Chapter 16
Vibrations and Waves
Vibrations/Oscillations
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Object at the end of a spring
Tuning fork
Pendulum
String of a violin
Atoms in a crystal
Source of a wave
Hooke’s Law

Fs = - k x
• Fs is the spring force
• k is the spring constant

It is a measure of the stiffness of the spring
• A large k indicates a stiff spring and a small k indicates
a soft spring
• x is the displacement of the object from its
equilibrium position

x = 0 at the equilibrium position
• The negative sign indicates that the force is
always directed opposite to the displacement
Hooke’s Law Force

The force always acts toward the
equilibrium position
• It is called the restoring force

The direction of the restoring force is
such that the object is being either
pushed or pulled toward the
equilibrium position
Hooke’s Law Applied to a
Spring – Mass System

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When x is positive
(to the right), F is
negative (to the
left)
When x = 0 (at
equilibrium), F is 0
When x is negative
(to the left), F is
positive (to the
right)
Motion of the Spring-Mass
System
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Assume the object is initially pulled to a
distance A and released from rest
As the object moves toward the
equilibrium position, F and a decrease, but
v increases
At x = 0, F and a are zero, but v is a
maximum
The object’s momentum causes it to
overshoot the equilibrium position
Motion of the Spring-Mass
System, cont
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The force and acceleration start to
increase in the opposite direction and
velocity decreases
The motion momentarily comes to a
stop at x = - A
It then accelerates back toward the
equilibrium position
The motion continues indefinitely
Simple Harmonic Motion

Motion that occurs when the net
force along the direction of motion
obeys Hooke’s Law
• The force is proportional to the
displacement and always directed
toward the equilibrium position

The motion of a spring mass system
is an example of Simple Harmonic
Motion
Simple Harmonic Motion, cont.


Not all periodic motion over the
same path can be considered Simple
Harmonic motion
To be Simple Harmonic motion, the
force needs to obey Hooke’s Law
Amplitude

Amplitude, A
• The amplitude is the maximum position
of the object relative to the equilibrium
position
• In the absence of friction, an object in
simple harmonic motion will oscillate
between the positions x = ±A
Period and Frequency

The period, T, is the time that it takes for
the object to complete one complete cycle
of motion
• From x = A to x = - A and back to x = A

The frequency, ƒ, is the number of
complete cycles or vibrations per unit time
•ƒ=1/T
• Frequency is the reciprocal of the period
• Unit: Hertz, 1 Hz = 1 cycle/s
Acceleration of an Object in
Simple Harmonic Motion
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Newton’s second law will relate force and
acceleration
The force is given by Hooke’s Law
F=-kx=ma
• a = -kx / m

The acceleration is a function of position
• Acceleration is not constant and therefore the
uniformly accelerated motion equation cannot
be applied
Elastic Potential Energy

A compressed spring has potential
energy
• The compressed spring, when allowed
to expand, can apply a force to an
object
• The potential energy of the spring can
be transformed into kinetic energy of
the object
Elastic Potential Energy, cont

The energy stored in a stretched or
compressed spring or other elastic
material is called elastic potential energy
• PEs = ½kx2

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The energy is stored only when the spring
is stretched or compressed
Elastic potential energy can be added to
the statements of Conservation of Energy
and Work-Energy
Energy in a Spring Mass
System
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A block sliding on a
frictionless system
collides with a light
spring
The block attaches
to the spring
The system oscillates
in Simple Harmonic
Motion
Energy Transformations
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The block is moving on a frictionless surface
The total mechanical energy of the system is the
kinetic energy of the block
Energy Transformations, 2
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The spring is partially compressed
The energy is shared between kinetic energy and
elastic potential energy
The total mechanical energy is the sum of the
kinetic energy and the elastic potential energy
Energy Transformations, 3
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The spring is now fully compressed
The block momentarily stops
The total mechanical energy is stored as
elastic potential energy of the spring
Energy Transformations, 4
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When the block leaves the spring, the total
mechanical energy is in the kinetic energy of the
block
The spring force is conservative and the total
energy of the system remains constant
Graphical Representation of
Motion
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When x is a maximum
or minimum, velocity
is zero
When x is zero, the
velocity is a maximum
When x is a maximum
in the positive
direction, a is a
maximum in the
negative direction
Period of a SHM

It can be shown that
m
Period  T  2
k
Simple Pendulum
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The simple
pendulum is
another example
of simple
harmonic motion
The force is the
component of the
weight tangent to
the path of
motion
• Ft = - m g sin θ
Simple Pendulum, cont
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In general, the motion of a pendulum
is not simple harmonic
However, for small angles, it
becomes simple harmonic
• In general, angles < 15° are small
enough
l
Period  T  2
g
Period of Simple Pendulum
l
Period  T  2
g
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This shows that the period is
independent of the amplitude
The period depends on the length of
the pendulum and the acceleration of
gravity at the location of the
pendulum
Measure g
Example
A body with a mass of 5.0 kg is
suspended by a spring, which
stretches 10 cm when the body is
attached. The body is then pulled
downward an additional 5 cm and
released. Find k, T, f, E, vmax and
amax.
Example
Period of a simple pendulum is 2s on
Earth. What would be the period of
the same pendulum on the moon?
(gmoon=1.67 m/s2)
Forced Vibrations and
Resonance
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Shaking (vibration) as specific
frequency
Pushing child on swing
glass
Tacoma Narrows Bridge
! Every object has its natural
frequencies
Wave Motion
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A wave is the motion of a disturbance
Mechanical waves require
• Some source of disturbance
• A medium that can be disturbed
• Some physical connection between or
mechanism though which adjacent portions
of the medium influence each other

All waves carry energy and momentum
Types of Waves – Traveling
Waves
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Flip one end of a
long rope that is
under tension and
fixed at one end
The pulse travels
to the right with a
definite speed
A disturbance of
this type is called a
traveling wave
Types of Waves – Transverse
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In a transverse wave, each element that is
disturbed moves in a direction
perpendicular to the wave motion
Types of Waves – Longitudinal
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In a longitudinal wave, the elements of
the medium undergo displacements
parallel to the motion of the wave
A longitudinal wave is also called a
compression wave
Other Types of Waves
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Waves may be a combination of
transverse and longitudinal
Mainly consider periodic sinusoidal
waves
Waveform – A Picture of a
Wave
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The brown curve is a
“snapshot” of the
wave at some
instant in time
The blue curve is
later in time
The high points are
crests of the wave
The low points are
troughs of the wave
Longitudinal Wave Represented as
a Sine Curve
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A longitudinal wave can also be represented as a
sine curve
Compressions correspond to crests and stretches
correspond to troughs
Also called density waves or pressure waves
Amplitude and Wavelength
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Amplitude is the
maximum
displacement of string
above the equilibrium
position
Wavelength, λ, is the
distance between two
successive points that
behave identically
Speed of a Wave
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v=ƒλ
• Is derived from the basic speed
equation of distance/time
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This is a general equation that can
be applied to many types of waves
Speed of a Wave on a String
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The speed of wave on a stretched
rope under some tension, F
v 
F
m
where m 
m
L
 m is called the linear density

The speed depends only upon the
properties of the medium through
which the disturbance travels
Example
Mass and length
of the string are
0.9 kg and 8 m.
What is the
speed of wave on
the string?
Superposition Principle
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Two traveling waves can meet and pass
through each other without being
destroyed or even altered
Waves obey the Superposition Principle
• If two or more traveling waves are moving
through a medium, the resulting wave is found
by adding together the displacements of the
individual waves point by point
• Actually only true for waves with small
amplitudes
Constructive Interference

Two waves, a and
b, have the same
frequency and
amplitude
• Are in phase

The combined
wave, c, has the
same frequency
and a greater
amplitude
Constructive Interference in a
String
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Two pulses are traveling in opposite directions
The net displacement when they overlap is the
sum of the displacements of the pulses
Note that the pulses are unchanged after the
interference
Destructive Interference
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Two waves, a and b,
have the same
amplitude and
frequency
They are 180° out of
phase
When they combine,
the waveforms cancel
Destructive Interference in a
String
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Two pulses are traveling in opposite directions
The net displacement when they overlap is
decreased since the displacements of the pulses
subtract
Note that the pulses are unchanged after the
interference
Chapter 17
Sound
Producing a Sound Wave
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Sound waves are longitudinal waves
traveling through a medium
A tuning fork can be used as an example
of producing a sound wave
Using a Tuning Fork to Produce
a Sound Wave
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A tuning fork will produce a
pure musical note
As the tines vibrate, they
disturb the air near them
As the tine swings to the
right, it forces the air
molecules near it closer
together
This produces a high density
area in the air
• This is an area of compression
Using a Tuning Fork, cont.
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
As the tine moves
toward the left, the air
molecules to the right
of the tine spread out
This produces an area
of low density
• This area is called a
rarefaction
Using a Tuning Fork, final
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
As the tuning fork continues to vibrate, a
succession of compressions and rarefactions
spread out from the fork
A sinusoidal curve can be used to represent the
longitudinal wave
• Crests correspond to compressions and troughs to
rarefactions
Categories of Sound Waves
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Audible waves
• Lay within the normal range of hearing of the
human ear
• Normally between 20 Hz to 20,000 Hz
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Infrasonic waves
• Frequencies are below the audible range
• Earthquakes are an example
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Ultrasonic waves
• Frequencies are above the audible range
• Dog whistles are an example
Applications of Ultrasound
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Can be used to produce images of
small objects
Widely used as a diagnostic and
treatment tool in medicine
• Ultrasounds to observe babies in the womb
• Cavitron Ultrasonic Surgical Aspirator (CUSA) used to
surgically remove brain tumors

Ultrasonic ranging unit for cameras
Speed of Sound, General
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The speed of sound is higher in solids
than in gases
The speed is slower in liquids than in
solids
Speed of Sound in Air
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331 m/s is the speed of sound at 0°C
and 1 atm
Changes with temperature
vT  331  0.6T (in m/s)
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T in °C
At 20 °C, 343 m/s
In other substances
in He: 1000 m/s
in Water: 1500 m/s
in Al: 5000 m/s
Intensity of Sound Waves
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The average intensity of a wave is the rate
at which the energy flows through a unit
area, A, oriented perpendicular to the
direction of travel of the wave
The rate of energy transfer is the power
Units are W/m2
Various Intensities of Sound
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Threshold of hearing
• Faintest sound most humans can hear
• About 1 x 10-12 W/m2

Threshold of pain
• Loudest sound most humans can tolerate
• About 1 W/m2

The ear is a very sensitive detector of
sound waves
• It can detect pressure fluctuations as small as
about 3 parts in 1010
Intensity Level of Sound Waves
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The sensation of loudness is
logarithmic in the human hear
β is the intensity level or the decibel
level of the sound
I
  10 log
Io
Io = 1 x 10-12 W/m2 is the threshold
of hearing
Various Intensity Levels
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Threshold of hearing is 0 dB
Threshold of pain is 120 dB
Jet airplanes are about 150 dB
Table of next slide lists intensity
levels of various sounds
• Multiplying a given intensity by 10 adds
10 dB to the intensity level
Example
Quite automobile: 10-7W/m2
1000 automobiles:?
Standing Waves
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When a traveling wave reflects back
on itself, it creates traveling waves in
both directions
The wave and its reflection interfere
according to the superposition
principle
With exactly the right frequency, the
wave will appear to stand still
• This is called a standing wave
Standing Waves, cont

A node occurs where the two traveling
waves have the same magnitude of
displacement, but the displacements are
in opposite directions
• Net displacement is zero at that point
• The distance between two nodes is ½λ

An antinode occurs where the standing
wave vibrates at maximum amplitude
Standing Waves on a String

Nodes must occur at the ends of the string
because these points are fixed
Standing Waves, cont.
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The pink arrows
indicate the direction
of motion of the parts
of the string
All points on the string
oscillate together
vertically with the
same frequency, but
different points have
different amplitudes of
motion
Standing Waves on a String,
final

The lowest
frequency of
vibration (b) is
called the
fundamental
frequency
nn
2L
L
 n 
2
n
v nv
fn 

 nf1
n 2 L
Standing Waves on a String –
Frequencies

ƒ1, ƒ2, ƒ3 form a harmonic series
• ƒ1 is the fundamental and also the first
harmonic
• ƒ2 is the second harmonic (1st overtone)

Waves in the string that are not in
the harmonic series are quickly
damped out
• In effect, when the string is disturbed, it
“selects” the standing wave frequencies
Example
A guitar has 0.6 m long string. Wave
speed on the string is 420 m/s. What
are the frequencies of the first few
harmonics?
Standing Waves in Air Columns


If one end of the air column is
closed, a node must exist at this end
since the movement of the air is
restricted
If the end is open, the elements of
the air have complete freedom of
movement and an antinode exists
Tube Open at Both Ends
Resonance in Air Column Open
at Both Ends

In a pipe open at both ends, the
natural frequency of vibration forms
a series whose harmonics are equal
to integral multiples of the
fundamental frequency
v
ƒn  n
 nƒ1
2L
n  1, 2, 3,
Tube Closed at One End

Closed pipe
Resonance in an Air Column
Closed at One End


The closed end must be a node
The open end is an antinode
v
fn  n
 nƒ1
4L

n  1, 3, 5,
There are no even multiples of the
fundamental harmonic
Example
An open organ pipe has a fundamental
frequency of 660 Hz at 0 C and 1
atm.
a.
Frequency of 2nd overtone?
b.
Fundamental at 20 C?
c.
Replacing air with He?
Beats
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Beats are alternations in loudness, due to
interference
Waves have slightly different frequencies and the
time between constructive and destructive
interference alternates
The beat frequency equals the difference in
frequency between the two sources:
ƒb  ƒ2  ƒ1
Doppler Effect

A Doppler effect is experienced
whenever there is relative motion
between a source of waves and an
observer.
• When the source and the observer are
moving toward each other, the observer
hears a higher frequency
• When the source and the observer are
moving away from each other, the
observer hears a lower frequency
Doppler Effect, General Case


Both the source and the observer could be
moving
v  vo
fo  f s (
)
v  vs
Use positive values of vo if observer
moving toward source
• Frequency appears higher

Use positive values of vs if source moving
away from observer
• Frequency appears lower
Example
Fs = 300 Hz and v=300m/s
a.
Observer moves 30m/s away from
source.
b.
Source moves 30 m/s toward
observer.
c.
Both moving
Police radar
“Redshift” of distant galaxies

Shock Waves


A shock wave
results when the
source velocity
exceeds the
speed of the
wave itself
The circles
represent the
wave fronts
emitted by the
source
Shock Waves, final


Shock waves carry
energy
concentrated on
the surface of the
cone, with
correspondingly
great pressure
variations
A jet produces a
shock wave seen
as a fog