Vibrating String

The fundamental vibrational mode of a stretched string is such that the wavelength is twice the length of the string.


Applying the basic wave relationship gives an expression for the fundamental frequency:

Calculation


Since the wave velocity is given by , the frequency expression

can be put in the form:


The string will also vibrate at all harmonics of the fundamental. Each of these harmonics will form a standing wave on the string.
String frequenciesString instrumentsIllustration with a slinkyMathematical form
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Wave Velocity in String

The velocity of a traveling wave in a stretched string is determined by the tension and the mass per unit length of the string.

The wave velocity is given by
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When the wave relationship is applied to a stretched string, it is seen that resonant standing wave modes are produced. The lowest frequency mode for a stretched string is called the fundamental, and its frequency is given by

From

velocity = sqrt ( tension / mass per unit length )
the velocity = m/s
when the tension = N = lb
for a string of length cm and mass/length = gm/m.
For such a string, the fundamental frequency would be Hz.

Any of the highlighted quantities can be calculated by clicking on them. If numerical values are not entered for any quantity, it will default to a string of 100 cm length tuned to 440 Hz. Default values will be entered for any quantity which has a zero value. Any quantities may be changed, but you must then click on the quantity you wish to calculate to reconcile the changes.

Derivation of wave speed
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Harmonics

An ideal vibrating string will vibrate with its fundamental frequency and all harmonics of that frequency. The position of nodes and antinodes is just the opposite of those for an open air column.

The fundamental frequency can be calculated from

where

and the harmonics are integer multiples.

Illustration with a slinky
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Vibrating String Frequencies

If you pluck your guitar string, you don't have to tell it what pitch to produce - it knows! That is, its pitch is its resonant frequency, which is determined by the length, mass, and tension of the string. The pitch varies in different ways with these different parameters, as illustrated by the examples below:

If you have a string with
starting pitch:
100 Hz
and change* to
the pitch
will be
double the length
50 Hz
four times the tension
200 Hz
four times the mass
50 Hz
*with the other parameters reset to their
original values.
Calculation
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