Notes for waves, J. Hedberg © 2015
8/27/15, 1:54 PM
Waves
Types of waves
Mechanical
Waves
Electromagnetic
Waves
These
waves
require a
medium. A
disturbance in the
medium propagates
through the medium
and this is called the
wave.
These are
oscillations of
electromagnetic
fields. These waves
don't require a
medium.
Matter Waves
Very small
particles have
wave-like properties.
(We'll have to save
that for later.)
Light
Radio waves
wi-fi
Water waves
Sound waves
Stadium
waves
(excited
people)
Wave phenomena are ubiquitous in the natural world. Nearly every discipline in science must be able to deal with waves
or wave-like behaviour. Consider a geologist studying earthquakes. The motions of the earth following tectonic activity
are waves. Understanding how they move through the earth's surface is paramount to dealing with these disasters. If
you're an electrical engineer, you'll need to understand electromagnetic waves thoroughly. Even economists can analyze
certain properties of the global economy in terms of wave physics.
Perhaps the most important feature of all waves, is that they do not move matter in the direction of travel, only energy is
transmitted. We'll see how this makes sense as we consider the various types of waves.
Mechanical Waves
These are the most familiar.
They require a medium.
We can use some of the physics from last term to analyze mechanical waves.
We'll arrive at some general properties of waves in the process. They can be
applied in some ways to the other types of waves.
A mechanical wave involves the motion of matter. However, as mentioned above, the material object that is
experiencing the wave motion doesn't travel in the direction of the wave motion. Some examples of mechanical waves
are sounds waves, ocean waves, and waves on a string. All of these require a medium. The medium must be an elastic
material, which means that if part of it is displaced, it will experience a restoring force that seeks to return the displaced
region back to equilibrium.
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A wave pulse
Play/Pause
The left end of the rope is being displaced in the vertical direction. This
displacement is propagated throughout the rest of the rope.
The wave form, a pulse in this case is seen to travel down the length of the rope.
In this example, the medium is the rope. We apply a disturbance to the medium by displacing part of the rope. Since the
rope is somewhat elastic, the tension in the rope seeks to restore the displaced section back to equilibrium.
A sinusoidal wave
Play/Pause
The left end of the rope is being displaced in the vertical direction. This
displacement is propagated throughout the rest of the rope. If the displacement
on the left end is continuously repeated, we observe the familiar sinusoidal
shape.
Now, instead of a single displacement, we keep oscillating one side of the rope in a sinusoidal fashion. This disturbance
is propagated down the length of the rope. We can see also that the original displacement is perpendicular to the
direction of travel of the wave. This is an important characteristic.
Play/Pause
Here's another familiar example of a mechanical wave. A spring (aka Slinky). A
compressed region propagates along the length of the spring.
This is another example of a mechanical wave. This time however, our original displacement is parallel to the direction
of travel for the wave. Still, the spring is an elastic medium, and we have a displacement of that medium which is
propagated.
Two types of mechanical waves
There are 2 general classes of
waves:
Longitudinal: The Particles move
parallel to the direction in which the
wave travels.
Play/Pause
Transverse: Particles in the medium
move perpendicular to the direction
that wave is traveling
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Imagine one small segment of the rope.
Let's look at its motion while the wave is
passing through that point.
We'll need to be able to quantify the
motion of this little blue region.
Play/Pause
Waves, a mathematical formulation
Imagine a small segment of the
rope, or spring, or water or
whatever medium. To fully describe
its motion, we'll need variables
which tell us where it is and when -essentially space and time. If we
imagine a rope, which we can say
only has two spatial dimensions,
then we just need three total
variables: x, y, and t .
y
x
We can therefore completely
describe the motion of a segment
using:
y = h(x, t),
where h is a function of space and
time.
y is simply the transverse
displacement of a small element of
the string.
Play/Pause
A basic wave
Here's a sinusoidal wave traveling in the
+x direction.
The dot shows one element of the
medium oscillating up and down,
sinusoidally.
It is important to note that in this situation, the segment of interest, represented by the little dot, is moving perpendicular
to the motion of the wave. Its motion is entirely in the y direction, though the wave is clearly moving in the x direction.
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So, let's use a sine function to describe the position of a small section of the
medium.
y(x, t) = A sin(kx − ωt)
Notice how this is an equation for the y position as a function of x and t . This equation describes the motion of the
material that the medium consists of. In this case of the wave on a string, it's a little bit of string. It moves up and down
(in the y direction) based on this equation. The two independent variables on the R.H.S are x and t . This tells us that the
motion is dependent on where the segment is, and what time value t has.
1.5
snapshot
1.0
0.5
y
Now, we have 3 variables in one
function. This means that in order to
plot this on a 2D graphs, we'll need
to keep one variable constant.
0.0
- 0.5
Depending on which one, we'll end
up with two different
representations of the same wave.
- 1.0
- 1.5
x
1.5
history
1.0
y
0.5
0.0
- 0.5
- 1.0
- 1.5
t
This is probably our first example of an equation that has three variables, 2 of which are considered independent,
meaning they can take on any value.
Amplitude
1.5
This plot shows two waves with
amplitudes which differ by a factor
of two.
1.0
0.5
y
Here we refer to the maximum
displacement of the elements away
from their equilibrium positions.
0.0
- 0.5
- 1.0
- 1.5
0
5
10
x
15
20
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Wavelength
1.5
Let's take a sine wave at time t = 0 .
Our previous formula will then be:
Now, the displacement is the same
at both ends of the wavelength
0.5
y
y(x, 0) = A sin(kx)
1.0
0.0
- 0.5
- 1.0
- 1.5
0
5
10
x
15
20
A sin(kx1 ) = A sin k(x1 + λ)
= A sin(kx1 + kλ)
This can only be true if
k=
2π
λ
We'll call k the wavenumber, which
we can see is inverse to the
wavelength.
The wavenumber k can be thought of as a spatial frequency. If k is large, then λ will be small. If the wavelength is
small, then there will be many repetitions of it in a given length. The temporal frequency is big if there many repetitions
in a given time, so this make sense by analogy.
Think of walking north-south in Manhattan as opposed to east-west. If you go north-south, you will encounter about 20
blocks in a mile. Whereas if you go east-west, there are on average only 7 blocks to a mile. Thus the spatial frequency of
blocks is greater in the north-south direction compared to the east-west. The distance between blocks would be
something like the wavelength in this analogy.
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Period and Frequency
1.5
0.5
We can perform a similar anaylisis
of a history graph, and obtain the
period of the wave. That is, how
long does it take for an element of
the string to make on full
oscillation. We'll also have what's
called the angular frequency. It's
the ω from our displacement
equation.
ω=
T
1.0
y
The wavelength and wavenumber
gave us spatial information about
the wave. What about temporal?
0.0
- 0.5
- 1.0
- 1.5
0
1
2
3
4
t(s)
2π
T
This is related to the frequency, f ,
by the following:
f =
1
ω
=
T
2π
.
The velocity of a segment.
y(x, t) = A sin(kx − ωt)
Here is our position equation for the material of the wave. If we wanted to find
the velocity of a given point, we would just need to take the time derivative of
this equation.
Play/Pause
Question:
Here is a snapshot of a traveling wave on a rope. It's moving in the −x
direction. Which of the labeled segments has the largest negative yvelocity value?
E.
All have zero y-velocity values.
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Question:
Here is a snapshot of a traveling wave on a rope. It's moving in the −x
direction. Which of the labeled segments has the largest positive yacceleration value?
E.
All have zero y-acceleration values.
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Speed of a traveling wave
Play/Pause
If we think about the crest of the
wave as it moves, its displacement
in the y axis is constant.
1.5
y(x, t) = A sin( kx − ωt )
⏟
constant
or, kx − ωt = constant
v
1.0
0.5
Let's take the time derivative of that
equation:
y
0.0
- 0.5
- 1.0
- 1.5
0
5
10
x
15
k
20
For that to be the case, the argument
of the sinusoidal term must be
constant as well.
dx
−ω = 0
dt
-ordx
ω
=v=
dt
k
rewriting in more familiar terms:
v = λf
Recall that the frequency, f , is just the inverse of the period, T . Using this, we can rearrange this expression:
v = λf =
λ
distance
⇒
T
time
We should be somewhat comforted by the fact that this is our familiar distance over time expression. (wavelegth is a
distance, period is a time. )
Note that this is just the velocity of the wave, not the velocity of the material in the wave. That velocity is given by the
derivative of the position equation:
dy
= ẏ(x, t) = v(x, t) = −Aω cos (kx − ωt)
dt
This is a critical distinction to make. The wave speed, sometimes also called c , does not change in time. dc/dt = 0 ,
whereas the transverse velocity of a given segment most certainly does change.
dẏ
= ÿ ≠ 0
dt
.
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Question:
Clicker
A longitudinal wave with an amplitude of 0.02 m moves horizontally
along a Slinky with a speed of 2 m/s. Which one of the following
statements concerning this wave is true?
a) Each particle in the Slinky moves a distance of 2 m each
second.
b) Each particle in the Slinky moves a vertical distance of 0.04 m
during each period of the wave.
c) Each particle in the Slinky moves a horizontal distance of 0.04
m during each period of the wave.
d) Each particle in the Slinky moves a vertical distance of 0.02 m
during each period of the wave.
e) Each particle in the Slinky has a wavelength of 0.04 m.
Wave speed on a real string.
What determines the speed of a traveling wave on a
stretched string?
We could show by looking at the tensions in the string
that the velocity must be determined by the tension, τ ,
and the linear density, μ.
v=
τ
‾‾
√μ
Here is a derivation of the wave speed based on the 2nd law.
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Question:
Which of the following actions would make a pulse travel faster down
a stretched string?
a) Use a heavier string of the same length, under the same tension.
b) Use a lighter string of the same length, under the same tension.
c) Move your hand up and down more quickly as you generate
the pulse.
d) Move your hand up and down a larger distance as you generate
the pulse.
e) Use a longer string of the same thickness, density, and tension.
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Question:
A thick heavy (i.e. not massless)rope is hanging from a very tall ceiling.
A person grabs the end of the rope and begins moving it back and forth
with a constant amplitude and frequency. A transverse wave moves up
the rope. Which of the following statements describing the speed of the
wave is true?
a)
b)
c)
d)
The speed of the wave decreases as it moves upward.
The speed of the wave increases as it moves upward.
The speed of the wave is constant as it moves upward.
The speed of the wave does not depend on the mass of the
rope.
e) The speed of the wave depends on its amplitude.
Example Problem:
Example Problem: Thick vs. Thin String
A wave travels along a string at speed v0 . What will the speed be if the
string is replaced by one made of the exact same material, but having twice
the radius. (The tension is the same)
Example Problem:
Example Problem: Wave Characteristics
A sinusoidal wave with an amplitude of 1.00 cm and a frequency of 100 Hz
travels at 200 m/s in the positive x-direction. At t=0s, the point x = 1.00 m is
on a crest of the wave.
a) Determine A, v, λ, k, f , ω, T, and ϕ for this wave.
b) Write the equation for the wave's displacement at it travels.
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Energy
An interesting aspect of waves is that they are a means of transferring energy,
but not matter.
Here is a frame from the earlier
animation. The vectors show the
velocity of each element of mass,
dm. At the crest, , the velocity is
zero. While, at the y = 0 point, ,
the velocity will be a maximum.
Kinetic energy is given by the
square of the velocity, thus we can
see that the K.E. of the dm element
will oscillate between a minimum
and maximum during the wave
travel.
Energy, cont'd
We should also consider the
potential energy of the element dm.
At point , the string is not
stretched at all, while at point ,
the string length is elongated as it
passes through the origin. This
change in length will change the
elastic potential energy.
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Quantify energy
The kinetic energy of a particle in motion is given by KE =
of a little section of a rope, dm, we write:
dK =
(ẏ =
dy
dt
1
mv2 . In the case
2
1
dmẏ2
2
which is the time derivative of y, or speed in the transverse direction)
Going back to our original definition for the y displacement:
y(x, t) = A sin (kx − ωt) , we can see that ẏ is just:
ẏ = −Aω cos (kx − ωt)
Therefore, our dK can be written:
dK =
1
1
dm(−Aω)2 cos 2 (kx − ωt) = μdx(−Aω)2 cos 2 (kx − ωt)
2
2
Quantify energy
If we continue, and take the derivative of dK with respect to time:
dK
1
= μvA2 ω2 cos 2 (kx − ωt)
2
dt
Now, the kinetic energy of one little element is clearly changing all the time, but
we can consider the average change in kinetic energy:
2 2
2
( dt )avg = 2 μvA ω [cos (kx − ωt)] avg
dK
1
The last term: [cos 2 (kx − ωt)] avg is equal to
rate of kinetic change:
1
, so in the end, for the average
2
2 2
( dt )avg = 4 μvA ω
dK
1
The wave also transmits elastic potential energy (since the rope is kinda
springy). This should be equal to the average kinetic energy since they are
conserved quantities. Thus, the total rate of energy transmission (aka Power) will
be twice what we figured for the kinetic energy:
Pavg =
1
μvA2 ω2
2
Sketch a graph that shows the power of a wave as a function of the linear density
of the string.
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The wave equation
This: y(x, t) = A sin(kx − ωt) was just a specific case of a wave. (A sinusoidal
traveling wave). We'll need a more general wave equation which can be used to
describe any travelling wave.
**Derive**
∂2 y
∂x 2
=
1 ∂2 y
v2 ∂t 2
Sound waves
Play/Pause
Question:
A bell is ringing inside of a sealed glass jar that is connected to a
vacuum pump. Initially, the jar is filled with air at atmospheric
pressure. What does one hear as the air is slowly removed from the jar
by the pump?
a)
b)
c)
d)
e)
The sound intensity gradually increases.
The sound intensity gradually decreases.
The sound intensity of the bell does not change.
The frequency of the sound gradually increases.
The frequency of the sound gradually decreases.
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Speed of Sound
We saw before that the speed of a wave (transverse) was equal to:
v=
τ
‾‾
√μ
In this case, τ was a type of 'elastic property' while μ would be classified as an
'inertial property'. i.e.
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
elastic proptery ‾
√ inertial property
=v
These two quantities only made sense in reference to a string, but the speed of
other types of waves can be determined by analogous considerations.
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Bulk Modulus
Δp = −B
ΔV
V
B is the Bulk modulus of a material. It tells us how the volume of given material
will change if pressure is applied to it. Steel for example, has a B = 1.6 × 1011 N/m
2 , while water, which is a bit more compressible, has a B = 2.2 × 109 N/m2
We'll use this parameter to calculate how compression waves travel in a medium
(i.e. sound waves)
Speed of Sound
Material
v (m/s)
Gases
Hydrogen (0°C) 1286
Helium (0°C)
972
Air (20°C)
343
Air (0°C)
331
Here is a table that lists the velocity
of sound in various materials.
The speed of sound is dictated by
the material properties of the
medium. Just like it was for the
string.
vsound =
Liquids
Sea water
1533
Water
1493
‾‾
‾B
√ρ
Solids
Diamond
12000
Pyrex glass
5640
Iron
5130
Aluminum
5100
Copper
3560
Gold
3240
Rubber
1600
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Example Problem:
A hammer taps on the end of a 4.00 m long metal bar at room temperature.
A microphone at the other end of the bar picks up two pulses of sound, one
that travels through the metal and one that travels through the air. The pulses
are separated in time by 9.00 ms. What is the speed of sound in this metal?
Question:
You are observing a thunderstorm. In the distance, you see a flash of
lightning. Five seconds later, you hear thunder. How far away was the
lightning flash?
a)
b)
c)
d)
e)
1 mile (1.6 km)
.5 mile (.8 km)
2 miles (3.2 km)
.25 miles (.4 km)
5 miles (8.0 km)
Traveling Sound Waves
Play/Pause
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To describe the longitudinal motion of an element of the medium, we can use a
sinusoidal function:
s(x, t) = sm cos(kx − ωt)
All of our wave parameters are still present: f , λ, ω, k, T
However, it's easier to work with pressure, p.
The volume of our element will just
be the length times its crosssectional area:
V = AΔx.
While the change in volume of this
element will be given by:
ΔV = AΔs.
We can substitute these back into
our formula for the bulk modulus:
Δp = −B
Δs
∂s
= −B
Δx
∂x
∂s
= −ksm sin(kx − ωt)
∂x
or
Δp = Bksm sin(kx − ωt)
Thus, we can see the pressure at a given location oscillates with time.
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Graphical Representation of sound waves
s
longitudinal displacement
x
We'll see plots like this a lot. Here we have a speaker producing a sound wave.
We'll plot the displacement of the elements of air as a function of position.
Understanding this plot is paramount. It looks upon first glance that an particle is oscillating up and down as the wave
propagates. This is not true. Sound is a longitudinal wave, so the displacements of the air molecules will be in the
direction of the wave motion.
Question:
A particle of dust is floating in the air approximately one half meter in
front of a speaker. The speaker is then turned on produces a constant
pure tone of 267 Hz, as shown. The sound waves produced by the
speaker travel horizontally. Which one of the following statements
correctly describes the subsequent motion of the dust particle?
dust particle
267 Hz
a) The particle of dust will oscillate left and right with a frequency
of 267 Hz.
b) The particle of dust will oscillate up and down with a
frequency of 267 Hz.
c) The particle of dust will be accelerated toward the right and
continue moving in that direction.
d) The particle of dust will move toward the right at constant
velocity.
e) The dust particle will remain motionless as it cannot be affected
by sound waves.
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Radially Propagating Waves
Here is our standard, circular wave pattern.
Play/Pause
We can approximate the circular waves as parallel wave fronts if we are far
enough away from the source. (far would mean d ≫ λ ).
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Wave fronts
plane waves with power = P
We'll consider an intensity, I , to
describe and quantify the loudness
of the waves.
v
area = a
I=
P
A
Here, P, is the rate of energy
transfer (power) and A is the area
over which we are considering.
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Imagine a sound source:
If it’s in the center of the room, the
waves will propagate outwards in a
circle.
These are two different ways of
representing the pressure waves:
a) on top, a model of the particles
b) underneath: a color gradient
where blue is high pressure and
white is low pressure.
Play/Pause
Play/Pause
Play/Pause
To find the power at some distance, r , from the source, we need to use the
surface area of a sphere at that distance:
I=
P
P
=
A
4πr 2
Example Problem:
A helium-neon laser emits 1 mW of light power into a 1.0 mm diameter laser
beam. What is the intensity of the laser beam?
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Intensity
To quantify how loud a given sound is, we'll define a term β : the sound intensity
level. It is calculated by comparing the intensity of the sound in question, I , to a
base level, the threshold of hearing: I0 .
β = (10dB) log
I
I0
The threshold of hearing I0 defined as I0 = 1.0 × 10−12 W/m2
β
The units of β are given in decibels.
If a sound has an intensity of 1.0 × 10−12 W/m2 , then we can see that it will
have an intensity level of 0 dB.
β = (10dB) log10
I0
= (10dB) log10 (1) = 0 dB
I0
A very loud sound, one that might damage your ears, can have an intensity of 10 W/m
2 . How many decibels is that?
Sound
Threshold of
Hearing
β
(dB)
I W/m2
0
1.0 × 10−12
A Whisper at
20
1m
1.0 × 10−10
Conversation
60
at 1m
1.0 × 10−6
Vacuum
Cleaner
1.0 × 10−4
80
We can see that there is wiiiide range of
intensities heard during a normal day out
and about. That is why it makes more
sense to use a log scale when describing
this phenomenon.
Home Stereo 110 0.1
Threshold of
PAIN
130 10
Example Problem:
If the sound intensity level at distance d of one trombone is β = 70 dB, what
is the sound intensity level of 99 identical trombones, all at distance d ?
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Question:
Nancy Reagan is a distance d in front of a speaker emitting sound
waves. She then moves to a position that is a distance 2d in front of the
speaker. By what percentage does the sound intensity decrease for
Nancy between the two positions?
a)
b)
c)
d)
e)
10%
25%
50%
75%
The sound intensity remains constant because it is not
dependent on the distance.
Example Problem:
The Grateful Dead and their crew built a speaker
system that was able to generate 26,400 Watts of
audio power. It was called the Wall of Sound. How
many decibels would this make at ¼ mile away
from the stage? (Assume a isotropic sound
distributioin)
Doppler
Play/Pause
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Doppler
Play/Pause
General Doppler Shift
f′=f
a)
b)
c)
d)
e)
v ± vD
v ± vS
f = emitted frequency
f ′ = detected frequency
v = speed of sound in air
vD = speed of detector
vS = speed of source
​When the motion of detector or source is toward the other, the sign on its speed
must give an upward shift in frequency. When the motion of detector or source is
away from the other, the sign on its speed must give a downward shift in
frequency.
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Question:
A child is swinging back and forth with a constant period and
amplitude. Somewhere in front of the child, a stationary horn is
emitting a constant tone of frequency f . Five points are labeled in the
drawing to indicate positions along the arc as the child swings. At
which position(s) will the child hear the lowest frequency for the sound
from the horn?
stationary
horn
1
4
2
a)
b)
c)
d)
e)
3
at 2 when moving toward 1
at 2 when moving toward 3
at 3 when moving toward 2
at 3 when moving toward 4
at both 1 and 4
A 2kHz sine wave generator is swung around in a circle with a rope of length 1m
at a speed of 100 rotations per minute. Find the highest and lowest frequencies
heard by stationary listeners out side the circle but in the plane of rotation.
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