For derivation of equation for stationary wave, y1=Asin(kx-omega t) and y2=Asin(kx+omega t) are considered. Why y1=Asin (kx-omega t) and y2=Asin(kx+omega t+pi) cannot be used to derive?

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Answer 1 · 2018-05-02T19:36:33

See below

# sin (kx-omega t) + sin(kx+omega t+pi)#

# = sin kx cos omega t - sin omega t cos kx + sin kx cos (omega t + pi) + sin (omega t + pi) cos kx #

Now:

#sin (alpha + pi) = sin alpha cos pi + sin pi cos alpha = - sin alpha#

#cos (alpha + pi) = cos alpha cos pi - sin alpha sin pi = - cos alpha#

So we now have:

# = sin kx cos omega t - sin omega t cos kx - sin kx cos omega t - sin omega t cos kx #

#= - 2cos kx sin omega t#

Whilst it's spatial and temporal phase may not suit, that certainly defines a stationary wave.