Chapter 15
Wave Motion
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Units of Chapter 15
Characteristics of Wave Motion
Types of Waves: Transverse and Longitudinal
Energy Transported by Waves
Mathematical Representation of a Traveling
Wave
The Wave Equation
The Principle of Superposition
Reflection and Transmission
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Units of Chapter 15
Interference
Standing Waves; Resonance
Refraction
Diffraction
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15-1 Characteristics of Wave Motion
All types of traveling waves transport energy.
Study of a single wave
pulse shows that it is
begun with a vibration
and is transmitted
through internal forces in
the medium.
Continuous waves start
with vibrations, too. If the
vibration is SHM, then the
wave will be sinusoidal.
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15-1 Characteristics of Wave Motion
Wave characteristics:
Amplitude, A
Wavelength,
Frequency, f and period, T
Wave velocity,
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15-2 Types of Waves: Transverse
and Longitudinal
The motion of particles in a wave can be either
perpendicular to the wave direction (transverse)
or parallel to it (longitudinal).
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15-2 Types of Waves: Transverse
and Longitudinal
Sound waves are longitudinal waves:
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15-2 Types of Waves: Transverse
and Longitudinal
The velocity of a transverse wave on a
cord is given by:
As expected, the
velocity increases
when the tension
increases, and
decreases when
the mass
increases.
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15-2 Types of Waves: Transverse
and Longitudinal
Example 15-2: Pulse on a wire.
An 80.0-m-long, 2.10-mm-diameter
copper wire is stretched between two
poles. A bird lands at the center point of
the wire, sending a small wave pulse out
in both directions. The pulses reflect at
the ends and arrive back at the birds
location 0.750 seconds after it landed.
Determine the tension in the wire.
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15-2 Types of Waves: Transverse
and Longitudinal
The velocity of a longitudinal wave depends
on the elastic restoring force of the medium
and on the mass density.
or
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15-2 Types of Waves: Transverse
and Longitudinal
Example 15-3: Echolocation.
Echolocation is a form of sensory perception
used by animals such as bats, toothed
whales, and dolphins. The animal emits a
pulse of sound (a longitudinal wave) which,
after reflection from objects, returns and is
detected by the animal. Echolocation waves
can have frequencies of about 100,000 Hz.
(a) Estimate the wavelength of a sea animals
echolocation wave. (b) If an obstacle is 100
m from the animal, how long after the animal
emits a wave is its reflection detected?
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15-2 Types of Waves: Transverse
and Longitudinal
Earthquakes produce both longitudinal and
transverse waves. Both types can travel through
solid material, but only longitudinal waves can
propagate through a fluidin the transverse
direction, a fluid has no restoring force.
Surface waves are waves that travel along the
boundary between two media.
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15-3 Energy Transported by Waves
By looking at the
energy of a particle of
matter in the medium
of a wave, we find:
Then, assuming the entire medium has the same
density, we find:
Therefore, the intensity is proportional to the
square of the frequency and to the square of the
amplitude.
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15-3 Energy Transported by Waves
If a wave is able to spread out threedimensionally from its source, and the medium is
uniform, the wave is spherical.
Just from geometrical
considerations, as long as
the power output is
constant, we see:
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15-3 Energy Transported by Waves.
Example 15-4: Earthquake intensity.
The intensity of an earthquake P wave
traveling through the Earth and detected
100 km from the source is 1.0 x 106 W/m2.
What is the intensity of that wave if
detected 400 km from the source?
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15-4 Mathematical Representation of
a Traveling Wave
Suppose the shape of a wave is given
by:
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15-4 Mathematical Representation of
a Traveling Wave
After a time t, the wave crest has traveled a
distance vt, so we write:
Or:
with
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15-4 Mathematical Representation of
a Traveling Wave
Example 15-5: A traveling wave.
The left-hand end of a long horizontal stretched cord
oscillates transversely in SHM with frequency f = 250 Hz
and amplitude 2.6 cm. The cord is under a tension of
140 N and has a linear density = 0.12 kg/m. At t = 0,
the end of the cord has an upward displacement of 1.6
cm and is falling. Determine (a) the wavelength of
waves produced and (b) the equation for the traveling
wave.
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15-5 The Wave Equation
Look at a segment of string under tension:
Newtons second law gives:
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15-5 The Wave Equation
Assuming small angles, and taking the limit
x 0, gives (after some manipulation):
This is the one-dimensional wave
equation; it is a linear second-order
partial differential equation in x and t.
Its solutions are sinusoidal waves.
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15-6 The Principle of Superposition
Superposition: The
displacement at any
point is the vector sum
of the displacements of
all waves passing
through that point at that
instant.
Fouriers theorem: Any
complex periodic wave
can be written as the
sum of sinusoidal waves
of different amplitudes,
frequencies, and phases.
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15-6 The Principle of Superposition
Conceptual Example 15-7:
Making a square wave.
At t = 0, three waves are
given by D1 = A cos kx, D2 =
-1/3A cos 3kx, and D3 = 1/5A
cos 5kx, where A = 1.0 m
and k = 10 m-1. Plot the
sum of the three waves
from x = -0.4 m to +0.4 m.
(These three waves are
the first three Fourier
components of a square
wave.)
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15-7 Reflection and Transmission
A wave reaching the end
of its medium, but where
the medium is still free
to move, will be reflected
(b), and its reflection will
be upright.
A wave hitting an obstacle will be
reflected (a), and its reflection will be
inverted.
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15-7 Reflection and Transmission
A wave encountering a denser medium will be
partly reflected and partly transmitted; if the
wave speed is less in the denser medium, the
wavelength will be shorter.
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15-7 Reflection and Transmission
Two- or three-dimensional waves can be
represented by wave fronts, which are curves
of surfaces where all the waves have the same
phase.
Lines perpendicular to
the wave fronts are
called rays; they point in
the direction of
propagation of the wave.
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15-7 Reflection and Transmission
The law of reflection: the angle of incidence
equals the angle of reflection.
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15-8 Interference
The superposition principle says that when two waves
pass through the same point, the displacement is the
arithmetic sum of the individual displacements.
In the figure below, (a) exhibits destructive interference
and (b) exhibits constructive interference.
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15-8 Interference
These graphs show the sum of two waves. In
(a) they add constructively; in (b) they add
destructively; and in (c) they add partially
destructively.
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15-9 Standing Waves; Resonance
Standing waves occur
when both ends of a
string are fixed. In that
case, only waves which
are motionless at the
ends of the string can
persist. There are nodes,
where the amplitude is
always zero, and
antinodes, where the
amplitude varies from
zero to the maximum
value.
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15-9 Standing Waves; Resonance
The frequencies of the
standing waves on a
particular string are
called resonant
frequencies.
They are also referred to
as the fundamental and
harmonics.
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15-9 Standing Waves; Resonance
The wavelengths and frequencies of standing
waves are:
and
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15-9 Standing Waves; Resonance
Example 15-8: Piano string.
A piano string is 1.10 m long and has a
mass of 9.00 g. (a) How much tension
must the string be under if it is to vibrate
at a fundamental frequency of 131 Hz? (b)
What are the frequencies of the first four
harmonics?
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15-9 Standing Waves; Resonance
Example 15-9: Wave forms.
Two waves traveling in
opposite directions on a string
fixed at x = 0 are described by
the functions
D1 = (0.20 m)sin(2.0x 4.0t) and
D2 = (0.20m)sin(2.0x + 4.0t)
(where x is in m, t is in s), and
they produce a standing wave
pattern. Determine (a) the
function for the standing wave,
(b) the maximum amplitude at x
= 0.45 m, (c) where the other
end is fixed (x > 0), (d) the
maximum amplitude, and
where it occurs.
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15-10 Refraction
If the wave enters a medium where the wave
speed is different, it will be refractedits wave
fronts and rays will change direction.
We can calculate the angle of
refraction, which depends on
both wave speeds:
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15-10 Refraction
The law of refraction works both waysa wave
going from a slower medium to a faster one
would follow the red line in the other direction.
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15-10 Refraction
Example 15-10: Refraction of an earthquake
wave.
An earthquake P wave passes across a
boundary in rock where its velocity increases
from 6.5 km/s to 8.0 km/s. If it strikes this
boundary at 30, what is the angle of
refraction?
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15-11 Diffraction
When waves encounter
an obstacle, they bend
around it, leaving a
shadow region. This is
called diffraction.
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15-11 Diffraction
The amount of diffraction depends on the size of
the obstacle compared to the wavelength. If the
obstacle is much smaller than the wavelength,
the wave is barely affected (a). If the object is
comparable to, or larger than, the wavelength,
diffraction is much more significant (b, c, d).
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Summary of Chapter 15
Vibrating objects are sources of waves, which
may be either pulses or continuous.
Wavelength: distance between successive
crests
Frequency: number of crests that pass a given
point per unit time
Amplitude: maximum height of crest
Wave velocity:
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Summary of Chapter 15
Transverse wave: oscillations perpendicular to
direction of wave motion
Longitudinal wave: oscillations parallel to
direction of wave motion
Intensity: energy per unit time crossing unit
area (W/m2):
Angle of reflection is equal to angle of
incidence
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Summary of Chapter 15
When two waves pass through the same region
of space, they interfere. Interference may be
either constructive or destructive.
Standing waves can be produced on a string
with both ends fixed. The waves that persist are
at the resonant frequencies.
Nodes occur where there is no motion;
antinodes where the amplitude is maximum.
Waves refract when entering a medium of
different wave speed, and diffract around
obstacles.
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