Explain the reflection of wave at rigid support.

Explanation

A pulse (wave) moving in +x-direction and reflecting wave from fixed support are shown in figure.
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If we suppose that the energy is not absorbed at end, then the shape of the reflected pulse will be same as incident but the phase will be changed by `180^(@)(pi).`
The reason behind it is the end is fixed. So that the displacement of pulse should be zero.
Supppose, the incident progressive wave displacement at .t. time is `y _(1) (x,t)= a sin (kx-omegat).`
Suppose, the displacement of reflected wave is `y _(i)`
According to superposition principle,
`y (x,t) =y _(i) (x,t) + y _(r)(x,t)`
But `y (x,t) =0`
`(because` displacemtn of fixed end is zero)
`therefore 0= y _(i) (x,t) + y _(r) (x,t)`
`therefore y _(r) (x,t) =-y _(i) (x,t)`
`=- a sin (kx - omega t )`
`therefore y _(r) (x,t) = a sin (kx - omega t + pi)`
Thus, when the pulse (wave) reaches to end the string applies force on fixed end.
According to third law of Netton the end will also apply the same force in opposite direction. Hence, there will be difference in phase of `pi` in reflected wave. It will be progressive in -x-direction.
Thus, the plane of reflected wave will be increased by `pi.` Hence, the shape of wave will be reversed. It means the crest will be through and through will be crest.