Two travelling waves {y_1} = Asin [k(x - ct)] | vedclass

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(D) The superposition of two waves travelling in opposite directions with the same frequency and amplitude results in a stationary wave.Using the prin

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$\pi /k$

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(D) The superposition of two waves travelling in opposite directions with the same frequency and amplitude results in a stationary wave.Using the principle of superposition,the resultant wave is $y = y_1 + y_2 = A\sin[k(x - ct)] + A\sin[k(x + ct)]$.Using the trigonometric identity $\sin(C) + \sin(D) = 2\sin(\frac{C+D}{2})\cos(\frac{C-D}{2})$,we get:$y = 2A\sin(kx)\cos(kct)$.Nodes occur where the displacement is zero for all time $t$,which happens when $\sin(kx) = 0$.This implies $kx = n\pi$,where $n = 0, 1, 2, \dots$.The positions of the nodes are $x = \frac{n\pi}{k}$.The distance between two adjacent nodes is the difference between consecutive positions:$\Delta x = \frac{(n+1)\pi}{k} - \frac{n\pi}{k} = \frac{\pi}{k}$.Alternatively,since $k = \frac{2\pi}{\lambda}$,the distance between adjacent nodes is $\frac{\lambda}{2} = \frac{\pi}{k}$.

Two travelling waves ${y_1} = A\sin [k(x - ct)]$ and ${y_2} = A\sin [k(x + ct)]$ are superimposed on a string. The distance between adjacent nodes is

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