Previously in Lesson 4, it was
mentioned that when an object is forced into resonance
vibrations at one of its natural frequencies, it vibrates in a manner such that a
standing wave pattern is formed within the object. Whether it is a guitar sting, a Chladni plate, or the air column enclosed within a trombone,
the vibrating medium vibrates in such a way that a standing wave pattern
results. Each natural frequency that an object or instrument produces has its
own characteristic vibrational mode or standing wave pattern. These patterns
are only created within the object or instrument at specific frequencies of
vibration; these frequencies are known as harmonic frequencies, or merely harmonics. At any frequency other than a harmonic frequency, the resulting
disturbance of the medium is irregular and non-repeating. For musical
instruments and other objects that vibrate in regular and periodic fashion, the
harmonic frequencies are related to each other by simple whole number ratios.
This is part of the reason why such instruments sound pleasant. We will see in this part of Lesson 4 why these whole number ratios
exist for a musical instrument.
First, consider a guitar string vibrating at its natural
frequency or harmonic frequency. Because the ends of the string are attached
and fixed in place to the guitar's structure (the bridge at one end and the
frets at the other), the ends of the string are unable to move. Subsequently,
these ends become nodes - points of no displacement. In between these two nodes
at the end of the string, there must be at least one antinode. The most
fundamental harmonic for a guitar string is the harmonic associated with a
standing wave having only one antinode positioned between the two nodes on the
end of the string. This would be the harmonic with the longest wavelength and the
lowest frequency. The lowest frequency produced by any particular instrument is
known as the fundamental frequency. The fundamental
frequency is also called the first harmonic of the
instrument. The diagram at the right shows the first harmonic of a guitar
string. If you analyze the wave pattern in
the guitar string for this harmonic, you will notice that there is not quite
one complete wave within the pattern. A complete
wave starts at the rest position, rises to a crest, returns to rest, drops to
a trough, and finally returns to the rest position before starting its next
cycle. (Caution: the use of the
words crest and trough to describe the pattern are only used to help identify
the length of a repeating wave cycle. A standing wave pattern is not actually a
wave, but rather a pattern of a wave. Thus, it does not consist of crests and
troughs, but rather nodes and antinodes. The pattern is the result of the interference
of two waves to produce these nodes and antinodes.) In this pattern, there is only
one-half of a wave within the length of the string. This is the case for the
first harmonic or fundamental frequency of a guitar string. The diagram below
depicts this length-wavelength relationship for the fundamental frequency of a
guitar string.
The second harmonic of a guitar string is produced by adding one more
node between the ends of the guitar string. And of course, if a node is added
to the pattern, then an antinode must be added as well in order to maintain an
alternating pattern of nodes and antinodes. In order to create a regular and
repeating pattern, that node must be located midway between the ends of the
guitar string. This additional node gives the second harmonic a total of three
nodes and two antinodes. The standing wave pattern for the second harmonic is
shown at the right. A careful investigation of the pattern reveals that there
is exactly one full wave within the length of the guitar string. For this
reason, the length of the string is equal to the length of the wave.
The third harmonic of a guitar string is produced by adding
two nodes between the ends of the guitar string. And of course, if two nodes
are added to the pattern, then two antinodes must be added as well in order to
maintain an alternating pattern of nodes and antinodes. In order to create a
regular and repeating pattern for this harmonic, the two additional nodes must
be evenly spaced between the ends of the guitar string. This places them at the
one-third mark and the two-thirds mark along the string. These additional nodes
give the third harmonic a total of four nodes and three antinodes. The standing
wave pattern for the third harmonic is shown at the right. A careful
investigation of the pattern reveals that there is more than one full wave
within the length of the guitar string. In fact, there are three-halves of a
wave within the length of the guitar string. For this reason, the length of the
string is equal to three-halves the length of the wave. The diagram below
depicts this length-wavelength relationship for the fundamental frequency of a
guitar string.
After a discussion of the first three harmonics, a pattern
can be recognized. Each harmonic results in an additional node and antinode,
and an additional half of a wave within the string. If the number of waves in a
string is known, then an equation relating the wavelength of the standing wave
pattern to the length of the string can be algebraically derived.
This information is summarized in the table below.
Harmonic # |
# of Waves in String |
# of Nodes |
# of Anti- nodes |
Length- Wavelength Relationship |
1 |
1/2 |
2 |
1 |
Wavelength = (2/1)*L |
2 |
1 or 2/2 |
3 |
2 |
Wavelength = (2/2)*L |
3 |
3/2 |
4 |
3 |
Wavelength = (2/3)*L |
4 |
2 or 4/2 |
5 |
4 |
Wavelength = (2/4)*L |
5 |
5/2 |
6 |
5 |
Wavelength = (2/5)*L |
The above discussion develops the mathematical relationship between the length
of a guitar string and the wavelength of the standing wave patterns for the
various harmonics that could be established within the string. Now these length-wavelength
relationships will be used to develop relationships for the ratio of the
wavelengths and the ratio of the frequencies for the various harmonics played
by a string instrument (such as a guitar string).
Consider an 80-cm long guitar string that has a fundamental
frequency (1st harmonic) of 400 Hz. For the first harmonic, the wavelength of
the wave pattern would be two times the length of the string (see table above); thus, the wavelength is 160 cm or 1.60 m. The speed of the
standing wave can now be determined from the wavelength and the frequency. The
speed of the standing wave is
speed = frequency • wavelength
speed = 400 Hz • 1.6 m
speed = 640 m/s
This speed of 640 m/s corresponds to the speed of any wave
within the guitar string. Since the speed of a wave is dependent upon the
properties of the medium (and not upon the properties of the wave),
every wave will have the same speed in this string regardless of its frequency
and its wavelength. So the standing wave pattern associated with the second
harmonic, third harmonic, fourth harmonic, etc. will also have this speed of
640 m/s. A change in frequency or wavelength will NOT cause a change in speed.
Using the table above, the wavelength
of the second harmonic (denoted by the symbol λ2) would be 0.8 m (the same as the length of the
string). The speed of the standing wave pattern (denoted by the symbol v) is still 640 m/s. Now the wave equation can be used to determine the
frequency of the second harmonic (denoted by the symbol f2).
speed = frequency • wavelength
frequency =
speed/wavelength
f2 = v / λ2
f2 = (640 m/s)/(0.8 m)
f2 = 800 Hz
This same process can be repeated for the third harmonic.
Using the table above, the wavelength of the third harmonic (denoted
by the symbol λ3) would be 0.533
m (two-thirds of the length of the string). The speed of the standing wave
pattern (denoted by the symbol v) is still 640 m/s. Now the
wave equation can be used to determine the frequency of the third harmonic
(denoted by the symbol f3).
speed = frequency • wavelength
frequency =
speed/wavelength
f3 = v / λ3
f3 = (640 m/s)/(0.533 m)
f3 = 1200 Hz
Now if you have been following along, you will have
recognized a pattern. The frequency of the second harmonic is two times the
frequency of the first harmonic. The frequency of the third harmonic is three
times the frequency of the first harmonic. The frequency of the nth harmonic
(where n represents the harmonic # of any of the harmonics) is n times the
frequency of the first harmonic. In equation form, this can be written as
fn = n • f1
The inverse of this pattern exists for the wavelength values
of the various harmonics. The wavelength of the second harmonic is one-half
(1/2) the wavelength of the first harmonic. The wavelength of the third
harmonic is one-third (1/3) the wavelength of the first harmonic. And the
wavelength of the nth harmonic is one-nth (1/n) the wavelength of the first
harmonic. In equation form, this can be written as
λn = (1/n) •
λ1
These relationships between wavelengths and frequencies of
the various harmonics for a guitar string are summarized in the table below.
Harmonic # |
Frequency (Hz) |
Wavelength (m) |
Speed (m/s) |
fn / f1 |
λn / λ1 |
1 |
400 |
1.60 |
640 |
1 |
1/1 |
2 |
800 |
0.800 |
640 |
2 |
1/2 |
3 |
1200 |
0.533 |
640 |
3 |
1/3 |
4 |
1600 |
0.400 |
640 |
4 |
1/4 |
5 |
2000 |
0.320 |
640 |
5 |
1/5 |
n |
n * 400 |
(2/n)*(0.800) |
640 |
n |
1/n |
The table above demonstrates that the individual frequencies
in the set of natural frequencies produced by a guitar string are related to
each other by whole number ratios. For instance, the first and
second harmonics have a 2:1 frequency ratio; the second
and the third harmonics have a 3:2 frequency ratio; the third
and the fourth harmonics have a 4:3 frequency ratio; and the
fifth and the fourth harmonic have a 5:4 frequency ratio. When the
guitar is played, the string, sound box and surrounding air vibrate at a set of
frequencies to produce a wave with a mixture of harmonics. The exact
composition of that mixture determines the timbre or quality of sound
that is heard. If there is only a single harmonic sounding out in the mixture
(in which case, it wouldn't be a mixture), then the sound is rather
pure-sounding. On the other hand, if there are a variety of frequencies
sounding out in the mixture, then the timbre of the sound is rather rich in
quality.
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1. Anna Litical cuts short sections
of PVC pipe into different lengths and mounts them in putty on the table. The
PVC pipes form closed-end air columns that sound out at different frequencies
when she blows over the top of them. The actual frequency of vibration is inversely
proportional to the wavelength of the sound; and thus, the frequency of
vibration is inversely proportional to the length of air inside the tubes.
Express your understanding of this resonance phenomenon by filling in the
following table.
The speed of wave is not dependent
upon wave properties such as wavelength and frequency. Thus, the speed of the
sound wave is 340 m/s for each of the four pipes.
For Pipe C, the frequency can be
determined from knowledge of the speed and the wavelength using the wave
equation: v = f • λ where λ is the wavelength. First, rearrange the
equation. Then substitute and solve as shown below.
f
= v / λ = (340 m/s) / (0.64 m) = 531 Hz
For all four pipes, the length of
the air column inside the pipe is one-fourth the wavelength of the wave. This
is evident when looking at the length - wavelength relationships for Pipes A
and B. Put in equation form: L = 0.25 • λ where λ is the wavelength.
For Pipe C:
L
= 0.25 • λ = 0.25 • (0.64 m) = 0.16 m
For Pipe D, the determination of
the wavelength demands that the L = 0.25 • λ equation be rearranged.
λ = 4 • L = 4 • 0.20 m = 0.80 m
2. In a rare moment of artistic brilliance, a Physics teacher pulls out
his violin bow and strokes a square metal plate to produce vibrations within
the plate. Often times, he places salt upon the plate and observes the standing
wave patterns established in the plate as it vibrates. Amazingly, the salt is
aligned along the locations of the plate that are not vibrating and far from
the locations of maximum vibration. The two most common standing wave patterns
are illustrated at the right. Compare the wavelength of pattern A to the
wavelength of pattern B. Suppose that the fundamental frequency of vibration is
nearly 1200 Hz. Estimate the frequency of vibration of the plate when it
vibrates in the second, third and fourth harmonics.
The frequencies of the various
harmonics are multiples of the frequency of the first harmonic. Each harmonic
frequency (fn) is given
by the equation fn = n • f1 where n is
the harmonic number and f1 is
the frequency of the first harmonic.
f2 = 2
• f1 = 2400 Hz
f3 = 3
• f1 = 3600 Hz
f4 = 4
• f1 = 4800 Hz
3. When a tennis racket strikes a tennis ball, the racket
begins to vibrate. There is a set of selected frequencies at which the racket
will tend to vibrate. Each frequency in the set is characterized by a
particular standing wave pattern. The diagrams below show the three of the more
common standing wave patterns for the vibrations of a tennis racket.
a. Compare the wavelength of pattern A to the wavelength of
pattern B. Make your comparison both qualitative and quantitative. Repeat for
pattern C.
The wavelength of A is bigger than
B which is bigger than C.
In
A, there is 1/4-th of a wave in the racket. In B, there is 3/4-ths of a wave in
the racket. In C, there is 4/4-ths of a wave in the racket. Thus, the wave in
Pattern A is 3 times the length of Wave in Pattern B and 4 times the length of
wave in Pattern C.
b. Compare the frequency of pattern A to the frequency of
pattern B. Make your comparison both qualitative and quantitative. Repeat for
pattern C.
The frequency of C is bigger than
B which is bigger than A.
In
A, there is 1/4-th of a wave in the racket. In B, there is 3/4-ths of a wave in
the racket. In C, there is 4/4-ths of a wave in the racket. Frequency and
wavelength are inversely related. The longer the wavelength, the lower the
frequency. Thus, wave B is 3 times the frequency of Wave A and wave C is 4
times the frequency of wave A.
c. When the racket vibrates as in pattern A, its frequency of
vibration is approximately 30 Hz. Determine the frequency of vibration of the
racket when it vibrates as in pattern B and pattern C.
Following
from the reasoning to the previous answer in part b of this question, wave B
must have a frequency of 90 Hz and wave C must have a frequency of 120 Hz.