Vibrating String

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

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.

This shows a resonant standing wave on a string. It is driven by a vibrator at 120 Hz.

For strings of finite stiffness, the harmonic frequencies will depart progressively from the mathematical harmonics. To get the necessary mass for the strings of an electric bass as shown above, wire is wound around a solid core wire. This allows the addition of mass without producing excessive stiffness.

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

Show

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

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.

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 T = string tension m = string mass L = string length and the harmonics are integer multiples.

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: