Question

A longitudinal wave is represented by $x=x_0\sin 2\pi \left(nt-\frac{x}{\lambda }\right)$. 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)

We have the longitudinal wave given by
$x=x_0\sin [2\pi (nt-x/\lambda )]$
$=x_0\sin \left(2\pi nt-\frac{2\pi }{\lambda }x\right)$
The maximum particle velocity
$=A\omega =-x_0(2\pi n)$
The wave velocity $=n\lambda$
Here we compare the given equation with the equation $x=x_0\sin (\omega t\pm kx)$
From the question, $2\pi n{x}_0=4n\lambda$
$\Rightarrow \lambda =\frac{\pi }{2}{x}_0$