Question

A standing wave given by the equation $y=2A\sin \left(\frac{\pi x}{L}\right)\sin \omega t$ is formed between the points $x=-L$ and $x=L$. The separation between two such particles, which have their maximum velocity that the maximum velocity of the antinodes and which are closest to the particle present at $x=0$, is L/k then find the value of k.

Answer 1

$\frac{\delta y}{\delta t}=2A\omega \sin \left(\frac{\pi x}{L}\right)\cos \omega t$
from the give condition $\Rightarrow 2A\omega \sin \left(\frac{\pi x}{L}\right)=A\omega$
$\Rightarrow \sin \frac{\pi x}{L}=\frac{1}{2}$
$\Rightarrow x=\pm \frac{L}{6}$
Hence $k=3$.