Physics - Wave on String Question with Solution | TestHub

The equation of the simple harmonic progressive wave is given by $y=a\sin 2\pi \left(nt-\frac{x}{\lambda }\right)$.

The wave velocity $v$ is given by $v=n\lambda$.

The particle velocity is obtained by differentiating the displacement with respect to time:

$v_p=\frac{\partial y}{\partial t}=a(2\pi n)\cos 2\pi \left(nt-\frac{x}{\lambda }\right)$

The maximum particle velocity is $v_{\max }=2\pi na$.

According to the given condition, the wave velocity is equal to $\frac{1}{6}$ of the maximum particle velocity:

$v=\frac{1}{6}{v}_{\max }$

Substituting the expressions for $v$ and $v_{\max }$:

$n\lambda =\frac{1}{6}(2\pi na)$

Canceling $n$ from both sides, we get:

$\lambda =\frac{\pi a}{3}$

Answer: $\frac{\pi a}{3}$