A wave represented by the equation ${y_1} = a\,\cos \,\left( {kx - \omega t} \right)$ is superimposed with another wave to form a stationary wave such that the point $x = 0$ is node. The equation for the other wave is
AIEEE 2012, Medium
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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 $y_{\text {incident }}=a \cos (k x-\omega t)$
$\Rightarrow y_{\text {incident }}=a \cos (-k x-\omega t+\pi)$
$=-a \cos (\omega t+k x)$
Hence equation for the other wave
path change of $\frac{\lambda}{2}$
So, if $y_{\text {incident }}=a \cos (k x-\omega t)$
$\Rightarrow y_{\text {incident }}=a \cos (-k x-\omega t+\pi)$
$=-a \cos (\omega t+k x)$
Hence equation for the other wave
$y=a \cos (k x+\omega t+\pi)$
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