In a standing wave on a string:

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(A) Standing waves are formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.The displacement

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In one time period,all the particles are simultaneously at rest twice.

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(A) Standing waves are formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.The displacement is given by: $y(x, t) = 2A \cos(kx) \sin(\omega t)$.For all particles to be simultaneously at rest,the velocity $v = \frac{\partial y}{\partial t} = 2A\omega \cos(kx) \cos(\omega t)$ must be zero for all $x$.This occurs when $\cos(\omega t) = 0$,which implies $\omega t = \frac{\pi}{2}, \frac{3\pi}{2}, \dots$Since $\omega = \frac{2\pi}{T}$,we have $t = \frac{T}{4}, \frac{3T}{4}, \dots$Within one time period $T$,this happens twice (at $t = \frac{T}{4}$ and $t = \frac{3T}{4}$). Thus,statement $(A)$ is correct.For all particles to be at their positive extremes simultaneously,the displacement $y(x, t) = 2A \cos(kx) \sin(\omega t)$ must be positive for all $x$. However,the term $\cos(kx)$ changes sign depending on the position $x$. Therefore,all particles cannot be at their positive extremes simultaneously. Thus,statement $(C)$ is incorrect.Therefore,the correct option is $(A)$.

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