Question

Discuss analytically, the different modes of vibration in the stretched string tied between two rigid supports.

Solution

Consider a string of length L which is stretched between two points A and B.
Let T be the tension in the string and 'v' be the velocity of wave in the string when plucked.

When the string is plucked, a progressive transverse wave be sent along the string toward right.
On reflection from B, it will travel along the string toward left.
Both incident and reflected waves superimpose in the string and give rise to stationary wave.
Incident and reflected wave are represented by,

space space space space space straight y subscript 1 equals rsin left parenthesis ωt minus kx right parenthesis

...(1)

straight y subscript 2 equals negative rsin left parenthesis ωt plus kx right parenthesis

...(2)
On superposition,

straight y equals straight y subscript 1 plus straight y subscript 2

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rightwards double arrow

straight y equals negative 2 rcosωt space sinkx

...(3)
which is the resultant stationary wave.
The string is rigidly tied at A and B, therefore the boundary conditions are,
At A: at x = 0; y = 0
At B: at x = L; y = 0 '
From equation (3), it follows that

straight y equals negative 2 rcosωt space sink left parenthesis 0 right parenthesis space equals space 0

The solution is trivial.

straight y equals negative 2 rcosωt space sinkL space equals space 0

rightwards double arrow

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rightwards double arrow

kL space equals space nπ

, when n is an integer.

rightwards double arrow

fraction numerator 2 straight pi over denominator straight lambda end fraction equals nπ over straight L

rightwards double arrow

straight lambda equals fraction numerator 2 straight L over denominator straight n end fraction

∴

space space space straight v equals v over lambda equals straight n fraction numerator v over denominator 2 straight L end fraction

Therefore,
1st mode of vibration i.e. n = 1

straight lambda equals 2 straight L

and

straight v equals fraction numerator v over denominator 2 straight L end fraction

This is known as fundamental mode of vibration.

Here, the point of minimum amplitude of vibration i.e., A and B are known as nodes. Point C is the point of maximum amplitude of vibration and is known as anti-node.
In first mode of vibration, the string vibrates in one segment.
2nd mode of vibration i.e. n = 2

straight lambda equals straight L

and

space space space straight v italic equals italic 2 fraction numerator v over denominator italic 2 L end fraction

Frequency of second mode of vibration is two times the fundamental frequency, and hence is known as second harmonic.
In second harmonic, the string vibrates in two segments.
Similarly, in n^th mode of vibration, the string vibrates in n segments with frequency n times the frequency of fundamental mode.
In n^th mode of vibration the number of nodes are n+1, and number of antinodes are n.

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Mechanical Properties Of Fluids

Discuss analytically, the different modes of vibration in the stretched string tied between two rigid supports.

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