Chapter 17
Waves
Wave Motion
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Fundamental to physics (as important
as particles)
A wave is the motion of a disturbance
All waves carry energy and momentum
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
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
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The speed depends only upon the
properties of the medium through
which the disturbance travels
Example
String vibrates at 10 hz and a
snapshot. Determine wavelength,
period, amplitude, speed.
Example
Mass and length
of the string are
0.9 kg and 8 m.
What is the
speed of wave on
the string?
Wave fronts & rays
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Wave fronts – locate
crests of waves
• Ripples from a pebble
dropping in a pond
• concentric arcs
• The distance between
successive wave fronts is
the wavelength
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Rays are the radial lines
pointing out from the
source and perpendicular
to the wave fronts
Plane Wave
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Far away from the
source, the wave
fronts are nearly
parallel planes
The rays are nearly
parallel lines
A small segment of
the wave front is
approximately a
plane wave
Reflection of Waves
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Waves reflect when they hit
boundaries
• Fixed end: wave inverts upon reflection
• Free end: no inversion
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
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Two waves, a and
b, have the same
frequency and
amplitude
• Are in phase
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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
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
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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
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An antinode occurs where the standing
wave vibrates at maximum amplitude
• The distance between two antinodes is ½λ
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Distance between node and antinode λ/4
Standing Waves on a String
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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
Resonance
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Can have resonance in strings (these
are actually standing waves)
Amplitude increases
How to determine resonance
frequencies?
Standing Waves on a String,
final
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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
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ƒ1, ƒ2, ƒ3 form a harmonic series
• ƒ1 is the fundamental and also the first
harmonic
• ƒ2 is the second harmonic (1st overtone)
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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?
Example
String 80 cm long is driven with
frequency of 120 Hz when both ends
fixed. There are 4 nodes in the
middle of the string. Find speed of
wave on string?
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
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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
Standing Waves in Air Columns
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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
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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
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Closed pipe
Resonance in an Air Column
Closed at One End
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The closed end must be a node
The open end is an antinode
v
fn  n
 nƒ1
4L
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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?