A wave represented by the given equation $y = a\cos (kx - \omega \,t)$ is superposed with another wave to form a stationary wave such that the point $x = 0$ is a node. The equation for the other wave is
AIEEE 2002,IIT 1988,AIIMS 1998, Medium
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(b) Since the point $x = 0$ is a node and reflection is taking place from point $x = 0.$
This means that reflection must be taking place from the fixed end and hence the reflected ray must suffer an additional phase change of $\pi$ or a path change of $\frac{\lambda }{2}$.
So, if ${y_{{\rm{incident}}}} = a\cos (kx - \omega \,t)$
==> ${y_{{\rm{reflected}}}} = a\cos ( - kx - \omega \,t + \pi )$$ = - a\cos (\omega \,t + kx)$
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