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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.
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Solution
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 = \[\frac{A}{2}\]
\[2\left( \frac{\omega}{2\pi} \right) = \frac{\omega}{\pi}\]
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