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Vibrating strings are the basis of any string instrument like guitar or piano.

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1 Speed of propagation of the wave

Let be the length of the string, its mass and the tension.

When the string is touched it bends as an arc of circle. Let be the radius and the angleThis article is about angles in geometry. For other articles, see Angle (disambiguation An angle (from the Lat. angulus a corner, a diminutive, of which the primitive form, angus does not occur in Latin; cognate are the Lat. angere, to compress into a ben under the arc. Then .

The string is recalled to its natural position by a forceIn physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. The concept appeared first in the second law of motion of classical mechanics. It is usually expressed by the equation F m · a where F is the force, which is equal to .

The force is also equal to the centripetal forceThe centripetal force is the force that causes an object to move in a circle. Its direction is towards the centre of the circle. In the case of an orbiting satellite the centripetal force is its weight and acts towards the satellite's primary; in the case , where is the speedFor alternate uses, see Speed (disambiguation). Speed (symbol: v is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t''. Speed is a scalar quantity with dimensions Length/ Time; the equiva of propagation of the wave in the string.

Let be the linear massLinear mass is a measure of mass per unit of length, and it is a characteristic of strings. The SI unit of linear mass is the kilogram per metre (kg/m). where mu is the linear mass of the string (measured in kilograms per metre) m is the total mass of the of the string. Then .

If we equal the two expressions of we have:

So

2 Frequency of the wave

Once we know the speed of propagation, it is almost immediate to find the frequency of the sound produced by the string. In fact we know that the speedFor alternate uses, see Speed (disambiguation). Speed (symbol: v is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t''. Speed is a scalar quantity with dimensions Length/ Time; the equiva of propagation of a wave is equal to the wavelengthThe wavelength is the distance between repeating units of a wave pattern. It is commonly designated by the greek letter lambda (λ). In a sine wave, the wavelength is the distance between peaks: The x axis represents distance, and I would be some va divided by the period , or multiplied by the frequency :

If the length of the string is , the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so is half of the wavelength of the fundamental harmonic.

Hence:

where is the tension, is the linear mass, and is the length of the vibrating part of the string. Therefore:

• the shorter is the string and the higher is the note
• the higher is the tension and the higher is the note
• the heavier is the string and the lower is the note