# 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.

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.

 Example measurements on a steel string
 String frequencies String instruments Illustration with a slinky Mathematical form
Index

Periodic motion concepts

Resonance concepts

 HyperPhysics***** Sound R Nave
Go Back

# 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

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.
Forsuch 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
Index

Periodic motion concepts

Resonance concepts

 HyperPhysics***** Sound R Nave
Go Back

# Harmonics

 An ideal vibrating string will vibrate with its fundamentalfrequency and all harmonics of that frequency. The positionof nodes and antinodes is justthe opposite of those for an open air column. The fundamental frequencycan be calculated fromwhereand the harmonicsare integer multiples.
Index

Periodic motion concepts

Resonance concepts

 HyperPhysics***** Sound R Nave
Go Back

# 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
Index

Periodic motion concepts

Resonance concepts

 HyperPhysics***** Sound R Nave
Go Back