Question

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

Answer 1

$y_1=A\sin [kx-kct]$

$y_2=A\sin [kx-kct]$

superposition of these two waves:

$y=y_1+y_2$

$\Rightarrow y=A\left[\sin \right(kx-kct)+\sin (kx+kct\left)\right]$

$\Rightarrow y=A\left[2\sin \right(\frac{kx-kct+kx+kct}{2})+\cos (\frac{kx-kct-kx-kct}{2}\left)\right]$

$\Rightarrow y=2A\left[\sin \right(kx)+\cos (kct\left)\right]$

$\Rightarrow y=2A\left[\right(\cos kct)+\sin (kx\left)\right]$ ( equation $1$ )

Comparing the euation $1$ by

$y=A\cos \left(\omega t\right)\sin \left(kx\right)$

$k=\frac{2\pi }{\lambda }$

$\therefore \lambda =\frac{2\pi }{k}$

Distance between two adjacent nodes$=\frac{\lambda }{2}=\frac{\pi }{k}$