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 superposed on a string. Find the distance between adjacent nodes.

Answer 1

Resultant motion will be $y=y_1+y_2=A[\sin \left(k\right(x-ct\left)\right)+\sin \left(k\right(x-ct\left)\right)]$

$\Rightarrow y=2A\sin kx.\cos kct$

$\Rightarrow y=A^{\prime }\cos kct$

Here $A^{\prime }=2A\sin kx$=New amplitude of the superpose motion. Node means the position of $x$ for which $A^{\prime }$ will be zero.

So, $A^{\prime }=0$

$\Rightarrow 2A\sin kx=0$

$\Rightarrow kx=0,\pi ,2\pi ,...$

Or, $x=0,\frac{\pi }{k},\frac{2\pi }{k},...$

So, distance between two consecutive node$=(\frac{\pi }{k}-0)=\frac{\pi }{k}$