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

Answer: (C)

$\lambda =\frac{\pi x_0}{2}$

The wave equation is given by:
$x=x_0\sin (2\pi nt-kx)$

Particle velocity, $v=\frac{dx}{dt}=2\pi n{x}_0\sin (2\pi nt-kx)$

Maximum particle velocity $=2\pi n{x}_0$, since maximum value of $\sin (2\pi nt-kx)=1$

Wave velocity, $v=n\lambda$

Given maximum velocity is equal to four times of wave velocity ie. $2\pi n{x}_0=4n\lambda \Rightarrow \lambda =\frac{\pi x_0}{2}$