The fundamental frequency of the string occurs when the only nodes are at the ends of the string. The fundamental frequency is also called the
first harmonic. The second harmonic has one additional node in the middle of the string, the third harmonic two such nodes, the fourth
harmonic three such nodes, and so on. Each harmonic is created by a particular mode of vibration of the string.
The fundamental frequency occurs when the wavelength of the standing wave is twice the length of the string, because two adjacent nodes
represent half a wave. In general, the wavelength of the nth harmonic is twice the length of the string divided by n, with n being a positive
integer. That is, the wavelength of the nth harmonic on a string of length L is Ȝn = 2L/n.
The frequency of a wave is its speed divided by the wavelength, which lets us restate the equation above in terms of frequency. The equation
to the right is the basic equation for harmonic frequencies. The harmonic frequencies are positive integer multiples of the fundamental
frequency v/2L. Let’s say the fundamental frequency ƒ 1 of a string is 30 hertz (Hz). The second harmonic ƒ 2 will be 60 Hz, the third harmonic
ƒ 3 90 Hz, and so forth.
You see these principles in play in musical instruments like the piano. Its strings are of different lengths, which is one factor in determining their
fundamental frequency. Other factors that you see in the equations are also employed in musical instruments to determine a string’s
fundamental frequency. Pianos have thicker and thinner strings. A string’s mass per unit length (its linear density) will partly determine its
fundamental frequency, by changing the wave speed on the string.
In addition, string instruments are “tuned” by changing the tension in a string. You will see a guitarist frequently adjusting her instrument by
turning a tuning peg, as you see at the top of this page, which determines the tension in a string. Along with linear density, this is the other
factor that determines wave speed. A guitarist will also press her finger on a single string to temporarily create a string with a specific length
and fundamental frequency.
A harmonic is sometimes called a resonance frequency or a natural frequency. Musicians also use other terms dealing with frequencies and
harmonics. An overtone or a partial is any frequency produced by a musical device that is higher than the fundamental frequency. Unlike a
harmonic, the frequency of an overtone does not necessarily bear any simple numerical relationship to the fundamental.
Some overtones are harmonics í that is, whole-number multiples of the fundamental í but some are not. Musical instruments such as drums,
whose modes of vibration can be very complex, create non-harmonic overtones. Although harmonic overtones are “simple” integer multiples of
the fundamental, they are often referred to by numbers that are, confusingly, “one off” from the numbers for harmonics: The first overtone is the
second harmonic, and so on. Each musical instrument has a characteristic set of overtones that creates its distinctive timbre.
Harmonics
Standing waves at specific wavelengths
Corresponding frequencies are
harmonics
First harmonic is fundamental frequency17.5 - Interactive problem: tune the string
Let’s say you play an unusual instrument in the orchestra, the alto pluck, an
instrument specifically designed for physics students. The concert is underway and
your big moment is coming up, when you get to play a particular note on the pluck.
You are supposed to play the G above middle C, a note that has a frequency of
392 Hz. You produce the correct note by setting the string length and the harmonic
number. Remember that harmonic numbers are multiples of the fundamental
frequency of the string. For this instrument, you are limited to harmonic numbers in
the range of one to four.
When you set a harmonic number higher than one, a finger will touch the string at a
position that will cause the string to vibrate with the harmonic number you selected.
It does so by creating a standing wave node at the appropriate location. For
instance, if you choose a harmonic number of four, there will be three nodes
between the two ends of the string, and the finger will be one-fourth of the way
along the string. If you see a musician such as a cellist performing, you will see that
he sometimes lightly places his fingers at locations along a string to create
harmonics in just this way. He also frequently presses a string firmly against the
fingerboard to create a different, shorter, string length.
The string length of the alto pluck can range from 1.00 to 2.00 meters, but within that range, you are restricted to values that will produce
frequencies found on the musical scale. You will find that as you click on the arrows for length, the values will move between appropriate string
lengths. The harmonic values are easy to set: Just choose a number from one to four!
The description above is reasonably complicated; it is impressive that musicians with some instruments must make similar determinations as
they play. In terms of approaching this problem, you will want to start with the equation
ƒn = nv/2L
that enables you to calculate the frequency of the nth harmonic. The wave speed in your stringed instrument is 588 m/s.