One end of a taut string of length $3 \ m$ along the $x$-axis is fixed at $x=0$. The speed of the waves in the string is $100 \ m / s$. The other end of the string is vibrating in the $y$-direction so that stationary waves are set up in the string. The possible waveform$(s)$ of these stationary waves is (are)
$(A)$ $y(t)=A \sin \frac{\pi x}{6} \cos \frac{50 \pi t}{3}$
$(B)$ $y(t)=A \sin \frac{\pi x}{3} \cos \frac{100 \pi t}{3}$
$(C)$ $y(t)=A \sin \frac{5 \pi x}{6} \cos \frac{250 \pi t}{3}$
$(D)$ $y(t)=A \sin \frac{5 \pi x}{2} \cos 250 \pi t$
IIT 2014, Advanced
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$V=100 \ m / s$
$Image$
Possible modes of vibration
$\ell=(2 n+1) \frac{\lambda}{4} $
$\lambda=\frac{12}{(2 n+1)} m $
$k=\frac{2 \pi}{\lambda}=\frac{2 \pi}{12 /(2 n+1)}=\frac{(2 n+1) \pi}{6} $
$\omega=v k=100(2 n+1) \frac{\pi}{6}=\frac{(2 n+1) 50 \pi}{3} $
$\text { if } \quad n=0 \quad k=\frac{\pi}{6} \quad \omega=\frac{50 \pi}{3} $
$\quad$ $n=1 \quad k=\frac{5 \pi}{6} \quad \omega=\frac{250 \pi}{3} $
$\quad$ $n=7 \quad k=\frac{5 \pi}{2} \quad \omega=250 \pi$
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