Question

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

• A. $\frac{c}{\pi }$
• B. $\frac{c}{2\pi }$
• C. $\frac{\pi }{2k}$
• D. $\frac{\pi }{k}$

Answer 1

The correct option is D $\frac{\pi }{k}$

Given two travelling waves $y_1=\text{A sin [k(x - ct)]}$ and $y_2=\text{A sin}\left[k\right(x+ct\left)\right]$
Since stationary wave is formed due to interference of two identical waves, moving in opposite directions.
$y_1+y_2=\text{A sin [k(x - ct)]}+\text{A sin [k(x + ct)]}=\text{2A sin [kx]}\text{cos [kct)]}$

Wave number $k=\frac{2\pi }{\lambda }$
$\therefore$ wavelength $\left(\lambda \right)=\frac{2\pi }{k}$
Distance between adjacent nodes will be,
$\Delta x=\frac{\lambda }{2}=\frac{\left(\frac{2\pi }{k}\right)}{2}$
$\therefore \Delta x=\frac{\pi }{k}$