Question

The equation \[y = A   \sin^2   \left( kx - \omega t \right)\] 
represents a wave motion with 

Options

• amplitude A, frequency \[\omega/2\pi\]

• amplitude A/2, frequency \[\omega/\pi\]

• amplitude 2A, frequency \[\omega/4\pi\]

• does not represent a wave motion.

Answer 1

amplitude A/2, frequency \[\omega/\pi\]
\[y = A \sin^2 \left( kx - \omega t \right)\]

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

\[y = A\left[ \frac{1 - \cos^2 \left( kx - \omega t \right)}{2} \right]\] 

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