Nicholas J. Giordano
www.cengage.com/physics/giordano
Chapter 12
Waves
Wave Motion
• A wave is a moving disturbance that transports
energy from one place to another without
transporting matter
• Questions about waves
• What is being disturbed?
• How is it disturbed?
• The motion associated with a wave disturbance
often has a repeating form, so wave motion has
much in common with simple harmonic motion
Introduction
Waves, String Example
• One example of a wave is a disturbance on a string
• Shaking the free end creates a disturbance that moves
horizontally along the string
• A single shake creates a wave pulse
• If the end of the string is shaken up and down in simple
harmonic motion, a periodic wave is produced
Section 12.1
Waves, String Example cont.
• The disturbances are examples of waves
• Portions of the string are moving so there is kinetic
energy associated with the wave
• There is elastic potential energy in the string as it
stretches
• The wave carries this energy as it travels
• The wave does not carry matter as it travels
• Pieces of the string do not move from one end of the
string to the other
Section 12.1
Analysis of The Wave Pulse
• A single pulse
propagates to the right
• The graph (part D in the
figure shown) shows the
displacement of point D
on the string
• It is perpendicular to
the direction of
propagation
• The wave transports
energy without
transporting matter
Section 12.1
Wave Terminology
• The “thing” being disturbed by the wave is its
medium
• When the medium is a material substance, the wave
is a mechanical wave
• In transverse waves the motion of the medium is
perpendicular to the direction of the propagation of
the wave
• The string was an example
• In longitudinal waves the motion of the medium is
parallel to the direction of the propagation of the
wave
Section 12.1
Example: Longitudinal Wave
• The spring is shaken
back and forth in the
horizontal direction
• At some places the coils
are compressed
• At other places the coils
are stretched
• This motion produces a
longitudinal wave
Section 12.1
Describing Periodic Waves
• Assume a person is
shaking the string so
that the end is
undergoing simple
harmonic motion
• The crest is the
maximum positive y
displacement
• The trough is the
maximum negative y
displacement
Section 12.2
Periodic vs. Nonperiodic Waves
• Nonperiodic waves
• The wave disturbance is limited to a small region of
space
• Periodic waves
• The wave extends over a very wide region of space
• The displacement of the medium varies in a repeating
and often sinusoidal pattern
• A periodic wave involves repeating motion as a
function of both space and time
Section 12.2
The Equation of a Wave
• Assume the displacement generating the wave in the
string vibrates as a simple harmonic oscillator with
yend = A sin (2 π ƒ t)
• The string’s displacement is given by
• λ is the symbol for wavelength
• This is a mathematical description of a periodic wave
• It shows the transverse displacement y of a point on
the string as it varies with time and location
Section 12.2
More Wave Terminology
• Periodic waves have a frequency
• The frequency is related to the “repeat time”
• The period is the time that a point takes to go from a
crest to the next crest in its motion
• Then ƒ = 1 / T
• Periodic waves have an amplitude
• Wave crests have y = + A
• Wave troughs have y = - A
Section 12.2
Wavelength
• The wavelength is the
“repeat distance” of the
wave
• Start at a given value of
y
• Advance x by a
distance equal to the
wavelength and y will
be at the same value
again
Section 12.2
Periodic Wave, Summary
• Periodic waves have both a repeat time and a repeat
distance
• A periodic wave is a combination of two simple
harmonic motions
• One is a function of time
• The other is a function of space
Section 12.2
Speed of a Wave
• The mathematical description of a wave contains
frequency, wavelength and amplitude
• The speed of a wave is
• This is based on the definitions of period and
wavelength
Section 12.2
Direction of a Wave
• To determine the direction of the wave, you can focus on
the motion of a crest
• As x becomes larger, the wave has moved to the right
and the wave velocity is positive and its equation is
• The equation of a wave moving to the left and having a
negative velocity is
Section 12.2
Interpreting the Equation of a Periodic
Wave
Displacement of the
medium as a
function of location
(x) and time (t)
Frequency
Amplitude
Wavelength
Section 12.2
Waves on a String
• Waves on a string are mechanical waves
• The medium that is disturbed is the string
• For a transverse wave on a string, the speed of the
wave depends on the tension in the string and the
string’s mass per unit length
• Mass / length = μ
• Tension will be denoted as FT to keep the tension
separate from the period
• The speed of the wave is
Section 12.3
Waves on a String, cont.
• The speed of the wave is independent of the
frequency of the wave
• The frequency will be determined by how rapidly the
end of the string is shaken
• The speed of transverse waves on a string is the
same for both periodic and nonperiodic waves
Section 12.3
Sound Waves
• Sound is a mechanical
wave that can travel
through almost any
material
• Travels in solids,
liquids, and gases
• Assume a speaker is
used to generate the
waves
Section 12.3
Sound Waves, cont.
• The speaker moves back and forth in the horizontal
•
•
•
•
direction
As it moves, it collides with nearby air molecules
The x component of the velocity of the air molecules
is affected by the speaker
The displacement of the air molecules associated
with the sound wave is also along the x direction
The result is a longitudinal wave
Section 12.3
Speed of Sound Waves
• The speed of sound depends on the properties of
•
•
•
•
•
the medium
At room temperature, the speed of sound in air is
approximately 343 m/s
The speed is independent of the frequency
The speed applies to both periodic and nonperiodic
waves
Sound waves in a liquid or solid are also longitudinal
The speed of sound is generally smallest for gases
and highest for solids
Section 12.3
Waves in a Solid
• Solids can support both
longitudinal and
transverse waves
• The longitudinal waves
are considered sound
waves
• The speed of the sound
depends on the solid’s
elastic properties
Section 12.3
Speed of Sound in a Solid
• For a thin bar of material, the speed of sound is
given by
• The speed of a transverse wave is more complicated
and depends on the shear modulus and other elastic
constants
• In general, the speed of the transverse wave is
slower than the speed of longitudinal waves
Section 12.3
Transverse Waves
• Transverse waves can travel through solids
• They cannot travel through liquids or gases
• The displacements in transverse waves involve a
shearing motion
• Liquids and gases flow and there is no restoring force
to produce the oscillations necessary for a transverse
wave
Section 12.3
Electromagnetic Waves
• Electromagnetic (em) waves are not mechanical
waves
• They are electric and magnetic disturbances that can
propagate even in a vacuum
• No mechanical medium is required
• The electric and magnetic fields are always
perpendicular to the direction of propagation
• So they are transverse waves
• EM waves are classified according to their frequency
• The speed of an em wave in a vacuum is 3.00 x 108
m/s
• It is independent of the frequency of the wave
Section 12.3
Speed of Waves, Summary
• The speed of a wave depends on the properties of
the medium through which it travels
• The speed varies widely
• From slow waves on a string
• To very fast em waves
• Generally, the wave speed is independent of both
frequency and amplitude
• There are cases in light and optics where the speed
does depend on the frequency
• The speed is the same for periodic and nonperiodic
waves
Section 12.3
Water Waves
• A water wave can be generated by dropping a rock
onto the surface
• The waves propagate outward
Section 12.3
Water Waves, cont.
• The motion of the
water’s surface is both
transverse and
longitudinal
• A bug on the surface
moves up and down as
well as backward and
forward
Section 12.3
Wave Fronts: Spherical Waves
• A spherical wave
travels away from its
source in a threedimensional fashion
• The wave crests form
concentric spheres
centered on the source
• The crests are also
called wave fronts
Section 12.4
Spherical Waves, cont.
• The direction of the wave propagation is always
perpendicular to the surface of a wave front
• The direction is indicated by rays
• Each wave carries energy as it travels away from the
source
• Power measures the energy emitted by the source per
unit time
• Units of power are J/s = W
• W for Watt
Intensity
• Intensity is the power carried by the wave over a unit
area of the wave front
• SI units of intensity is W/m2
• Once a wave front is emitted, its energy remains the
same
• The intensity falls as the wave moves farther from
the source
• The area is becoming larger
Section 12.4
Intensity, cont.
• At a distance r from the source, the surface area of
the sphere is 4πr2
• The intensity is
• The intensity falls with distance as
Section 12.4
Plane Waves
• Wave fronts are not always spherical
• Another type is a plane wave
• In a perfect plane wave, each crest and trough extend over an infinite
plane in space
• The intensity is approximately constant over long propagation
distances
• Intensity is ideally independent of distance
Section 12.4
Intensity and Amplitude
• The intensity of a wave is related to its amplitude
• Spring example
• The potential energy is ½ k x2
• For a wave on a spring, the displacement is
proportional to the amplitude
• Therefore, the energy and intensity are proportional to
the square of the amplitude
Section 12.4
Superposition
• Waves generally propagate independently of one
another
• A wave can travel though a particular region of
space without affecting the motion of another wave
traveling though the same region
• This is due to the Principle of Superposition
• When two (or more) waves are present, the
displacement of the medium is equal to the sum of the
displacements of the individual waves
• The presence of one wave does not affect the
frequency, amplitude, or velocity of the other wave
Section 12.5
Constructive Interference
• Two wave pulses are
•
•
•
•
traveling toward each other
They have equal and
positive amplitudes
At C, the two waves
completely overlap and the
amplitude is twice the
amplitude of the individual
waves
The emerging pulses are
unchanged
This is an example of
constructive interference
Section 12.5
Destructive Interference
• Two pulses are traveling
•
•
•
•
toward each other
They have equal and
opposite amplitudes
At C, the two waves
completely overlap, total
displacement is zero
The emerging pulses are
unchanged
This is an example of
destructive interference
Section 12.5
Interference
• Constructive interference causes the waves to
produce a displacement that is larger than the
displacements of either of the individual waves
• Destructive interference causes the waves to
produce a displacement that is smaller than the
displacements of either of the individual waves
• In either case, the energy of each wave is contained
in the kinetic energy of the medium
• The waves can interfere, even destructively, and still
carry energy independently
Section 12.5
Interference of Periodic Waves
• The crests of the waves travel away from the initial source
• There is constructive interference where the wave crests
overlap
• There is destructive interference where a crest and trough
overlap
• The result shows an interference pattern with regions of
constructive and destructive interference
Section 12.5
Reflection
• Reflection changes the
propagation direction of
the wave
• Rays can be used to
indicate the direction of
energy flow
• The rays change direction
when a wave reflects from
the boundary of the
medium
• The wave is inverted as it
reflects from a fixed end
Section 12.6
Example: Reflection of Light
• The light wave from a
laser reflects from a
mirror
Section 12.6
Reflection – Light Ray Details
• The rays make an initial
angle of θi with a line
drawn perpendicular to
the surface
• The perpendicular
component of the wave’s
velocity reverses direction
• The parallel component of
the wave’s velocity is not
affected by the reflection
• The angle of incidence will
equal the angle of
reflection: θi = θr
Section 12.6
Reflection – Free Surface
• The end of the string is
attached to a ring that is
free to move up and down
• When the wave is
reflected, it is not inverted
• The properties of the
medium at the boundary
will determine if the
reflected wave will be
inverted or not
Section 12.6
Radar
• An application of wave
reflection is radar
• A radio wave pulse is sent
from a transmitting
antenna and reflects from
some distant object
• A portion of the reflected
wave will arrive back at
the original transmitter,
where it is detected
Section 12.6
Radar, cont.
• Radar determines the distance to the object by
measuring the time delay between the original and
reflected signals
• By using a rotating antenna, the direction of the
object can also be detected
• The amplitude of the reflected rays gives information
about the size of the object
• A larger object reflects more of the wave energy and
gives a larger signal at the detecting antenna
Section 12.6
Refraction
• If the rays follow bent paths in a medium, they are
said to be refracted
• The frequency of the wave stays the same
• It is determined by the source
• The change in direction of the wave is due to a
change in its speed
Section 12.7
Standing Waves
• Waves may travel back and forth along a string of
length L
• If the string has both ends held in fixed positions, the
displacement at both ends must be zero
• These conditions can be satisfied by a periodic wave
only for certain wavelengths
• For these wavelengths, a standing wave can be
produced
• It is called a standing wave because the outline of the
wave appears stationary
Section 12.8
Standing Waves, cont.
• The standing wave is
obtained by the
interference of two
waves traveling in
opposite directions
• The waves travel along
the string and are
reflected from the ends
Section 12.8
Standing Waves, final
• Points where the string
displacement is zero
are called nodes
• Points where the
displacement is largest
are called antinodes
• Many standing waves
may “fit” into the length
of the string
Section 12.8
Harmonics
• The longest possible wavelength corresponds to the
smallest possible frequency
• This frequency is called the fundamental
frequency, ƒ1
• The next longest wavelength is called the second
harmonic
• The pattern of wavelengths and frequencies is
Section 12.8
Harmonics, cont
• Combining the frequency and wavelength equations
gives other expressions for the frequency:
• This is for standing waves on a string with fixed ends
• The allowed standing wave frequencies are integer
multiples of the fundamental frequency
Section 12.8
Musical Tones
• Many musical instruments use strings as a vibrating
element
• Your fingers press down on the string and changes its
length
• The string vibrates with all the possible standing wave
pattern frequencies
• The pitch of note is determined by its fundamental
frequency
• Two notes whose fundamental frequencies differ by
a factor of 2 are said to be separated by an octave
Section 12.8
Seismic Waves
• Seismic waves propagate through the Earth
• Their source can be any large mechanical
disturbance such as an earthquake
• There are three types of seismic waves
Section 12.9
Types of Seismic Waves
• S waves
• S for shear
• Transverse waves
• The displacement of the solid Earth is perpendicular to
the direction of propagation
• P waves
• P for pressure
• Longitudinal sound waves
• Surface waves
• Similar to water waves but travel through the surface
of the Earth
• Seismic waves can be detected by a seismograph
Section 12.9
Structure of the Earth
• Seismic waves can help
determine the interior
structure of the Earth
• S waves do not
propagate through the
core
• So the core contains a
liquid
• Both S and P waves are
refracted
Section 12.9
Structure of the Earth, cont.
• Analysis of the waves led to the following structure:
• Inner core
• Outer core
• Mantle
• Crust
• Many characteristics of these sections also were
obtained from the study of seismic waves
Section 12.9