Standing Waves
Consider a piece of string, fixed at both ends, into which waves have been introduced. The
configuration is rich with interference. A wave traveling toward one end of the string reflects off
the fixed end and interferes with the waves that were trailing it. Then it reflects off the other end
and interferes with them again. This is true for every wave and it repeats itself continuously.
For any given length, linear density, and tension of the string, there are certain frequencies for
which, at, at least one point on the string, the interference is always constructive. When this is
the case, the oscillations at that point (or those points) on the string are maximal and the string is
said to have standing waves in it. Again, standing waves result from the interference of the
reflected waves with the transmitted waves and with each other. A point on the string at which
the interference is always constructive is called an antinode. Any string in which standing waves
exist also has at least one point at which the interference is always destructive. Such a point on
the string does not move from its equilibrium position. Such a point on the string is called a
node.
It might seem that it would be a daunting task to determine the frequencies that result in standing
waves. Suppose you want to investigate whether a point on a string could be an antinode.
Consider an instant in time when a wave crest is at that position. You need to find the conditions
that would make it so that in the time it takes for the crest to travel to one fixed end of the string,
reflect back as a trough and arrive back at the location in question; a trough, e.g. one that was
trailing the original crest, propagates just the right distance so that it arrives at the location in
question at the same time. As illustrated in the next chapter, the analysis that yields the
frequencies of standing waves is easier than these timing considerations would suggest.
2
24
Chapter 31 Strings, Air Columns
31 Strings, Air Columns
Be careful not to jump to any conclusions about the wavelength of a standing
wave. Folks will do a nice job drawing a graph of Displacement vs. Position
Along the Medium and then interpret it incorrectly. For instance, look at the
diagram on this page. Folks see that a half wavelength fits in the string segment
and quickly write the wavelength as
1
λ = L . But this equation says that a whole
2
wavelength fits in half the length of the string. This is not at all the case. Rather
than recognizing that the fraction 1 is relevant and quickly using that fraction in
2
an expression for the wavelength, one needs to be more systematic. First write
what you see, in the form of an equation, and then solve that equation for the
wavelength. For instance, in the diagram below we see that one half a
wavelength λ fits in the length L of the string. Writing this in equation form
yields 1 λ = L . Solving this for λ yields λ = 2L .
2
One can determine the wavelengths of standing waves in a straightforward manner and obtain
the frequencies from
v = λf
where the wave speed v
is determined by the tension and linear mass density of the string. The
method depends on the boundary conditions—the conditions at the ends of the wave medium.
(The wave medium is the substance [string, air, water, etc.] through which the wave is traveling.
The wave medium is what is “waving.”) Consider the case of waves in a string. A fixed end
forces there to be a node at that end because the end of the string cannot move. (A node is a
point on the string at which the interference is always destructive, resulting in no oscillations.
An antinode is a point at which the interference is always constructive, resulting in maximal
oscillations.) A free end forces there to be an antinode at that end because at a free end the wave
reflects back on itself without phase reversal (a crest reflects as a crest and a trough reflects as a
trough) so at a free end you have one and the same part of the wave traveling in both directions
along the string. The wavelength condition for standing waves is that the wave must “fit” in the
string segment in a manner consistent with the boundary conditions. For a string of length L
fixed at both ends, we can meet the boundary conditions if half a wavelength is equal to the
length of the string.
L
Such a wave “fits” the string in the sense that whenever a zero-displacement part of the wave is
aligned with one fixed end of the string another zero-displacement part of the wave is aligned
with the other fixed end of the string.
2
25
Chapter 31 Strings, Air Columns
Since half a wavelength fits in the string segment we have:
1 λ = L
2
λ = 2L
Given the wave speed v, the frequency can be solved for as follows:
v = λf
v
f =
λ
v
f =
2L
It should be noted that despite the fact that the wave is called a standing wave and the fact that it
is typically depicted at an instant in time when an antinode on the string is at its maximum
displacement from its equilibrium position, all parts of the string (except the nodes) do oscillate
about their equilibrium position.
Note that, while the interference at the antinode, the point in the middle of the string in the case
at hand, is always as constructive as possible, that does not mean that the string at that point is
always at maximum displacement. At times, at that location, there is indeed a crest interfering
with a crest, but at other times, there is a zero displacement part of the wave interfering with a
zero-displacement part of the wave, at times a trough interfering with a trough, and at times, an
intermediate-displacement part of the wave interfering with the same intermediate-displacement
part of the wave traveling in the opposite direction. All of this corresponds to the antinode
oscillating about its equilibrium position.
2
26
Chapter 31 Strings, Air Columns
The λ = 2L wave is not the only wave that will fit in the string. It is, however, the longest
wavelength standing wave possible and hence is referred to as the fundamental. There is an
entire sequence of standing waves. They are named: the fundamental, the first overtone, the
second overtone, the third overtone, etc, in order of decreasing wavelength, and hence,
increasing frequency.
Fundamental
L
1
λ = L so λ = 2L
2
1st Overtone
λ =
L
2nd Overtone
3 λ =
2
L so λ =
L
2
3
Each successive waveform can be obtained from the preceding one by including one more node.
A wave in the series is said to be a harmonic if its frequency can be expressed as an integer times
the fundamental frequency. The value of the integer determines which harmonic (1st, 2nd, 3rd,
etc.) the wave is. The frequency of the fundamental wave is, of course, 1 times itself. The
number 1 is an integer so the fundamental is a harmonic. It is the 1st harmonic.
2
27
Chapter 31 Strings, Air Columns
Starting with the wavelengths in the series of diagrams above, we have, for the frequencies,
using v = λf which can be rearranged to read
v
f =
λ
The Fundamental
λ
= 2L
FUND
v
f
=
FUND
λ FUND
v
f
=
FUND
2L
The 1st Overtone
λ
= L
O.T.
1st
v
f
=
O.T.
1st
λ O.T.
1st
v
f
=
O.T.
1st
L
The 2nd Overtone
2
λ
= L
2nd O.T.
3
v
f
=
O.T.
2nd
λ
O.T.
2nd
v
f
=
2nd O.T.
2 L
3
3 v
f
=
O.T.
2nd
2 L
2
28
Chapter 31 Strings, Air Columns
v
Expressing the frequencies in terms of the fundamental frequency f
=
we have
FUND
2L
v
v
f
=
= 1
= 1f
FUND
FUND
2L
2L
v
v
f
=
= 2
= 2f
O.T.
1st
FUND
L
2L
3 v
v
f
=
= 3
= 3f
2nd O.T.
FUND
2 L
2L
Note that the fundamental is (as always) the 1st harmonic; the 1st overtone is the 2nd harmonic;
and the 2nd overtone is the 3rd harmonic. While it is true for the case of a string that is fixed at
both ends (the system we have been analyzing), it is not always true that the set of all overtones
plus fundamental includes all the harmonics. For instance, consider the following example:
Example 31-1
An organ pipe of length L is closed at one end and open at the other.
Given that the speed of sound in air is v , find the frequencies of the
s
fundamental and the first three overtones.
Solution
L
Fundamental
1 λ = L so λ = 4L
4
L
1st Overtone
3 λ =
4
L so λ =
L
4
3
2
29
Chapter 31 Strings, Air Columns
L
2nd Overtone
5 λ =
4
L so λ =
L
4
5
L
3rd Overtone
7 λ =
4
L so λ =
L
4
7
In the preceding sequence of diagrams, a graph of displacement vs. position along the pipe, for
an instant in time when the air molecules at an antinode are at their maximum displacement from
equilibrium, is a more abstract representation then the corresponding graph for a string. The
sound wave in air is a longitudinal wave, so, as the sound waves travel back and forth along the
length of the pipe, the air molecules oscillate back and forth (rather than up and down as in the
case of the string) about their equilibrium positions. Thus, how high up on the graph a point on
the graph is, corresponds to how far to the right (using the viewpoint from which the pipe is
depicted in the diagrams) of its equilibrium position the thin layer of air molecules, at the
corresponding position in the pipe, is. It is conventional to draw the waveform right inside the
outline of the pipe. The boundary conditions are that a closed end is a node and an open end is
an antinode.
Starting with the wavelengths in the series of diagrams above, we have, for the frequencies,
using v = λf which can be rearranged to read
s
vs
f =
λ
The Fundamental
λ
= 4L
FUND
vs
f
=
FUND
λ FUND
vs
f
=
FUND
4L
2
30
Chapter 31 Strings, Air Columns
The 1st Overtone
4
λ
= L
O.T.
1st
3
vs
f
=
O.T.
1st
λ O.T.
1st
vs
f
=
O.T.
1st
4 L
3
3 vs
f
=
O.T.
1st
4 L
The 2nd Overtone
4
λ
= L
2nd O.T.
5
vs
f
=
2nd O.T.
λ
O.T.
2nd
vs
f
=
O.T.
2nd
4 L
5
5 vs
f
=
O.T.
2nd
4 L
The 3rd Overtone
4
λ
= L
O.T.
3rd
7
vs
f
=
O.T.
3rd
λ
O.T.
3rd
vs
f
=
O.T.
3rd
4 L
7
7 vs
f
=
O.T.
3rd
4 L
2
31
Chapter 31 Strings, Air Columns
v
Expressing the frequencies in terms of the fundamental frequency
s
f
=
we have
FUND
4L
v
v
s
s
f
=
= 1
= 1f
FUND
FUND
4L
4L
3 v
v
s
s
f
=
= 3
= 3f
O.T.
1st
FUND
4 L
4L
5 v
v
s
s
f
=
= 5
= 5f
O.T.
2nd
FUND
4 L
4L
7 v
v
s
s
f
=
= 7
= 7f
O.T.
3rd
FUND
4 L
4L
Note that the frequencies of the standing waves are odd integer multiples of the fundamental
frequency. That is to say that only odd harmonics, the 1st, 3rd, 5th, etc. occur in the case of a pipe
closed at one end and open at the other.
Regarding, Waves, in a Medium that is in Contact with a 2nd Medium
Consider a violin string oscillating at its fundamental frequency, in air. For convenience of
discussion, assume the violin to be oriented so that the oscillations are up and down.
Each time the string goes up it pushes air molecules up. This results in sound waves in air. The
violin with the standing wave in it can be considered to be the “something oscillating” that is the
cause of the waves in air. Recall that the frequency of the waves is identical to the frequency of
the source. Thus, the frequency of the sound waves in air will be identical to the frequency of
the waves in the string. In general, the speed of the waves in air is different from the speed of
waves in the string. From v = λf , this means that the wavelengths will be different as well.
2
32
Chapter 32 Beats, The Doppler Effect
32 Beats, The Doppler Effect
Some people get mixed up about the Doppler Effect. They think it’s about position
rather than about velocity. (It is really about velocity.) If a single frequency sound
source is coming at you at constant speed, the pitch (frequency) you hear is higher
than the frequency of the source. How much higher depends on how fast the source
is coming at you. Folks make the mistake of thinking that the pitch gets higher as
the source approaches the receiver. No. That would be the case if the frequency
depended on how close the source was to the receiver. It doesn’t. The frequency
stays the same. The Doppler Effect is about velocity, not position. The whole time
the source is moving straight at you, it will sound like it has one single unchanging
pitch that is higher than the frequency of the source. Now duck! Once the object
passes your position and it is heading away from you it will have one single
unchanging pitch that is lower than the frequency of the source.
Beats
Consider two sound sources, in the vicinity of each other, each producing sound waves at its own
single frequency. Any point in the air-filled region of space around the sources will receive
sound waves from both the sources. The amplitude of the sound at any position in space will be
the amplitude of the sum of the displacements of the two waves at that point. This amplitude
will vary because the interference will alternate between constructive interference and
destructive interference. Suppose the two frequencies do not differ by much. Consider the
displacements at a particular point in space. Let’s start at an instant when two sound wave crests
are arriving at that point, one from each source. At that instant the waves are interfering
constructively, resulting in a large total amplitude. If your ear were at that location, you would
find the sound relatively loud. Let’s mark the passage of time by means of the shorter period, the
period of the higher-frequency waves. One period after the instant just discussed, the next crest
(call it the second crest) from the higher-frequency source is at the point in question, but the peak
of the next crest from the lower-frequency source is not there yet. Rather than a crest interfering
with a crest, we have a crest interfering with an intermediate-displacement part of the wave. The
interference is still constructive but not to the degree that it was. When the third crest from the
higher-frequency source arrives, the corresponding crest from the lower-frequency source is even
farther behind. Eventually, a crest from the higher-frequency source is arriving at the point in
question at the same time as a trough from the lower-frequency source. At that instant in time,
the interference is as destructive as it gets. If your ear were at the point in question, you would
find the sound to be inaudible or of very low volume. Then the trough from the lower-frequency
source starts “falling behind” until, eventually a crest from the higher-frequency source is
interfering with the crest preceding the corresponding crest from the lower-frequency source and
the interference is again as constructive as possible.
To a person whose ear is at a location at which waves from both sources exist, the sound gets
loud, soft, loud, soft, etc. The frequency with which the loudness pattern repeats itself is called
the beat frequency. Experimentally, we can determine the beat frequency by timing how long it
takes f