Vibration of Stretched Strings

In physics, the word 'string' is used in a more general sense than what it normally denotes. In olden days, musical instruments employed strings of twisted intestines of animals, such as cat-gut. Nowadays, the strings of musical instruments like the veena, violin and guitar are made of metal wires. An ideal string is an infinitely thin, perfectly flexible cord of uniform area of cross-section. It should offer no resistance to bending and there should be no change in its length during vibration. In practice, a thin, long metal wire of uniform cross- section, stretched between two fixed supports can be considered as a string. Anyone who has played a musical instrument knows that

• a thick heavy string, when made to vibrate, has a lower natural pitch than a thin one

• a short string has a higher pitch than a long one and

• the tighter a string is stretched, higher is its pitch

• a stretched string can be excited by striking with light felt mallets as in a piano or by bowing with resined horsehair as in a violin or by plucking with finger nails or picks as in a veena or guitar.

In all the above cases, transverse vibrations are produced in the string. The velocity of the transverse wave that travels along the length of the string, depends on the nature of the string and its state of tension.

Let LAM represent a portion of a stretched string in which a transverse wave is travelling towards the right, with a velocity V. If the string is drawn towards the left with the same velocity, the wave becomes stationary. Let PQ represent a small element of this portion of the string. It is in the form of an arc with its centre of curvature at O. Let For the sake of clarity in the diagram q has been shown to be large, but it will be quite small in practice. Let the tension in the string be T at P or Q. The tensions will be along the tangents, meeting at A. Join AO. AO represents the radius of the circular arc PQ, which is represented by r.

If m is the mass per unit length of the string, then length of the arc PQ = m . PQ

The components of the tensions at P and Q along the radius will add up, while those perpendicular to it will cancel out. Therefore, the resultant tension in the element PQ is 2T sin q acting along AO and this provides the necessary centripetal force, making the particles of the string trace a circular path.

The fundamental mode of vibration of a stretched string is shown in the figure. It has two nodes at the ends and an antinode in the middle. If L is the length of the vibrating segment between the two nodes, then

Substituting for v from equation 1.47 we get

"For a given string under constant tension, the frequency of vibration is inversely proportional to the length of the string”.

"For a given string of constant length, the frequency of vibration is directly proportional to the square root of the tension”.

"For a string of constant length and under a constant tension, the frequency of vibration is inversely proportional to the square root of its mass per unit length”.

If M is the mass and L is the length of the string, then

If d is the diameter of the wire, then

Substituting in equation (1.48), we get

The law of mass may be put into two additional laws, for strings of circular cross-section, as given below.

"For a string of a given material and length and under a constant tension, the frequency is inversely proportional to its diameter”.

"For a string of a given length and diameter and under constant tension, the frequency is inversely proportional to the square root of the density of the material of the string”.