then on. In general, while the block is oscillating, the energy
E = K + U
is partly kinetic energy
1
2
K =
v
m
and partly spring potential energy
1
2
U = k x . The amount of
2
2
each varies, but the total remains the same. At time 0, the K in E = K + U is zero since the
velocity of the block is zero. So, at time 0:
E = U
2
1
E = k x
2
max
An endpoint in the motion of the block is a particularly easy position at which to calculate the
total energy since all of it is potential energy.
As the spring contracts, pulling the block toward the wall, the speed of the block increases so, the
kinetic energy increases while the potential energy
1
2
U = kx
2
decreases because the spring
becomes less and less stretched. On its way toward the equilibrium position, the system has both
kinetic and potential energy
2
00
Chapter 28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion
E = K + U
with the kinetic energy K increasing and the potential energy U decreasing. Eventually the block
reaches the equilibrium position. For an instant, the spring is neither stretched nor compressed
and hence it has no potential energy stored in it. All the energy (the same total that we started
with) is in the form of kinetic energy,
1
2
K =
v
m
.
2
0
E = K + U
E = K
The block keeps on moving. It overshoots the equilibrium position and starts compressing the
spring. As it compresses the spring, it slows down. Kinetic energy is being converted into
spring potential energy. As the block continues to move toward the wall, the ever-the-same
value of total energy represents a combination of kinetic energy and potential energy with the
kinetic energy decreasing and the potential energy increasing. Eventually, at its closest point of
approach to the wall, its maximum displacement in the –x direction from its equilibrium position,
at its turning point, the block, just for an instant has a velocity of zero. At that instant, the kinetic
energy is zero and the potential energy is at its maximum value:
0
E = K + U
E = U
Then the block starts moving out away from the wall. Its kinetic energy increases as its potential
energy decreases until it again arrives at the equilibrium position. At that point, by definition,
the spring is neither stretched nor compressed so the potential energy is zero. All the energy is in
the form of kinetic energy. Because of its inertia, the block continues past the equilibrium
position, stretching the spring and slowing down as the kinetic energy decreases while, at the
same rate, the potential energy increases. Eventually, the block is at its starting point, again just
for an instant, at rest, with no kinetic energy. The total energy is the same total as it has been
throughout the oscillatory motion. At that instant, the total energy is all in the form of potential
energy. The conversion of energy, back and forth between the kinetic energy of the block and
the potential energy stored in the spring, repeats itself over and over again as long as the block
continues to oscillate (with—and this is indeed an idealization—no loss of mechanical energy).
A similar description, in terms of energy, can be given for the motion of an ideal (no air
resistance, completely unstretchable string) simple pendulum. The potential energy, in the case
of the simple pendulum, is in the form of gravitational potential energy U =
gy
m
rather than
spring potential energy. The one value of total energy that the pendulum has throughout its
oscillations is all potential energy at the endpoints of the oscillations, all kinetic energy at the
midpoint, and a mix of potential and kinetic energy at locations in between.
2
01
Chapter 29 Waves: Characteristics, Types, Energy
29 Waves: Characteristics, Types, Energy
Consider a long taut horizontal string of great length. Suppose one end is in the hand of a person
and the other is fixed to an immobile object. Now suppose that the person moves her hand up
and down. The person causes her hand, and her end of the string, to oscillate up and down. To
discuss what happens, we, in our mind, consider the string to consist of a large number of very
short string segments. It is important to keep in mind that the force of tension of a string
segment exerted on any object, including another segment of the string, is directed away from
the object along the string segment that is exerting the force. (The following discussion and
diagrams are intentionally oversimplified. The discussion does correctly give the gross idea of
how oscillations at one end of a taut string can cause a pattern to move along the length of the
string despite the fact that the individual bits of string are essentially doing nothing more than
moving up and down.
The person is holding one end of the first segment. She first moves her hand upward.
This tilts the first segment so that the force of tension that it is exerting on the second segment
has an upward component.
This, in turn, tilts the second segment so that its force of tension on the third segment now has an
upward component. The process continues with the 3rd segment, the 4th segment, etc.
2
02
Chapter 29 Waves: Characteristics, Types, Energy
After reaching the top of the oscillation, the person starts moving her hand downward. She
moves the left end of the first segment downward, but by this time, the first four segments have
an upward velocity. Due to their inertia, they continue to move upward. The downward pull of
the first segment on the left end of the second segment causes it to slow down, come to rest,
and eventually start moving downward. Inertia plays a huge role in wave propagation. “To
propagate” means “to go” or “to travel.” Waves propagate through a medium.
2
03
Chapter 29 Waves: Characteristics, Types, Energy
2
04
Chapter 29 Waves: Characteristics, Types, Energy
Crest
Trough
Each very short segment of the string undergoes oscillatory motion like that of the hand, but for
any given section, the motion is delayed relative to the motion of the neighboring segment that is
closer to the hand. The net effect of all these string segments oscillating up and down, each with
the same frequency but slightly out of synchronization with its nearest neighbor, is to create a
disturbance in the string. Without the disturbance, the string would just remain on the original
horizontal line. The disturbance moves along the length of the string, away from the hand. The
disturbance is called a wave. An observer, looking at the string from the side sees crests and
troughs of the disturbance, moving along the length of the string, away from the hand. Despite
appearances, no material is moving along the length of the string, just a disturbance. The illusion
that actual material is moving along the string can be explained by the timing with which the
individual segments move up and down, each about its own equilibrium position, the position it
was in before the person started making waves.
Wave Characteristics
In our pictorial model above, we depicted a hand that was oscillating but not undergoing simple
harmonic motion. If the oscillations that are causing the wave do conform to simple harmonic
motion, then each string segment making up the string will experience simple harmonic motion
(up and down). When individual segments making up the string are each undergoing simple
harmonic motion, the wave pattern is said to vary sinusoidally in both time and space. We can
tell that it varies sinusoidally in space because a graph of the displacement y, the distance that a
given point on the string is above its equilibrium position, versus x, how far from the end of the
string the point on the string is; for all points on the string; is sinusoidal.
2
05
Chapter 29 Waves: Characteristics, Types, Energy
y [meters]
A BIT OF STRING’S DISPLACEMENT ABOVE OR BELOW THAT BIT’S
EQUILIBRIUM POSITION VS. THE POSITION OF THE BIT OF STRING
ALONG THE LENGTH OF THE STRING
0
x [meters]
We say that the wave varies sinusoidally with time because, for any point along the length of the
string, a graph of the displacement of that point from its equilibrium position vs. time is
sinusoidal:
y [meters]
A BIT OF STRING’S DISPLACEMENT ABOVE OR BELOW THAT BIT’S
EQUILIBRIUM POSITION VS. TIME
0
t [seconds]
There are a number of ways of characterizing the wave-on-a-string system. You could probably
come up with a rather complete list yourself: the rate at which the oscillations are occurring, how
long it takes for a given tiny segment of the string to complete one oscillation, how big the
oscillations are, the smallest length of the unique pattern which repeats itself in space, and the
speed at which the wave pattern travels along the length of the string. Physicists have, of course,
given names to the various quantities, in accordance with that important lowest level of scientific
activity—naming and categorizing the various characteristics of that aspect of the natural world
which is under study. Here are the names:
2
06
Chapter 29 Waves: Characteristics, Types, Energy
Amplitude
Any particle of a string with waves traveling through it undergoes oscillations. Such a particle
goes away from its equilibrium position until it reaches its maximum displacement from its
equilibrium position. Then it heads back toward its equilibrium position and then passes right
through the equilibrium position on its way to its maximum displacement from equilibrium on
the other side of its equilibrium position. Then it heads back toward the equilibrium position and
passes through it again. This motion repeats itself continually as long as the waves are traveling
through the location of the particle of the string in question. The maximum displacement of any
particle along the length of the string, from that point’s equilibrium position, is called the
amplitude y
of the wave.
max
The amplitude can be annotated on both of the two kinds of graphs given above (Displacement
vs. Position, and Displacement vs. Time). Here we annotate it on the Displacement vs. Position
graph:
y [meters]
DISPLACEMENT VS. POSITION
Amplitude ymax
0
Peak-to-Peak Amplitude
x [meters]
The peak-to-peak amplitude, a quantity that is often easier to measure than the amplitude itself,
has also been annotated on the graph. It should be obvious that the peak-to-peak amplitude is
twice the amplitude.
2
07
Chapter 29 Waves: Characteristics, Types, Energy
Period
The amount of time that it takes any one particle along the length of the string to complete one
oscillation is called the period T. Note that the period is completely determined by the source of
the waves. The time it takes for the source of the waves to complete one oscillation is equal to
the time it takes for any particle of the string to complete one oscillation. That time is the period
of the wave. The period, being an amount of time, can only be annotated on the Displacement
vs. Time graph (not on the Displacement vs. Position Along the String graph).
DISPLACEMENT VS. TIME
y [meters]
Period T
0
t [seconds]
Frequency
The frequency f is the number-of-oscillations-per-second that any particle along the length of
the string undergoes. It is the oscillation rate. Since it is the number-of-oscillations-per-second
and the period is the number-of-seconds-per-oscillation, the frequency f is simply the reciprocal
1
of the period T: f =
.
T
Amplitude, period, and frequency are quantities that you learned about in your study of
oscillations. Here, they characterize the oscillations of a point on a string. Despite the fact that
the string as a whole is undergoing wave motion, the fact that the point itself, any point along the
length of the string, is simply oscillating, means that the definitions of amplitude, period, and
frequency are the same as the definitions given in the chapter on oscillations. Thus, our
discussion of amplitude, period, and frequency represents a review. Now, however, it is time to
move on to something new, a quantity that does not apply to simple harmonic motion but does
apply to waves.
2
08
Chapter 29 Waves: Characteristics, Types, Energy
Wavelength
The distance over which the wave pattern repeats itself once, is called the wavelength λ of the
wave. Because the wavelength is a distance measured along the length of the string, it can be
annotated on the Displacement vs. Position Along the String graph (but not on the Displacement
vs. Time graph):
DISPLACEMENT VS. POSITON
y [meters]
Wavelength λ
0
x [meters]
Wave Velocity
The wave velocity is the speed and direction with which the wave pattern is traveling. (It is
NOT the speed with which the particles making up the string are traveling in their up and down
motion.) The direction part is straightforward, the wave propagates along the length of the
string, away from the cause (something oscillating) of the wave. The wave speed (the constant
speed with which the wave propagates) can be expressed in terms of other quantities that we
have just discussed.
2
09
Chapter 29 Waves: Characteristics, Types, Energy
To get at the wave speed, what we need to do is to correlate the up-and-down motion of a point
on the string, with the motion of the wave pattern moving along the string. Consider the
following Displacement vs. Position graph for a wave traveling to the right. In the diagram, I
have shaded in one cycle of the wave, marked off a distance of one wavelength, and drawn a dot
at a point on the string whose motion we shall keep track of.
y [meters]
Wavelength λ
0
x [meters]
Now, let’s allow some time to elapse, just enough time for the wave to move over one quarter of
a wavelength.
y [meters]
Wavelength λ
0
x [meters]
In that time we note that the point on the string marked by the dot has moved from its
equilibrium position to its maximum displacement from equilibrium position. As the wave has
moved over one quarter of a wavelength, the point on the string has completed one quarter of an
oscillation.
2
10
Chapter 29 Waves: Characteristics, Types, Energy
Let’s allow the same amount of time to elapse again, the time it takes for the wave to move over
one quarter of a wavelength.
y [meters]
Wavelength λ
0
x [meters]
At this point, the wave has moved a total of a half a wavelength over to the right, and the point
on the string marked by the dot has moved from its equilibrium position up to its position of
maximum positive displacement and back to its equilibrium position; that is to say, it has
completed half of an oscillation.
Let’s let the same amount of time elapse again, enough time for the wave pattern to move over
another quarter of a wavelength.
y [meters]
Wavelength λ
0
The wave has moved over a total distance of three quarters of a wavelength and the point on the
string that is marked with a dot has moved on to its maximum negative (downward)
displacement from equilibrium meaning that it has completed three quarters of an oscillation.
2
11
Chapter 29 Waves: Characteristics, Types, Energy
Now we let the same amount of time elapse once more, the time it takes for the wave to move
over one quarter of a wavelength.
y [meters]
Wavelength λ
0
At this point, the wave has moved over a distance equal to one wavelength and the point on the
string marked by a dot has completed one oscillation. It is that point of the string whose motion
we have been keeping track of that gives us a handle on the time. The amount of time that it
takes for the point on the string to complete one oscillation is, by definition, the period of the
wave. Now we know that the wave moves a distance of one wavelength λ in a time interval
equal to one period T. For something moving with a constant speed (zero acceleration), the
speed is simply the distance traveled during a specified time interval divided by the duration of
that time interval. So, we have, for the wave speed v :
λ
v =
(29-1)
T
One typically sees the formula for the wave speed expressed as
v = λf
(29-2)
1
where the relation f =
between frequency and period has been used to eliminate the period.
T
Equation 29-2 (v = λf ) suggests that the wave speed depends on the frequency and the
wavelength. This is not at all the case. Indeed, as far as the wavelength is concerned, it is the
other way round—the oscillator that is causing the waves determines the frequency, and the
corresponding wavelength depends on the wave speed. The wave speed is predetermined by the
characteristics of the string—how taut it is, and how much mass i