A wave represented by the equation y_1 = a co | vedclass

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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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$a \cos(kx + \omega t + \pi)$

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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 $y_1 = a \cos(kx - \omega t)$.The reflected wave must travel in the opposite direction,so its phase term must be $(kx + \omega t)$.Since $x = 0$ is a node,the resultant displacement at $x = 0$ must be zero for all $t$.Let the second wave be $y_2 = a \cos(kx + \omega t + \phi)$.The resultant wave is $y = y_1 + y_2 = a \cos(kx - \omega t) + a \cos(kx + \omega t + \phi)$.At $x = 0$,$y = a \cos(-\omega t) + a \cos(\omega t + \phi) = a \cos(\omega t) + a \cos(\omega t + \phi) = 0$.This implies $\cos(\omega t) = -\cos(\omega t + \phi) = \cos(\omega t + \phi + \pi)$.Thus,$\phi + \pi = 0$ or $\phi = \pi$.Therefore,the equation for the other wave is $y_2 = a \cos(kx + \omega t + \pi)$.

$A$ wave represented by the equation $y_1 = a \cos(kx - \omega t)$ is superimposed 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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