The equation of a standing wave in a string fixed at both ends is given as $y=2 A \sin k x \cos \omega t$ The amplitude and frequency of a particle vibrating at the mid of an antinode and a node are respectively
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(d)
$y=2 A \sin k x \cdot \cos \omega t$
In a standing waves the function of amplitude $\left(A_y\right)$ is given by $A_y=2 A \sin k x$
At mid-point of node and antinode $x=\frac{\lambda}{8}$
$A_y=2 A \sin \frac{2 \pi}{\lambda} \times \frac{\lambda}{8}\left[k=\frac{2 \pi}{\lambda}\right]$
$\text { or } A_y=\frac{2 A}{\sqrt{2}}$
$\therefore A_y=\sqrt{2} A$
Frequency is same at all points $=\frac{\omega}{2 \pi}$
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