Question

A progressive wave moving along x-axis represented by $y=A\sin \left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$. The wavelength $\left(\lambda \right)$ at which the maximum particle velocity is $3$ times the wave velocity is

• A. $\frac{A}{3}$
• B. $\frac{2A}{3\pi }$
• C. $\left(\frac{3}{4}\right)\pi A$
• D. $\left(\frac{2}{3}\right)\pi A$

Answer 1

The correct option is D $\left(\frac{2}{3}\right)\pi A$

$\Rightarrow y=A\sin \left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$
$\omega =kv$

$\Rightarrow v=\omega /k=\frac{v_{p,max}/A}{k}$

$\because (v_p{)}_{max}=3v$
$\Rightarrow k=\frac{3}{A}⟹\frac{2\pi }{\lambda }=\frac{3}{A}$

$\Rightarrow \lambda =\frac{2\pi }{3}A$