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

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(B) For a stationary wave to have a node at $x = 0$,the resultant displacement $y = y_1 + y_2$ must be zero at $x = 0$ for all $t$.Given $y_1 = a \cos

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

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(B) For a stationary wave to have a node at $x = 0$,the resultant displacement $y = y_1 + y_2$ must be zero at $x = 0$ for all $t$.Given $y_1 = a \cos(kx - \omega t)$.At $x = 0$,$y_1 = a \cos(-\omega t) = a \cos(\omega t)$.For a node at $x = 0$,the second wave $y_2$ must satisfy $y_1 + y_2 = 0$ at $x = 0$,so $y_2 = -a \cos(\omega t)$.Since the wave is traveling in the opposite direction ($-x$ direction),the wave function must be of the form $y_2 = a \cos(kx + \omega t + \phi)$.At $x = 0$,$y_2 = a \cos(\omega t + \phi) = -a \cos(\omega t) = a \cos(\omega t + \pi)$.Comparing the arguments,we get $\phi = \pi$.Thus,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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