Question

The equation $y=Asi{n}^2(kx-\omega t)$ represents a wave motion with

• A. amplitude A, frequency $\frac{\omega }{2\pi }$
• B. amplitude $\frac{A}{2}$, frequency $\frac{\omega }{\pi }$
• C. amplitude 2A, frequency$\frac{\omega }{4\pi }$
• D. does not represent a wave motion

Answer 1

The correct option is B

amplitude $\frac{A}{2}$, frequency $\frac{\omega }{\pi }$

$y=Asi{n}^2(kx-\omega t)$

This is not ideal equation of wave we don't have$si{n}^2\theta$ in ideal equation so let's remove the square.

Use trigonometry

$cos2\theta =co{s}^2\theta -si{n}^2\theta$

$=1-2si{n}^2\theta$

$si{n}^2\theta =\frac{1-cos2\theta }{2}$

$y=A\frac{\{1-cos2(kx-wt{)}^2\}}{2}$

$y=\frac{A}{2}-\frac{A}{2}cos(2kx-2\omega t)$

$y^{\prime }=(y-\frac{A}{2})=-\frac{A}{2}cos(2kx-2\omega t)$

So amplitude =$-\frac{A}{2}$

angular frequency$\left({\omega }^{\prime }\right)=2\omega$

frequency(f)=$\frac{\omega }{2\pi }=\frac{2\omega }{2\pi }=\frac{\omega }{\pi }$