Question

What are nodes and antinodes? Find the position of nodes and antinodes in a stretched string between two points vibrating in nth harmonic.

Solution

The position or points in the string where the amplitude of vibration is maximum are called antinodes.
Nodes are the points in the string where the particles in the string do not vibrate.

The stationary wave produced in the string is given by,

straight y equals Asinkx equals Asin fraction numerator 2 straight pi over denominator straight lambda end fraction straight x

Position of nodes:
These are the positions where y = 0.
i.e.

Asin fraction numerator 2 straight pi over denominator straight lambda end fraction straight x equals 0

rightwards double arrow

space space space fraction numerator 2 straight pi over denominator straight lambda end fraction straight x equals mπ

i.e.,

straight x equals straight m straight lambda over 2

where m = 1, 2, 3...n

Let L be the length of the string.
The wavelength of wave when string vibrates in nth modes is,

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

∴

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Thus the position of nodes is,

space space space straight L over straight n comma space 2 straight L over straight n comma space 3 straight L over straight n comma space.......... straight n straight L over straight n

Position of antinodes:
These are the positions where

straight y equals plus-or-minus straight A

i.e.

Asin fraction numerator 2 straight pi over denominator straight lambda end fraction straight x equals plus-or-minus straight A

rightwards double arrow

fraction numerator 2 straight pi over denominator straight lambda end fraction straight x equals fraction numerator 2 straight m minus 1 over denominator 2 end fraction straight pi

rightwards double arrow

straight x equals fraction numerator 2 straight m minus 1 over denominator 4 end fraction straight lambda equals fraction numerator 2 straight m minus 1 over denominator 2 end fraction open parentheses straight L over straight n close parentheses

where m = 1, 2, 3...n
Position of nodes is,

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