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(n t-x / \lambda)] $
$=x_{0} \sin \left(2 \pi n t-\frac{2 \pi}{\lambda} x\right)$
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 $x=x_{0} \sin (\omega t \pm k x)$
From the question, $2 \pi n x_{0}=4 n \lambda$
$\Rightarrow \lambda=\frac{\pi}{2} x_{0}$