2/22/2019
Fundamental Frequency and Harmonics
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The Physics Classroom(/) » Physics Tutorial(/class) » Sound Waves and Music(/class/sound) » Fundamental Frequency and Harmonics
Sound Waves and Music - Lesson 4 - Resonance and Standing Waves
Fundamental Frequency and Harmonics
Natural Frequency(/class/sound/Lesson-4/Natural-Frequency)
Forced Vibration(/class/sound/Lesson-4/Forced-Vibration)
Standing Wave Patterns(/class/sound/Lesson-4/Standing-Wave-Patterns)
Fundamental Frequency and Harmonics
Previously in Lesson 4(http://www.physicsclassroom.com/Class/sound/u11l4c.cfm), it was mentioned that
when an object is forced into resonance
vibrations(http://www.physicsclassroom.com/Class/sound/u11l4b.cfm#resonance) 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(http://www.physicsclassroom.com/Class/sound/u11l4c.cfm#chladni), 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(http://www.physicsclassroom.com/Class/sound/u11l3a.cfm#music). We will see in this part of
Lesson 4 why these whole number ratios exist for a musical instrument.
Recognizing the Length-Wavelength Relationship
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Fundamental Frequency and Harmonics
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(http://www.physicsclassroom.com/Class/waves/u10l2a.cfm#wavelength) 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(http://www.physicsclassroom.com/Class/waves/u10l4b.cfm) 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 lengthwavelength relationship for the fundamental frequency of a guitar string.
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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
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Fundamental Frequency and Harmonics
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.
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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
#
1
2
3
4
5
# of
Waves
in String
1/2
1 or 2/2
3/2
2 or 4/2
5/2
# of
Nodes
2
3
4
5
6
# of
Antinodes
1
2
3
4
5
LengthWavelength
Relationship
Wavelength = (2/1)*L
Wavelength = (2/2)*L
Wavelength = (2/3)*L
Wavelength = (2/4)*L
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).
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Fundamental Frequency and Harmonics
Determining the Harmonic Frequencies
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(http://www.physicsclassroom.com/Class/sound/U11L4d.cfm#table)); thus, the wavelength is 160
cm or 1.60 m. The speed(http://www.physicsclassroom.com/Class/waves/u10l2e.cfm) 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(http://www.physicsclassroom.com/Class/waves/u10l2d.cfm#media) (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(http://www.physicsclassroom.com/Class/sound/U11L4d.cfm#table), 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(http://www.physicsclassroom.com/Class/sound/U11L4d.cfm#table), 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
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Fundamental Frequency and Harmonics
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
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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
#
1
2
3
4
5
n
Frequency
(Hz)
400
800
1200
1600
2000
n * 400
Wavelength
(m)
1.60
0.800
0.533
0.400
0.320
(2/n)*(0.800)
Speed
(m/s)
640
640
640
640
640
640
fn / f1
λn / λ1
1
2
3
4
5
n
1/1
1/2
1/3
1/4
1/5
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(http://www.physicsclassroom.com/Class/sound/u11l3a.cfm#music). For instance, the first and
second harmonics have a 2:1 frequency
ratio(http://www.physicsclassroom.com/Class/sound/u11l2a.cfm#octave); the second and the third
harmonics have a 3:2 frequency ratio(http://www.physicsclassroom.com/Class/sound/u11l2a.cfm#table);
the third and the fourth harmonics have a 4:3 frequency
ratio(http://www.physicsclassroom.com/Class/sound/u11l2a.cfm#table); and the fifth and the fourth
harmonic have a 5:4 frequency ratio(http://www.physicsclassroom.com/Class/sound/u11l2a.cfm#table).
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.
In Lesson 5(http://www.physicsclassroom.com/Class/sound/u11l5a.cfm), these same principles of
resonance and standing waves will be applied to other types of instruments besides guitar strings.
Investigate!
The harmonics of an instrument, when played together, sound good. Use the Timbre widget below to
investigate this principle. Use the frequencies provided and try some combinations of your own.
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