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(B) stationary wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.Given the inciden

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$y = -a \cos (kx + \omega t)$

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(B) stationary wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.Given the incident wave is $y_1 = a \cos (kx - \omega t)$.For a point $x = 0$ to be a node,the resultant displacement at $x = 0$ must be zero for all time $t$.Let the second wave be $y_2 = a \cos (kx + \omega t + \phi)$.The resultant wave is $y = y_1 + y_2 = a [\cos (kx - \omega t) + \cos (kx + \omega t + \phi)]$.Using the identity $\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$,we get:$y = 2a \cos (kx + \phi/2) \cos (\omega t + \phi/2)$.At $x = 0$,$y = 2a \cos (\phi/2) \cos (\omega t + \phi/2)$.For this to be a node (zero displacement) for all $t$,we must have $\cos (\phi/2) = 0$,which implies $\phi/2 = \pi/2$,or $\phi = \pi$.Substituting $\phi = \pi$ into the equation for the second wave:$y_2 = a \cos (kx + \omega t + \pi) = -a \cos (kx + \omega t)$.

$A$ wave represented by the 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:

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