Question

A longitudinal wave is represented by $x=x_{0}sin2\pi (nt-\frac{\pi }{\lambda })$. 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 $\lambda =\frac{\pi x_0}{2}$

We have the longitudinal wave given by

$x=x_{0}sin\left[2\pi \right(nt-\frac{x}{\lambda }\left)\right]$

$=x_{0}sin\left(\begin{array}{c}2\pi nt-\frac{2\pi }{\lambda }x\end{array}\right)$

The maximum particle velocity

$=A\omega =-x_0\left(2\pi n\right)$

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, $2\pi n{x}_0=4n\lambda$

$\Rightarrow \lambda =\frac{\pi }{2}{x}_0$