Question

A longitudinal wave is represented by $x=x_{0}sin2\pi (nt-\frac{x}{\lambda })$. The maximum particle velocity will be four times the wave velocity if:

• A. $\lambda =\frac{\pi x_0}{4}$
• B. $\lambda =2\pi x_0$
• C. $\lambda =\frac{\pi x_0}{2}$
• D. $\lambda =4\pi x_0$

Answer 1

The correct option is C $\lambda =\frac{\pi x_0}{2}$

$x=x_{0}sin2\pi [nt-\frac{x}{\lambda }]$
Particle velocity $=\frac{dy}{dt}=2\pi x_{0}ncos2\pi [nt-\frac{x}{\lambda }]$
$V_{max}$ of particle $=2\pi n{x}_0$
$V_p=4V_w$
$2\pi n{x}_0=4n\lambda$
$\lambda =\frac{\pi x_0}{2}$