor. This causes the air molecules to the
  right of the string to be compressed into a small
  region of space. As the vibrating string moves in
  the reverse direction (leftward), it lowers the
  pressure of the air immediately to its right,
  thus causing air molecules to move back leftward.
  The lower pressure to the right of the string
  causes air molecules in that region immediately
  to the right of the string to expand into a large
  region of space. The back and forth vibration of
  the string causes individual air molecules (or a
  layer of air molecules) in the region immediately
  to the right of the string to continually vibrate
  back and forth horizontally. The molecules move
  rightward as the string moves rightward and then
  leftward as the string moves leftward. These back
  and forth vibrations are imparted to adjacent
  neighbors by particle-to-particle interaction.
  Other surrounding particles begin to move
  rightward and leftward, thus sending a wave to
  the right.

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• Since air molecules (the particles of the medium)
  are moving in a direction which is parallel to
  the direction which the wave moves, the sound
  wave is referred to as a longitudinal wave. The
  result of such longitudinal vibrations is the
  creation of compressions and rarefactions within
  the air.
• Regardless of the source of the sound wave -
  whether it be a vibrating string or the vibrating
  tines of a tuning fork - sound waves traveling
  through air are longitudinal waves. And the
  essential characteristic of a longitudinal wave
  which distinguishes it from other types of waves
  is that the particles of the medium move in a
  direction parallel to the direction of energy
  transport.

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26.3 Media That Transmit Sound

• Any elastic material can transmit sound.
• Steel is a very good conductor of sound.
• Water is not as good a conductor as steel, but
  is better than air.
• Air is a poor conductor of sound

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26.4 Speed of Sound

• The Speed of Sound
• A sound wave is a pressure disturbance which
  travels through a medium by means of
  particle-to-particle interaction. As one particle
  becomes disturbed, it exerts a force on the next
  adjacent particle, thus disturbing that particle
  from rest and transporting the energy through the
  medium. Like any wave, the speed of a sound wave
  refers to how fast the disturbance is passed from
  particle to particle. While frequency refers to
  the number of vibrations which an individual
  particle makes per unit of time, speed refers to
  the distance which the disturbance travels per
  unit of time. Always be cautious to distinguish
  between the two often confused quantities of
  speed (how fast...) and frequency (how often...).
• Since the speed of a wave is defined as the
  distance which a point on a wave (such as a
  compression or a rarefaction) travels per unit of
  time, it is often expressed in units of
  meters/second (abbreviated m/s). In equation
  form, this is
• speed distance/time

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• The faster a sound wave travels, the more
  distance it will cover in the same period of
  time. If a sound wave is observed to travel a
  distance of 700 meters in 2 seconds, then the
  speed of the wave would be 350 m/s. A slower wave
  would cover less distance - perhaps 660 meters -
  in the same time period of 2 seconds and thus
  have a speed of 330 m/s. Faster waves cover more
  distance in the same period of time.
• Factors Affecting Wave Speed
• The speed of any wave depends upon the properties
  of the medium through which the wave is
  traveling. Typically there are two essential
  types of properties which affect wave speed -
  inertial properties and elastic properties.
  Elastic properties are those properties related
  to the tendency of a material to maintain its
  shape and not deform whenever a force or stress
  is applied to it. A material such as steel will
  experience a very small deformation of shape (and
  dimension) when a stress is applied to it. Steel
  is a rigid material with a high elasticity. On
  the other hand, a material such as a rubber band
  is highly flexible when a force is applied to
  stretch the rubber band, it deforms or changes
  its shape readily. A small stress on the rubber
  band causes a large deformation.

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• Steel is considered to be a stiff or rigid
  material, whereas a rubber band is considered a
  flexible material. At the particle level, a stiff
  or rigid material is characterized by atoms
  and/or molecules with strong attractions for each
  other. When a force is applied in an attempt to
  stretch or deform the material, its strong
  particle interactions prevent this deformation
  and help the material maintain its shape. Rigid
  materials such as steel are considered to have a
  high elasticity. (Elastic modulus is the
  technical term). The phase of matter has a
  tremendous impact upon the elastic properties of
  the medium. In general, solids have the strongest
  interactions between particles, followed by
  liquids and then gases. For this reason,
  longitudinal sound waves travel faster in solids
  than they do in liquids than they do in gases.
  Even though the inertial factor may favor gases,
  the elastic factor has a greater influence on the
  speed (v) of a wave, thus yielding this general
  pattern
• vsolids gt vliquids gt vgases
• Inertial properties are those properties related
  to the material's tendency to be sluggish to
  changes in it's state of motion. The density of a
  medium is an example of an inertial property.

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• The greater the inertia (i.e., mass density) of
  individual particles of the medium, the less
  responsive they will be to the interactions
  between neighboring particles and the slower that
  the wave will be. As stated above, sound waves
  travel faster in solids than they do in liquids
  than they do in gases. However, within a single
  phase of matter, the inertial property of density
  tends to be the property which has a greatest
  impact upon the speed of sound. A sound wave will
  travel faster in a less dense material than a
  more dense material. Thus, a sound wave will
  travel nearly three times faster in Helium as it
  will in air. This is mostly due to the lower mass
  of Helium particles as compared to air particles.
• The speed of a sound wave in air depends upon the
  properties of the air, namely the temperature and
  the pressure. The pressure of air (like any gas)
  will affect the mass density of the air (an
  inertial property) and the temperature will
  affect the strength of the particle interactions
  (an elastic property). At normal atmospheric
  pressure, the temperature dependence of the speed
  of a sound wave through air is approximated by
  the following equation
• v 331 m/s (0.6 m/s/C)T
• where T is the temperature of the air in degrees
  Celsius. Using this equation to determine the
  speed of a sound wave in air at a temperature of
  20 degrees Celsius yields the following solution.

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26.5 Loudness

• v 331 m/s (0.6 m/s/C)T
• v 331 m/s (0.6 m/s/C)(20 C)
• v 331 m/s 12 m/s
• v 343 m/s

While the intensity of a sound is a very
objective quantity which can be measured with
sensitive instrumentation, the loudness of a
sound is more of a subjective response which will
vary with a number of factors. The same sound
will not be perceived to have the same loudness
to all individuals. Age is one factor which
effects the human ear's response to a sound.
Quite obviously, your grandparents do not hear
like they used to. The same intensity sound would
not be perceived to have the same loudness to
them as it would to you. Furthermore, two sounds
with the same intensity but different frequencies
will not be perceived to have the same loudness.
Because of the human ear's tendency to amplify
sounds having frequencies in the range from 1000
Hz to 5000 Hz, sounds with these intensities seem
louder to the human ear. Despite the distinction
between intensity and loudness, it is safe to
state that the more intense sounds will be
perceived to be the loudest sounds.
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26.6 Natural Frequency

• Nearly all objects, when hit or struck or plucked
  or strummed or somehow disturbed, will vibrate.
  If you drop a meter stick or pencil on the floor,
  it will begin to vibrate. If you pluck a guitar
  string, it will begin to vibrate. If you blow
  over the top of a pop bottle, the air inside will
  vibrate. When each of these objects vibrate, they
  tend to vibrate at a particular frequency or a
  set of frequencies. The frequency or frequencies
  at which an object tends to vibrate with when
  hit, struck, plucked, strummed or somehow
  disturbed is known as the natural frequency of
  the object. If the amplitude of the vibrations
  are large enough and if natural frequency is
  within the human frequency range, then the
  vibrating object will produce sound waves which
  are audible.
• All objects have a natural frequency or set of
  frequencies at which they vibrate. The quality or
  timbre of the sound produced by a vibrating
  object is dependent upon the natural frequencies
  of the sound waves produced by the objects.

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• Some objects tend to vibrate at a single
  frequency and they are often said to produce a
  pure tone. A flute tends to vibrate at a single
  frequency, producing a very pure tone. Other
  objects vibrate and produce more complex waves
  with a set of frequencies which have a whole
  number mathematical relationship between them
  these are said to produce a rich sound. A tuba
  tends to vibrate at a set of frequencies which
  are mathematically related by whole number
  ratios it produces a rich tone. Still other
  objects will vibrate at a set of multiple
  frequencies which have no simple mathematical
  relationship between them. These objects are not
  musical at all and the sounds which they create
  could be described as noise. When a meter stick
  or pencil is dropped on the floor, it vibrates
  with a number of frequencies, producing a complex
  sound wave which is clanky and noisy.

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26.7 Forced Vibrations

• If you were to take a guitar string and stretch
  it to a given length and a given tightness and
  have a friend pluck it, you would hear a noise
  but the noise would not even be close in
  comparison to the loudness produced by an
  acoustic guitar. On the other hand, if the string
  is attached to the sound box of the guitar, the
  vibrating string is capable of forcing the sound
  box into vibrating at that same natural
  frequency. The sound box in turn forces air
  particles inside the box into vibrational motion
  at the same natural frequency as the string. The
  entire system (string, guitar, and enclosed air)
  begins vibrating and forces surrounding air
  particles into vibrational motion. The tendency
  of one object to force another adjoining or
  interconnected object into vibrational motion is
  referred to as a forced vibration. In the case of
  the guitar string mounted to the sound box, the
  fact that the surface area of the sound box is
  greater than the surface area of the string,
  means that more surrounding air particles will be
  forced into vibration. This causes an increase in
  the amplitude and thus loudness of the sound.

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• This same principle of a forced vibration is
  often demonstrated in a Physics classroom using a
  tuning fork. If the tuning fork is held in your
  hand and hit with a rubber mallet, a sound is
  produced as the tines of the tuning fork set
  surrounding air particles into vibrational
  motion. The sound produced by the tuning fork is
  barely audible to students in the back rows of
  the room. However, if the tuning fork is set upon
  the whiteboard panel or the glass panel of the
  overhead projector, the panel begins vibrating at
  the same natural frequency of the tuning fork.
  The tuning fork forces surrounding glass (or
  vinyl) particles into vibrational motion. The
  vibrating whiteboard or overhead projector panel
  in turn forces surrounding air particles into
  vibrational motion and the result is an increase
  in the amplitude and thus loudness of the sound.
  This principle of forced vibration explains why
  demonstration tuning forks are mounted on a sound
  box, why a commercial music box mechanism is
  mounted on a sounding board, why a guitar
  utilizes a sound box,
• and why a piano string is attached to a
  sounding
• board. A louder sound is always produced when
  
• an accompanying object of greater surface
  area
• is forced into vibration at the same natural
  frequency.

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26.8 Resonance

• Now consider a related situation which resembles
  another common Physics demonstration. Suppose
  that a tuning fork is mounted on a sound box and
  set upon the table and suppose a second tuning
  fork/sound box system having the same natural
  frequency (say 256 Hz) is placed on the table
  near the first system. Neither of the tuning
  forks is vibrating. Suppose the first tuning fork
  is struck with a rubber mallet and the tines
  begin vibrating at its natural frequency - 256
  Hz. These vibrations set its sound box and the
  air inside the sound box vibrating at the same
  natural frequency of 256 Hz. Surrounding air
  particles are set into vibrational motion at the
  same natural frequency of 256 Hz and every
  student in the classroom hears the sound. Then
  the tines of the tuning fork are grabbed to
  prevent their vibration and remarkably the sound
  of 256 Hz is still being heard. Only now the
  sound is being
• produced by the second tuning fork - the
• one which wasn't hit with the mallet. Amazing!!
• The demonstration is often repeated to
• assure that the same surprising results are
• observed. They are! What is happening?

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• In this demonstration, one tuning fork forces
  another tuning fork into vibrational motion at
  the same natural frequency. The two forks are
  connected by the surrounding air particles. As
  the air particles surrounding the first fork (and
  its connected sound box) begin vibrating, the
  pressure waves which it creates begin to impinge
  at a periodic and regular rate of 256 Hz upon the
  second tuning fork (and its connected sound box).
  The energy carried by this sound wave through the
  air is tuned to the frequency of the second
  tuning fork. Since the incoming sound waves share
  the same natural frequency as the second tuning
  fork, the tuning fork easily begins vibrating at
  its natural frequency. This is an example of
  resonance - when one object vibrating at the same
  natural frequency of a second object forces that
  second object into vibrational motion.
• The result of resonance is always a large
  vibration. Regardless of the vibrating system, if
  resonance occurs, a large vibration results.

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26.9 Interference

• 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. As mentioned in the last
  chapter, if two upward displaced pulses having
  the same shape meet up with one another while
  traveling in opposite directions along a medium,
  the medium will take on the shape of an upward
  displaced pulse with twice the amplitude of the
  two interfering pulses. This type of interference
  is known as constructive interference. If an
  upward displaced pulse and a downward displaced
  pulse having the same shape meet up with one
  another while traveling in opposite directions
  along a medium, the two pulses will cancel each
  other's effect upon the displacement of the
  medium and the medium will assume the equilibrium
  position. This type of interference is known as
  destructive interference.

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• The diagrams below show two waves - one is blue
  and the other is red - interfering in such a way
  to produce a resultant shape in a medium the
  resultant is shown in green. In two cases (on the
  left and in the middle), constructive
  interference occurs and in the third case (on the
  far right, destructive interference occurs.
• But how can sound waves which do not possess
  upward and downward displacements interfere
  constructively and destructively? Sound is a
  pressure wave which consists of compressions and
  rarefactions. As a compression passes through a
  section of a medium, it tends to pull particles
  together into a small region of space, thus
  creating a high pressure region. And as a
  rarefaction passes through a section of a medium,
  it tends to push particles apart, thus creating a
  low pressure region. The interference of sound
  waves causes the particles of the medium to
  behave in a manner that reflects the net effect
  of the two individual waves upon the
• particles.

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• The animation below shows two sound waves
  interfering constructively in order to produce
  very large oscillations in pressure at a variety
  of anti-nodal locations. Note that compressions
  are labeled with a C and rarefactions are labeled
  with an R.
• Now if two sound waves interfere at a given
  location in such a way that the compression of
  one wave meets up with the rarefaction of a
  second wave, destructive interference results.
  The net effect of a compression (which pushes
  particles together) and a rarefaction (which
  pulls particles apart) upon the particles in a
  given region of the medium is to not even cause a
  displacement of the particles. The tendency of
  the compression to push particles together is
  canceled by the tendency of the rarefactions to
  pull particles apart the particles would remain
  at their rest position as though there wasn't
  even a disturbance passing through them. This is
  a form of destructive interference.

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• Now if a particular location along the medium
  repeatedly experiences the interference of a
  compression and rarefaction followed up by the
  interference of a rarefaction and a compression,
  then the two sound waves will continually cancel
  each other and no sound is heard. The absence of
  sound is the result of the particles remaining at
  rest and behaving as though there were no
  disturbance passing through it. Amazingly, in a
  situation such as this, two sound waves would
  combine to produce no sound. As mentioned in in
  the last chapter when talking about standing
  waves, locations along the medium where
  destructive interference continually occurs are
  known as nodes.
• Two Source Sound Interference
• A popular Physics demonstration involves the
  interference of two sound waves from two
  speakers. The speakers are set approximately 1
  meter apart and produced identical tones. The two
  sound waves traveled through the air in front of
  the speakers, spreading our through the room in
  spherical fashion. A snapshot in time of the
  appearance of these waves is shown in the diagram
  on the next page.

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• In the diagram, the compressions of a wavefront
  are represented by a thick line and the
  rarefactions are represented by thin lines. These
  two waves interfere in such a manner as to
  produce locations of some loud sounds and other
  locations of no sound. Of course the loud sounds
  are heard at locations where compressions meet
  compressions or rarefactions meet rarefactions
  and the "no sound" locations appear wherever the
  compressions of one of the waves meet the
  rarefactions of the other wave. If you were to
  plug one ear and turn the other ear towards the
  place of the speakers and then slowly walk across
  the room parallel to the plane of the speakers,
  then you would encounter an amazing phenomenon.
  You would alternatively hear loud sounds as you
  approached anti-nodal locations and virtually no
  sound as you approached nodal locations.

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• Destructive interference of sound waves becomes
  an important issue in the design of concert halls
  and auditoriums. The rooms must be designed in
  such as way as to reduce the amount of
  destructive interference. Interference can occur
  as the result of sound from two speakers meeting
  at the same location as well as the result of
  sound from a speaker meeting with sound reflected
  off the walls and ceilings. If the sound arrives
  at a given location such that compressions meet
  rarefactions, then destructive interference will
  occur resulting in a reduction in the loudness of
  the sound at that location. One means of reducing
  the severity of destructive interference is by
  the design of walls, ceilings, and baffles that
  serve to absorb sound rather than reflect it.
• The destructive interference of sound waves can
  also be used advantageously in noise reduction
  systems. Ear phones have been produced which can
  be used by factory and construction workers to
  reduce the noise levels on their jobs. Such ear
  phones capture sound from the environment and use
  computer technology to produce a second sound
  wave which one-half cycle out of phase. The
  combination of these two sound waves within the
  headset will result in destructive interference
  and thus reduce a worker's exposure to loud
  noise.

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26.10 Beats

• A final application of physics to the world of
  music pertains to the topic of beats. Beats are
  the periodic and repeating fluctuations heard in
  the intensity of a sound when two sound waves of
  very similar frequencies interfere with one
  another. The diagram illustrates the wave

interference pattern resulting from two waves
(drawn in red and blue) with very similar
frequencies. A beat pattern is characterized by
a wave whose amplitude is changing at a regular
rate. Observe that the beat pattern (drawn in
green) repeatedly oscillates from zero amplitude
to a large amplitude, back to zero amplitude
throughout the pattern. Points of constructive
interference (C.I.) and destructive interference
(D.I.) are labeled on the diagram. When
constructive interference occurs between two
crests or two troughs, a loud sound is heard.
This corresponds to a peak on the beat pattern
(drawn in green).
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• When destructive interference between a crest and
  a trough occurs, no sound is heard this
  corresponds to a point of no displacement on the
  beat pattern. Since there is a clear relationship
  between the amplitude and the loudness, this beat
  pattern would be consistent with a wave which
  varies in volume at a regular rate.
• A piano tuner frequently utilizes the phenomenon
  of beats to tune a piano string. She will pluck
  the string and tap a tuning fork at the same
  time. If the two sound sources - the piano string
  and the tuning fork - produce detectable beats
  then their frequencies are not identical. She
  will then adjust the tension of the piano string
  and repeat the process until the beats can no
  longer be heard. As the piano string becomes more
  in tune with the tuning fork, the beat frequency
  will be reduced and approach 0 Hz. When beats are
  no longer heard, the piano string is tuned to the
  tuning fork that is, they play the same
  frequency. The process allows a piano tuner to
  match the strings' frequency to the frequency of
  a standardized set of tuning forks.
• Important Note Many of the previous diagrams
  represent a sound wave by a sine wave. Such a
  wave more closely resembles a transverse wave and
  may mislead people into thinking that sound is a
  transverse wave. Sound is not a transverse wave,
  but rather a longitudinal wave. Nonetheless, the
  variations in pressure with time take on the
  pattern of a sine wave and thus a sine wave is
  often used to represent the pressure-time
  features of a sound wave.