Wave Motion 2 Question 29
29. 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 $100 ms^{-1}$. 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 wave is (are)
(2014 Adv.)
(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) $y(t)=A \sin \frac{5 \pi x}{2} \cos 250 \pi t$
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Answer:
Correct Answer: 29. (a, c, d)
Solution:
- There should be a node at $x=0$ and antinode at $x=3 m$.
$$ \begin{array}{ll} \text { Also, } & v=\frac{\omega}{k}=100 m / s . \\ \therefore & y=0 \text { at } x=0 \\ \text { and } & y= \pm A \text { at } x=3 m . \end{array} $$
Only (a), (c) and (d) satisfy the condition.
