Chapter 14 Lecture
Essential University Physics
Richard Wolfson
2nd Edition
Wave Motion
© 2012 Pearson Education, Inc.
Slide 14-1
In this lecture you’ll learn
• To explain waves as traveling
disturbances that transport
energy but not matter
• To describe waves
quantitatively in terms of
frequency, period, wavelength,
and amplitude
• To describe specific types of
waves
– Waves on strings
– Sound waves
• To describe interference,
reflection, and standing waves
• To describe the Doppler effect
and shock waves
© 2012 Pearson Education, Inc.
Slide 14-2
What’s a Wave?
• A wave is a traveling disturbance that transports
energy but not matter.
– Mechanical waves are disturbances of a material
medium.
• The medium moves briefly as the wave goes by, but the
medium itself isn’t transported any distance.
• The wave propagates as the disturbance of the medium is
communicated to adjacent parts of the medium.
– Electromagnetic waves, including light, do not require a
medium.
• Nevertheless, they share many of the properties of
mechanical waves.
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Slide 14-3
Longitudinal and Transverse Waves
• In a longitudinal wave, the
disturbance is parallel to the
wave motion.
Longitudinal wave on a
mass-spring system:
• In a transverse wave,
the disturbance is
perpendicular to the wave
motion.
Transverse wave on a
mass-spring system:
• Some waves, like surface
waves on water, involve both
longitudinal and transverse
motions.
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Slide 14-4
Properties of Waves
• Wavelength  is the distance
over which a wave repeats in
space.
• Period T is the time for a
complete cycle of the wave at a
fixed position:
– Frequency f = 1/T
• Amplitude A is the peak value of
the wave disturbance.
• Wave speed is the rate at which
the wave propagates:
v = /T = f
• Wave equation:
2 y 1 2 y
 2 2
2
x
v t
© 2012 Pearson Education, Inc.
Slide 14-5
Simple Harmonic Waves
• A simple harmonic wave has a sinusoidal shape.
– A simple harmonic wave is described by a sinusoidal
function of space and time:
y  x t   A  kx  t 
• y measures the wave disturbance at position x and time t.
• k = 2p / is the wave number, a measure of the rate at
which the wave varies in space.
•  = 2pf = 2p /T is the angular frequency, a measure of the
rate at which the wave varies in time.
• The wave speed is v = f =  /k.
These two waves have the same
speed. How do their wavenumbers,
angular frequencies, and periods
compare?
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Slide 14-6
Waves on Strings
• On strings, fibers, long springs, cables, wires, etc.,
tension provides the restoring force that helps
transverse waves propagate.
– Newton’s second law gives
mv 2 2 R v 2
2 F 

 2 v 2
R
R
where F is the tension and 
is the mass per unit length.
– The speed of such waves is
v=
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F
m
Slide 14-7
Wave Power and Intensity
• The power carried by a wave is proportional to the wave
speed and to the square of the wave amplitude.
– Details depend on the type of wave; for waves on a string, the
average power is P = 1 mw 2 A2 v.
2
• Wave intensity is the power per unit
area.
– In a plane wave, the intensity remains
constant.
• The plane wave is a good
approximation to real waves far
from their source.
– A spherical wave spreads in three
dimensions, so its intensity drops as
the inverse square of the distance
P
P
from its source:
I=
© 2012 Pearson Education, Inc.
A
=
4p r
2
Slide 14-8
Sound
• Sound waves are longitudinal mechanical waves that
propagate through gases, liquids, and solids.
– Sound waves in air involve small
changes in air pressure and density,
associated with back-and-forth motion
of the air as the wave passes.
– Sound intensity levels are measured
in decibels, a logarithmic unit based
on comparison with a reference
intensity I0= 10–12 W/m2: b = 10 log I I 0 .
(
)
– The human ear
responds to a
broad range of
sound intensities
and frequencies.
© 2012 Pearson Education, Inc.
Slide 14-9
Wave Interference
• Unlike particles, two waves can be in the same place at
the same time.
• When they are, they interfere.
– In most cases, the waves superpose, or simply add.
• When wave crests coincide, the interference is constructive.
• When crests coincide with troughs, the interference is
destructive.
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Slide 14-10
Interference Phenomena
• When waves of slightly different
frequencies interfere, they
alternate between constructive
and destructive interference.
– This results in beats at the difference
of their frequencies.
• Interference from two closely
spaced sources results in patterns
of high- and low-amplitude waves.
– The photo shows such an
interference pattern with water waves.
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Slide 14-11
Wave Reflection
• Waves reflect at an interface
with a different medium.
– The outgoing wave interferes with
the incoming wave.
– The reflected wave is inverted,
depending on properties of the
second medium.
– The diagram shows waves on a
string reflecting at clamped and
free ends.
– More generally, a wave is partially
reflected and partially transmitted
at an interface between different
media.
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Slide 14-12
Partial Reflection and Refraction
• Partial reflection of light
occurs at an interface
between air and glass.
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• Waves striking an interface
at an oblique angle undergo
refraction, a change in the
direction of propagation.
Slide 14-13
Standing Waves
• Waves on a confined medium reflect at both ends.
– The result is standing waves that oscillate but don’t propagate.
– The length of the medium restricts the allowed wavelengths and
frequencies to discrete values.
On a string
clamped at both
ends, the string
length must be an
integer multiple of
a half-wavelength:
L = m/2,
with m an integer.
© 2012 Pearson Education, Inc.
On a string
clamped at one
end, the string
length must be an
odd integer
multiple of a halfwavelength:
L = m/4,
with m odd.
Slide 14-14
Standing Waves in Musical Instruments
• Stringed instruments are analogous to the strings of the
previous slide: The string length determines the allowed
wavelengths and, together with the wave speed, the allowed
frequencies.
• Wind instruments are analogous, with sound waves in their
air columns.
– Wind instruments are typically open at one end or both.
© 2012 Pearson Education, Inc.
Slide 14-15
The Doppler effect
• When a wave source moves through the wave medium, a
stationary observer experiences a shift in wavelength and
frequency.
– The frequency increases for an approaching source.
– The frequency decreases for a receding source.
– The shifted frequencies are given by f   f
1  u v  .
• Here u is the source speed and v is the wave speed.
• A similar effect occurs for a moving observer, but there’s no wavelength
shift.
• The Doppler effect for light is similar but slightly different because light
has no medium. The formula above applies only for speeds much less
than light.
© 2012 Pearson Education, Inc.
Slide 14-16
Shock Waves
• Shock waves occur when a wave source moves through
the medium at a speed u greater than the wave speed v.
– The ratio u/v is called the Mach number.
– Mach angle  is defined by sin = u/v.
– Examples include sonic booms from aircraft, wakes of boats,
and astrophysical bodies moving through interplanetary and
interstellar gas.
© 2012 Pearson Education, Inc.
Slide 14-17
Summary
• A wave is a traveling disturbance that carries energy but not matter.
– Mechanical waves involve the disturbance of a material medium.
• These include sound waves.
– Electromagnetic waves, including light, have no medium.
– Simple harmonic waves
are sinusoidal in shape.
y xt A
kx  t
– The speed of a wave follows from its frequency and wavelength or from
its angular frequency and wavenumber: v = f =  /k.
– Important wave phenomena include
•
•
•
•
•
Reflection and refraction
Interference
Standing waves
The Doppler effect
Shock waves
© 2012 Pearson Education, Inc.
Slide 14-18