The equation $y = A{\cos ^2}\left( {2\pi \;nt - 2\pi \frac{x}{\lambda }} \right)$ represents a wave with
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(a) The given equation can be $x$ written as
$y = \frac{A}{2}\cos \,\left( {4\pi nt - \frac{{4\pi x}}{\lambda }} \right) + \frac{A}{2}$ $\left( \because cos^2 \theta = \frac{{1+ cos2\theta}}{{2}} \right)$
Hence amplitude $ = \frac{A}{2}$ and frequency $ = \frac{\omega }{{2\pi }} = \frac{{4\pi n}}{{2\pi }} = 2n$
and wave length $ = \frac{{2\pi }}{k} = \frac{{2\pi }}{{4\pi /\lambda }} = \frac{\lambda }{2}$.
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