Engineering
Physics
Basics of Simple Harmonic Motion
Energy in Prograssive Wave
Standing Wave on String
Question

A standing wave is produced in a uniform string stretched between two fixed ends.

The phase difference (at different points of the string) between the two progressive waves due to which this standing wave is formed is either 0 or π.

There is maximum strain in the string when it’s elements have maximum acceleration.

The force constant of SHM of all elements (of the same but small length) except at nodes is same.

The speeds of all the particles will be in same phase.

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Solution
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General equation of standing wave is

            y = (2A sin kx) constant ;

            Vp = –(2A ω sin kx) sin ωt ;

            ap = –(2A ω2 sin kx) constant ;

Force constant, k=(dm)(ap)y=−(dm)ω2 is same for all elements except at nodes.

Velocity of particles on one half of wave will be in same phase and differ with the other half by π, but speed of all particles will be in same phase. Strain is maximum where potential energy is maximum which is possible at extreme position of each element that is maximum acceleration. The phase difference between two progressive waves is π so as to form standing wave.