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(D) The given equation $y(x, t) = 10 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$ is of the form $y(x, t) = 2A \sin(kx) \cos(\omega t)$,which

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$B$ and $C$

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(D) The given equation $y(x, t) = 10 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$ is of the form $y(x, t) = 2A \sin(kx) \cos(\omega t)$,which represents a standing (stationary) wave.Comparing the given equation with the standard form,we get $k = \frac{2 \pi}{3}$ and $\omega = 120 \pi$.The frequency of the standing wave is $f = \frac{\omega}{2 \pi} = \frac{120 \pi}{2 \pi} = 60 \ Hz$. Thus,statement $(B)$ is correct.$A$ standing wave is formed by the superposition of two waves of the same frequency and wavelength travelling in opposite directions.The wavelength is $\lambda = \frac{2 \pi}{k} = \frac{2 \pi}{2 \pi / 3} = 3 \ m$.The wave speed is $v = \frac{\omega}{k} = \frac{120 \pi}{2 \pi / 3} = 180 \ m/s$. Thus,statement $(C)$ is correct.Since the string length is $L = 1.5 \ m$ and $\lambda = 3 \ m$,we have $L = \frac{\lambda}{2}$. This corresponds to the fundamental mode of a string fixed at both ends. Therefore,reflection occurs from a fixed end,not a free end. Statement $(D)$ is incorrect.Hence,statements $(B)$ and $(C)$ are correct.

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