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Question

(C) The resultant wave function at $x = 0$ is $y = y_1 + y_2$.Substituting $x = 0$ into the wave functions:$y_1 = A \cos(-kvt) = A \cos(kvt)$$y_2 = A

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For destructive interference at $x = 0, \phi = \pi$.

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(C) The resultant wave function at $x = 0$ is $y = y_1 + y_2$.Substituting $x = 0$ into the wave functions:$y_1 = A \cos(-kvt) = A \cos(kvt)$$y_2 = A \cos(kvt + \phi)$Using the superposition principle,$y = A [\cos(kvt) + \cos(kvt + \phi)]$.Using the trigonometric identity $\cos C + \cos D = 2 \cos(\frac{C+D}{2}) \cos(\frac{C-D}{2})$:$y = 2A \cos(kvt + \frac{\phi}{2}) \cos(\frac{\phi}{2})$.For constructive interference,the amplitude must be maximum,which occurs when $|\cos(\frac{\phi}{2})| = 1$,implying $\frac{\phi}{2} = n\pi$,or $\phi = 2n\pi$ (where $n = 0, 1, 2, ...$).For destructive interference,the amplitude must be zero,which occurs when $\cos(\frac{\phi}{2}) = 0$,implying $\frac{\phi}{2} = (2n+1)\frac{\pi}{2}$,or $\phi = (2n+1)\pi$ (where $n = 0, 1, 2, ...$).Checking the options,for $n=0$ in the destructive interference condition,$\phi = \pi$. Thus,option $C$ is correct.

Two waves are propagating along a taut string that coincides with the $x$-axis. The first wave has the wave function $y_1 = A \cos[k(x - vt)]$ and the second has the wave function $y_2 = A \cos[k(x + vt) + \phi]$.

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