Waves

1
Waves

• References
• Conceptual Physics, Paul G. Hewitt, 10th edition,
  Addison Wesley publisher
• http//www.physicsclassroom.com/Class/waves/wavest
  oc.html
• http//www.physicsclassroom.com/Class/sound/soundt
  oc.html

• By Sandrine Colson-Inam, Ph.D

2
Outline

• The Nature of a Wave
• Properties of a Wave
• Behavior of Waves
• Standing Waves

3
The Nature of a Wave

• Waves are everywhere sound waves, light waves,
  radio waves, microwaves, water waves, sine waves,
  cosine waves, telephone chord waves, stadium
  waves, earthquake waves, waves on a string, and
  slinky waves.
• Water ripples form waves. The water wave
• has a crest and a through and travels from
• one location to another
• Slinky waves.

4
What is a Wave?

• A wave can be described as a disturbance that
  travels through a medium from one location to
  another location.
• When the slinky is stretched from end to end and
  is held at rest, it assumes a natural position
  known as the equilibrium or rest position.
• The act of moving the first coil of the slinky in
  a given direction and then returning it to its
  equilibrium position creates a disturbance in the
  slinky.
• A pulse is a single disturbance moving through a
  medium from one location to another location.
• The repeating and periodic disturbance which
  moves through a medium from one location to
  another is referred to as a wave.
• A medium is a substance or material which carries
  the wave.
• Waves are said to be an energy transport
  phenomenon. As a disturbance moves through a
  medium from one particle to its adjacent
  particle, energy is being transported from one
  end of the medium to the other.
• In conclusion, a wave can be described as a
  disturbance which travels through a medium,
  transporting energy from one location (its
  source) to another location without transporting
  matter. Each individual particle of the medium is
  temporarily displaced and then returns to its
  original equilibrium positioned.

5
Categories of Waves

• A transverse wave is a wave in which particles of
  the medium move in a direction perpendicular to
  the direction which the wave moves. If a slinky
  is stretched out in a horizontal direction across
  the classroom, and a pulse is introduced into the
  slinky on the left end by vibrating the first
  coil up and down, then energy will begin to be
  transported through the slinky from left to
  right. As the energy is transported from left to
  right, the individual coils of the medium will be
  displaced upwards and downwards. In this case,
  the particles of the medium move perpendicular to
  the direction which the pulse moves. This type of
  wave is a transverse wave. Transverse waves are
  always characterized by particle motion being
  perpendicular to wave motion. EX ROPE
• A longitudinal wave is a wave in which particles
  of the medium move in a direction parallel to the
  direction which the wave moves. If a slinky is
  stretched out in a horizontal direction across
  the classroom, and a pulse is introduced into the
  slinky on the left end by vibrating the first
  coil left and right, then energy will begin to be
  transported through the slinky from left to
  right. As the energy is transported from left to
  right, the individual coils of the medium will be
  displaced leftwards and rightwards. In this case,
  the particles of the medium move parallel to the
  direction which the pulse moves. This type of
  wave is a longitudinal wave. Longitudinal waves
  are always characterized by particle motion being
  parallel to wave motion. FOR EXAMPLE SOUND WAVE

6
Categories of Waves continues

• A surface wave is a wave in which particles of
  the medium undergo a circular motion. Surface
  waves are neither longitudinal nor transverse. In
  longitudinal and transverse waves, all the
  particles in the entire bulk of the medium move
  in a parallel and a perpendicular direction
  (respectively) relative to the direction of
  energy transport. In a surface wave, it is only
  the particles at the surface of the medium which
  undergo the circular motion. The motion of
  particles tend to decrease as one proceeds
  further from the surface.
• An electromagnetic wave is a wave which is
  capable of transmitting its energy through a
  vacuum (i.e., empty space). Electromagnetic waves
  are produced by the vibration of electrons within
  atoms on the Sun's surface. These waves
  subsequently travel through the vacuum of outer
  space, subsequently reaching Earth. Were it not
  for the ability of electromagnetic waves to
  travel to Earth, there would undoubtedly be no
  life on Earth. All light waves are examples of
  electromagnetic waves. Light waves are the topic
  of another unit at The Physics Classroom. While
  the basic properties and behaviors of light will
  be discussed, the detailed nature of an
  electromagnetic wave is quite complicated and
  beyond the scope of The Physics Classroom
• A mechanical wave is a wave which is not capable
  of transmitting its energy through a vacuum.
  Mechanical waves require a medium in order to
  transport their energy from one location to
  another. A sound wave is an example of a
  mechanical wave. Sound waves are incapable of
  traveling through a vacuum. Slinky waves, water
  waves, stadium waves, and telephone chord waves
  are other examples of mechanical waves each
  requires some medium in order to exist. A slinky
  wave requires the coils of the slinky a water
  wave requires water a stadium wave requires fans
  in a stadium and a telephone chord wave requires
  a telephone chord.

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Properties of Waves The Anatomy of a Transverse
Wave

• Dashed line equilibrium or rest position. The
  position of the rope if there was no disturbance.
• Crest the point on the medium which exhibits
  the maximum amount of positive or upwards
  displacement from the rest position
• Trough the point on the medium which exhibits
  the maximum amount of negative or downwards
  displacement from the rest position
• Amplitude refers to the maximum amount of
  displacement of a a particle on the medium from
  its rest position. In a sense, the amplitude is
  the distance from rest to crest.
• Wavelength of a wave is simply the length of one
  complete wave cycle. A wave has a repeating
  pattern (wave cycle). the diagram above, the
  wavelength is the distance from A to E, or the
  distance from B to G, or the distance from E to
  J, or the distance from D to I, or the distance
  from C to H. Any one of these distance
  measurements would suffice in determining the
  wavelength of this wave.

8
Properties of Waves The Anatomy of a
Longitudinal Wave
                                        \

• A compression is a point on a medium through
  which a longitudinal wave is traveling which has
  the maximum density. A region where the coils are
  spread apart, thus maximizing the distance
  between coils, is known as a rarefaction.
• A rarefaction is a point on a medium through
  which a longitudinal wave is traveling which has
  the minimum density. Points A, C and E on the
  diagram above represent compressions and points
  B, D, and F represent rarefactions.
• In the case of a longitudinal wave, a wavelength
  measurement is made by measuring the distance
  from a compression to the next compression or
  from a rarefaction to the next rarefaction. On
  the diagram above, the distance from point A to
  point C or from point B to point D would be
  representative of the wavelength.

9
Frequency and Period of a Wave

• The frequency of a wave refers to how often the
  particles of the medium vibrate when a wave
  passes through the medium. Frequency refers to
  how often something happens period refers to the
  time it takes something to happen. Frequency is a
  rate quantity period is a time quantity.
• The quantity frequency is often confused with the
  quantity period. Period refers to the time which
  it takes to do something. When an event occurs
  repeatedly, then we say that the event is
  periodic and refer to the time for the event to
  repeat itself as the period. The period of a wave
  is the time for a particle on a medium to make
  one complete vibrational cycle. Period, being a
  time, is measured in units of time such as
  seconds, hours, days or years.

10
Energy Transport and the Amplitude of a Wave

• A wave is an energy transport phenomenon which
  transports energy along a medium without
  transporting matter.
• The energy is imparted to the medium by the
  person as he/she does work upon the first coil to
  give it kinetic energy.
• In fact, a high energy pulse would likely do some
  rather noticeable work upon your hand upon
  reaching the end of the medium the last coil of
  the medium would displace you hand in the same
  direction of motion of the coil. For the same
  reasons, a high energy ocean wave does
  considerable damage to the piers along the
  shoreline when it crashes upon it.
• The amount of energy carried by a wave is related
  to the amplitude of the wave. A high energy wave
  is characterized by a high amplitude a low
  energy wave is characterized by a low amplitude.

11
The Speed of a Wave

• 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,
• EXAMPLE
• 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
• If an echo is heard one second after the holler
  and reflects off canyon walls which are a
  distance of 170 meters away, then what is the
  speed of the wave?
• 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.

12
The Wave Equation

• The diagrams at the right show several
  "snapshots" of the production of a wave within a
  rope. The motion of the disturbance along the
  medium after every one-fourth of a period is
  depicted. Observe that it takes that from the
  first to the last snapshot, the hand has made one
  complete back-and-forth motion. A period has
  elapsed. Observe that during this same amount of
  time, the disturbance has moved a distance equal
  to one complete wavelength. So in a time of one
  period, the wave has moved a distance of one
  wavelength. Combining this information with the
  equation for speed (speeddistance/time), it can
  be said that the speed of a wave is also the
  wavelength/period.
• Wave speed is dependent upon medium properties
  and independent of wave properties.

Speed Wavelength Frequency
v f  
13
Behavior of Waves Boundary Behavior

• As a wave travels through a medium, it will often
  reach the end of the medium and encounter an
  obstacle or perhaps another medium through which
  it could travel.
• The behavior of a wave (or pulse) upon reaching
  the end of a medium is referred to as boundary
  behavior. When one medium ends, another medium
  begins the interface of the two media is
  referred to as the boundary and the behavior of a
  wave at that boundary is described as its
  boundary behavior.

14
Fixed Rope

• If a pulse is introduced at the left end of the
  rope, it will travel through the rope towards the
  right end of the medium. This pulse is called the
  incident pulse since it is incident towards
  (i.e., approaching) the boundary with the pole.
  When the incident pulse reaches the boundary, two
  things occur
• A portion of the energy carried by the pulse is
  reflected and returns towards the left end of the
  rope. The disturbance which returns to the left
  after bouncing off the pole is known as the
  reflected pulse.
• A portion of the energy carried by the pulse is
  transmitted to the pole, causing the pole to
  vibrate.
• One observes the reflected pulse off the fixed
  end, there are several notable observations.
  First the reflected pulse is inverted. Other
  notable characteristics of the reflected pulse
  include
• the speed of the reflected pulse is the same as
  the speed of the incident pulse
• the wavelength of the reflected pulse is the same
  as the wavelength of the incident pulse
• the amplitude of the reflected pulse is less than
  the amplitude of the incident pulse
• Since the speed of a wave (or pulse) is dependent
  upon the medium through which it travels, two
  pulses in the same medium will have the same
  speed.

15
Free-end Rope
16
Less to More Dense Medium

• Upon reaching the boundary, the usual two
  behaviors will occur.
• A portion of the energy carried by the incident
  pulse is reflected and returns towards the left
  end of the thin rope. The disturbance which
  returns to the left after bouncing off the
  boundary is known as the reflected pulse.
• A portion of the energy carried by the incident
  pulse is transmitted into the thick rope. The
  disturbance which continues moving to the right
  is known as the transmitted pulse.
• Comparisons can also be made between the
  characteristics of the transmitted pulse and
  those of the reflected pulse. Once more there are
  several noteworthy characteristics.
• the transmitted pulse (in the more dense medium)
  is traveling slower than the reflected pulse (in
  the less dense medium)
• the transmitted pulse (in the more dense medium)
  has a smaller wavelength than the reflected pulse
  (in the less dense medium)
• the speed and the wavelength of the reflected
  pulse are the same as the speed and the
  wavelength of the incident pulse

17
More to Less Dense Medium

• Comparisons between the characteristics of the
  transmitted pulse and the reflected pulse lead to
  the following observations.
• the transmitted pulse (in the less dense medium)
  is traveling faster than the reflected pulse (in
  the more dense medium)
• the transmitted pulse (in the less dense medium)
  has a larger wavelength than the reflected pulse
  (in the more dense medium)
• the speed and the wavelength of the reflected
  pulse are the same as the speed and the
  wavelength of the incident pulse

18
Summary of Boundary Wave Behavior

• The boundary behavior of waves can be summarized
  by the following principles
• the wave speed is always greatest in the least
  dense medium,
• the wavelength is always greatest in the least
  dense medium,
• the frequency of a wave is not altered by
  crossing a boundary,
• the reflected pulse becomes inverted when a wave
  in a less dense medium is heading towards a
  boundary with a more dense medium,
• the amplitude of the incident pulse is always
  greater than the amplitude of the reflected
  pulse.
• All the observations discussed can be explained
  by the simple application of these principles.

19
Reflection, Refraction, and Diffraction

• Reflection
• If a linear object attached to an oscillator bobs
  up and down within the water, it becomes a source
  of straight waves. These straight waves have
  alternating crests and troughs. As viewed on the
  sheet of paper below the tank, the crests are the
  bright lines stretching across the paper and the
  troughs are the dark lines. These waves will
  travel through the water until they encounter an
  obstacle - such as the wall of the tank or an
  object placed within the water. The diagram at
  the right depicts a series of straight waves
  approaching a long barrier extending at an angle
  across the tank of water. The direction which
  these wavefronts (straight-line crests) are
  traveling through the water is represented by the
  blue arrow. The blue arrow is called a ray and is
  drawn perpendicular to the wavefronts. Upon
  reaching the barrier placed within the water,
  these waves bounce off the water and head in a
  different direction. The diagram below shows the
  reflected wavefronts and the reflected ray.
  Regardless of the angle at which the wavefronts
  approach the barrier, one general law of
  reflection holds true the waves will always
  reflect in such a way that the angle at which
  they approach the barrier equals the angle at
  which they reflect off the barrier. This is known
  as the law of reflection. i r

20
Reflection on Curved Surfaces
              The discussion above pertains to
the reflection of
21
Refraction

• Refraction of waves involves a change in the
  direction of waves as they pass from one medium
  to another.
• As water waves are transmitted from deep water
  into shallow water, the speed decreases, the
  wavelength decreases, and the direction changes.
• Law of refraction

22
Diffraction

• Diffraction involves a change in direction of
  waves as they pass through an opening or around a
  barrier in their path.
• The amount of diffraction (the sharpness of the
  bending) increases with increasing (longer)
  wavelength and decreases with decreasing
  wavelength. In fact, when the wavelength of the
  waves are smaller than the obstacle, no
  noticeable diffraction occurs.
• Diffraction is observed of light waves but only
  when the waves encounter obstacles with extremely
  small wavelengths (such as particles suspended in
  our atmosphere).

23
Interference of Waves - When two waves meet

• Wave interference is the phenomenon which occurs
  when two waves meet while traveling along the
  same medium.
• There are two types of wave interference
• Constructive
• Destructive

24
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 when a
  crest meets a crest but it is also observed when
  a trough meets a trough as shown in the diagram
  below.

25
Destructive Interference

• 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.
• In the situation 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 disturbance in the medium.
• If two interfering waves do not need to have
  equal amplitudes in opposite directions then
  destructive interference does not occur.

26
After Interference

• Yet waves meet, produce a net resulting shape of
  the medium, and then continue on doing what they
  were doing before the interference.

27
Principle of Superposition (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.
28
The Doppler Effect

• 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.

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.
29
Traveling Waves vs. Standing Waves

• A mechanical wave is a disturbance which is
  created by a vibrating object and subsequently
  travels through a medium from one location to
  another, transporting energy as it moves. The
  mechanism by which a mechanical wave propagates
  itself through a medium involves particle
  interaction one particle applies a push or pull
  on its adjacent neighbor, causing a displacement
  of that neighbor from the equilibrium or rest
  position. As a wave is observed traveling through
  a medium, a crest is seen moving along from
  particle to particle. This crest is followed by a
  trough which is in turn followed by the next
  crest. In fact, one would observe a distinct wave
  pattern (in the form of a sine wave) traveling
  through the medium. This sine wave pattern
  continues to move in uninterrupted fashion until
  it encounters another wave along the medium or
  until it encounters a boundary with another
  medium. This type of wave pattern which is seen
  traveling through a medium is sometimes referred
  to as a traveling wave.
• Traveling waves are observed when a wave is not
  confined to a given space along the medium. The
  most commonly observed traveling wave is an ocean
  wave.

30
Standing Wave

• It is possible however to have a wave confined to
  a given space in a medium and still produce a
  regular wave pattern which is readily discernible
  amidst the motion of the medium. For instance, if
  an elastic rope is held end to end and vibrated
  at just the right frequency, a wave pattern would
  be produced which assumes the shape of a sine
  wave and is seen to change over time. The wave
  pattern is only produced when one end of the rope
  is vibrated at just the right frequency. When the
  proper frequency is used, the interference of the
  incident wave and the reflected wave occur in
  such a manner that there are specific points
  along the medium which appear to be standing
  still. Because the observed wave pattern is
  characterized by points which appear to be
  standing still, the pattern is often called a
  standing wave pattern.

31
Formation of Standing Waves

• A standing wave pattern is a vibrational pattern
  created within a medium when the vibrational
  frequency of the source causes reflected waves
  from one end of the medium to interfere with
  incident waves from the source in such a manner
  that specific points along the medium appear to
  be standing still. Because the observed wave
  pattern is characterized by points which appear
  to be standing still, the pattern is often called
  a "standing wave pattern." Such patterns are only
  created within the medium at specific frequencies
  of vibration these frequencies are known as
  harmonic frequencies, or merely harmonics. At any
  frequency other than a harmonic frequency, the
  interference of reflected and incident waves
  results in a resulting disturbance of the medium
  which is irregular and non-repeating.
• A standing wave pattern is an interference
  phenomenon. It is formed as the result of the
  perfectly time interference of two waves passing
  through the same medium. A standing wave pattern
  is not actually a wave rather it is the pattern
  resulting from the presence of two waves
  (sometimes more) of the same frequency with
  different directions of travel within the same
  medium.

32
Standing Wave Nodes and Anti-Nodes

• One characteristic of every standing wave pattern
  is that there are points along the medium which
  appear to be standing still. These points,
  sometimes described as points of no displacement,
  are referred to as nodes. There are other points
  along the medium which undergo vibrations between
  a large positive and and large negative
  displacement. These are the points which undergo
  the maximum displacement during each vibrational
  cycle of the standing wave. In a sense, these
  points are the opposite of nodes, and so they are
  called antinodes. A standing wave pattern always
  consist of an alternating pattern of nodes and
  antinodes.

33
Harmonics and Patterns
First Harmonic Standing Wave Pattern
Second Harmonic Standing Wave Pattern

• A variety of actual wave patterns could be
  produced, with each pattern characterized by a
  distinctly different number of nodes. Such
  standing wave patterns can only be produced
  within the medium when it is vibrated at certain
  frequencies. There are several frequencies with
  which the snakey can be vibrated to produce the
  patterns. Each frequency is associated with a
  different standing wave pattern. These
  frequencies and their associated wave patterns
  are referred to as harmonics.

A pattern with three nodes and two antinodes is
referred to as the second harmonic
A pattern with two nodes and one antinode is
referred to as the first harmonic
34
QUESTION!!
Third Harmonic Standing Wave Pattern
                                                  
                                                  
                                                  
                                                  
                                                  
                                                  
                                                  
                                                  
                                                  
                                                  
                                

• How many nodes and antinodes in the third
  harmonic?

Third Harmonic Standing Wave Pattern
35
Mathematics of Standing Waves

• Consider the first harmonic standing wave pattern
  for a vibrating rope as shown below.
• The pattern for the first harmonic reveals a
  single antinode in the middle of the rope. This
  antinode position along the rope vibrates up and
  down from a maximum upward displacement from rest
  to a maximum downward displacement as shown. The
  vibration of the rope in this manner creates the
  appearance of a loop within the string.
• In comparing the standing wave pattern for the
  first harmonic with its single loop to the
  diagram of a complete wave, it is evident that
  there is only one-half of a wave stretching
  across the length of the string. That is, the
  length of the string is equal to one-half the
  length of a wave. Put in the form of the equation
  above.

36
More Harmonics
37
Sinusoidal Nature of Waves
Physical waves, or mechanical waves, form through
the vibration of a medium, be it a string, the
Earth's crust, or particles of gases and fluids.
Waves have mathematical properties that can be
analyzed to understand the motion of the wave.
A wave having a form which, if plotted, would be
the same as that of a trigonometric sine or
cosine function. The sine wave may be thought of
as the projection on a plane of the path of a
point moving around a circle at uniform speed. It
is characteristic of one-dimensional vibrations
and one-dimensional waves having no dissipation.
The sine wave is the basic function employed in
harmonic analysis. It can be shown that any
complex motion in a one-dimensional system can be
described as the superposition of sine waves
having certain amplitude and phase relationships.
The technique for determining these relationships
is known as Fourier analysis.
38
Sinusoidal Nature of Waves

• This wave pattern occurs often in nature,
  including ocean waves, sound waves, and light
  waves.
• A cosine wave is said to be "sinusoidal", because
  cos(x) sin(x p / 2), which is also a sine
  wave with a phase-shift of p/2. Because of this
  "head start", it is often said that the cosine
  function leads the sine function or the sine lags
  the cosine.
• The human ear can recognize single sine waves as
  sounding clear because sine waves are
  representations of a single frequency with no
  harmonics some sounds that approximate a pure
  sine wave are whistling, a crystal glass set to
  vibrate by running a wet finger around its rim,
  and the sound made by a tuning fork.
• To the human ear, a sound that is made up of more
  than one sine wave will either sound "noisy" or
  will have detectable harmonics this may be
  described as a different timbre.

39
Sinusoidal Nature of Waves

• A simple travelling wave with a single frequency
  is sinusoidal.
• At t 0, y A sin (2p/l x) where y is the
  displacement of the wave (longitudinal or
  transverse) at position x, A is the amplitude of
  the wave, and l is the wavelength.
• If the wave is moving to the right with
  velocity v. At time t, each part of the wave has
  moved to the right at distance vt.
• y A sin (2p/l (x vt))
• If the wave is moving left
• y A sin (2p/l (x vt))