Question

The equation $y=A{\sin }^2\left(kx-\omega t\right)$ represents a wave motion with
(a) amplitude A, frequency $\omega /2\pi$
(b) amplitude A/2, frequency $\omega /\pi$
(c) amplitude 2A, frequency $\omega /4\pi$
(d) does not represent a wave motion.

Answer 1

(b) amplitude A/2, frequency $\omega /\pi$

$y=A{\sin }^2\left(kx-\omega t\right)$

$$\begin{gathered} \left[{\cos }^2\theta =1-2{\sin }^2\theta \\ {\sin }^2\theta =\frac{1-{\cos }^2\theta }{2}\right] \end{gathered}$$

$$\begin{gathered} y=A\left[\frac{1-{\cos }^2\left(kx-\omega t\right)}{2}\right] \\ y=\frac{A}{2}\left[1-{\cos }^2\left(kx-\omega t\right)\right] \end{gathered}$$
Thus, we have:
Amplitude = $\frac{A}{2}$
Frequency = $2\left(\frac{\omega }{2\pi }\right)=\frac{\omega }{\pi }$