Download sinusoidal wave

General Physics I
Spring 2011
Traveling Waves and Sound
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Mechanical Waves
• A mechanical wave is a disturbance that propagates
(travels) through a medium (solid, liquid, or gas). As the
wave propagates, particles that compose the medium
move about their equilibrium positions, but there is no
overall transport of the medium with the wave. The wave
travels at a fixed speed that depends on the properties of
the medium.
• Most waves fall into two categories: transverse and
longitudinal.
• A transverse wave is one in which the particles of the
medium move transverse, or perpendicular, to the
direction of propagation of the wave. A wave generated on
a string is a transverse wave.
• A longitudinal wave is one in which the particles of the
medium move along the same direction as the wave
travels. A sound wave is a longitudinal wave.
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Mechanical Waves
Water waves are neither purely transverse nor purely
longitudinal. Each water element describes a circle as the wave
passes it position.
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Other Types of Waves
• Electromagnetic waves are disturbances that consist of electric
and magnetic fields and travel at the speed of light. These
waves do not need a medium for propagation; they can travel in
a vacuum. Examples are microwaves and visible light.
• Matter waves are a representation of the behavior of material
objects (usually very small, such as electrons) according to the
subject of quantum mechanics. Quantum mechanics precisely
and accurately explains the behavior of nuclei, atoms, and even
large groups of atoms such as a solid material.
• A common characteristic of all waves is that they transport
energy.
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Traveling Mechanical Waves
• A mechanical wave propagates in a medium because adjacent
pieces of the medium exert forces on each other. Consider a
transverse wave pulse traveling to the right on a string. When
the leading edge of the pulse reaches a given point in the string
(see below), the tension forces acting on the tiny string piece
(called a string element) at that point produce a net force and
acceleration in the vertically upward direction. Thus, the
element moves upward. At a later time when the curvature of
the string at the position of the element is downward, the net
force is downward and so the element slows down because its
velocity is still upward.
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Traveling Mechanical Waves
• When the peak (crest) of the pulse arrives at the position of the
string element, the element cannot go any higher, so its velocity
must be zero. Because of the downward curvature of the string
at that instant, the net force is downward and so the element
will move downward immediately after the crest passes. The
element continues to move down, speeding up initially then
slowing down as the trailing edge of the pulse approaches.
When the trailing edge passes, the element remains
permanently at rest, as it was before the pulse arrived.
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Speed of Transverse Waves on a String
• If a transverse wave travels in an elastic string of mass m and
length L in which the tension is Ts, the wave speed is given by
v s tr in g
where
T
= µs ,
µ = m.
L
is the linear mass density (mass per unit length) of the string.
• We see that if the tension increases, the wave speed increases.
This makes sense because a greater tension produces greater
accelerations of string elements as they move up and down.
Thus, the pulse passes any given point on the string faster.
• If the mass per unit length increases, the wave speed
decreases. This is because a greater mass gives smaller
accelerations of string elements. Thus, the pulse moves more
slowly.
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Workbook: Chapter 15, Question 3, 4
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Question
9
Graphical Description of Waves
• Consider a wave pulse traveling to
the right along an elastic string. The
top figure shows a “snapshot” of the
wave at a single instant in time. It
shows the individual vertical
displacements (y) of the elements of
the string at their horizontal positions
(x) along the direction of wave
motion, all at the same instant in time.
It shows a “profile” of the wave. The
snapshot graph is a graph of vertical
displacement of string elements
versus horizontal position!
• Each snapshot below the top one
shows the pulse at a later time. Note
that the pulse does not change its
shape as it moves.
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Graphical Description of Waves
• Another way to graphically
represent a wave is to plot the
vertical displacement of a single
string element as a function of time.
Thus, the graph is a plot of y versus
time (t) at a single value of x. This is
precisely the motion that was
explored on pages 5 and 6. A y
versus t graph for a single particle is
called a history graph.
• The lower graph is the history graph
corresponding to the string element
at position x1 in the upper graph.
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Graphical Description of Waves
t=0s
t=1s
t=2s
t=3s
t=4s
t=5s
t=6s
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Graphical Description of Waves
• Note that the string element
reaches the maximum
displacement quickly because the
front of the pulse rises steeply, so
there is a short time interval
between the arrival of the leading
edge at x1 and the arrival of the
peak. The displacement of the
string element takes a longer time
to go back to zero because the
back of the pulse has a gentler
slope than the front.
• Remember that the history graph
is a plot of vertical displacement of
a string element versus time!
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Workbook: Chapter 15, Questions 6(a), 7, 10
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Solution
6
15
Solution
7.
16
Solution
10.
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Snapshot and History Graph Problems
• To generate a history graph given a snapshot graph:
(1) Find the time at which the leading edge of the pulse arrives
at the given position.
(2) Convert the positions of the pulse in the snapshot graph to
times using ∆t = ∆x / vwave.
• To generate a snapshot graph given a history graph:
(1) Find the position of the leading edge of the pulse at the
given time.
(2) Convert the times of the pulse in the history graph to
positions using ∆x = vwave∆t.
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Sinusoidal Waves
• If one particle of a medium is
disturbed by, e.g., applying a
force, a wave disturbance will
propagate outward from the
source particle. If the applied
force is a linear restoring force
so that the particle moves with
simple harmonic motion (SHM),
then a sinusoidal wave will
propagate. The passage of a
sinusoidal wave causes a
continuous periodic disturbance
in the medium. All the particles
of the medium where the wave
exists will move with SHM.
Waves Simulation (PhET)
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Traveling Sinusoidal Waves
• The top picture shows a history graph
for one particle of a medium in which a
sinusoidal wave is traveling. The
amplitude (A) of the SHM (i.e.,
maximum displacement from
equilibrium) is equal to the amplitude of
the wave. The period of the SHM is
equal to the period of the wave. It
follows that the frequency of the SHM
(f = 1/T) is equal to the frequency of
the wave.
• The lower picture shows a snapshot
graph of a traveling sinusoidal wave in
a medium (e.g., a string). We see that
the vertical particle displacements also
vary sinusoidally with the particle
positions. (There is a particle at each
position along the x axis.)
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Traveling Sinusoidal Waves
• The wavelength (λ) of the wave is the
distance over which the wave repeats
itself along the direction of travel.
• Since the wave moves without
changing its shape (like a rigid wire
frame), the wave will travel a distance
equal to one wavelength in the same
amount of time it takes for the wave
to repeat itself, which is one period.
(The next slide shows this more
clearly).
• Since ∆x = vx∆t , we have λ = vT or,
v = λ /T , which gives
v = λ f . (Sinusoidal waves)
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Traveling Sinusoidal Waves
v
• During one period of the wave, a crest moves
a distance equal to one wavelength along the
direction of wave motion. The particle at the
position of the original crest has completed
one cycle of SHM. (Remember that the
particles of the medium do not move in the
direction of wave motion! They simply
oscillate as the wave passes their positions.)
• Particles that are separated by a distance
equal to one wavelength oscillate exactly in
step (or in phase). They have exactly the
same vertical displacements. Particles that
are separated by a distance that is less than
one wavelength will be at different stages in
their SHM cycles at any given instant in time
and so their vertical displacements will be
different.
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Mathematical Description
• The displacement of a sinusoidal traveling wave moving along
the positive x direction is given by







2
π
x
2
π
t
y( x,t) = Acos
−
.
T
λ


The equation gives the displacement of the particle located at
position x at time instant t. The expression comes from the fact
that a sinusoidal wave exhibits sinusoidal behavior as a
function of both time t and position x. Note that a sine function
could also have been used.
• For a sinusoidal traveling wave moving in the negative x
direction, the displacement is given by







2
π
x
2
π
t
y( x,t) = Acos
+
.
T
λ


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Workbook: Chapter 15, Questions 12, 13
Textbook: Chapter 15, Problem 62
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Longitudinal Waves
• A longitudinal wave is generated
by applying a periodic force to a
particle in a medium so that the
particle oscillates with SHM with a
displacement along the same line
that neighboring particles are
located on.
• In a longitudinal wave, there are
regions where the particles are
closer to together than when the
medium is undisturbed. These
regions are called compressions.
There are also regions where the
particles are farther apart than
when the medium is undisturbed.
These regions are rarefactions.
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Sound Waves
• A sound wave is a longitudinal wave.
As a sinusoidal sound wave
propagates, the pressure in the
medium varies sinusoidally with
position. The high pressure areas
(crests) are compressions and the
low pressure areas (troughs) are
rarefactions. Note that the equilibrium
pressure is not zero.
• Sound travels in solids, liquids, and
gases. A medium is necessary for
sound waves to exist. Sound waves
generally have the greatest speeds in
solids, where the atoms/molecules
are closest together and interatomic
forces are generally strongest.
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Light Waves
• Visible light is just one type of
electromagnetic wave.
Electromagnetic waves are due to
oscillating electric and magnetic
fields. These fields can exist in
vacuum, so light does not need a
medium to propagate. (This is
fortunate; otherwise no sunlight
would reach us across the vacuum
of space.) Electromagnetic (EM)
waves are transverse waves. In
vacuum, all EM waves travel at the
speed of light, which is 3.00×108 m/s.
• The various types of
electromagnetic waves comprise
the electromagnetic spectrum,
which is shown in the picture.
Microwaves, x-rays, etc.
correspond to different
wavelength ranges. All types
have the same speed in
vacuum.
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