Question

One end of a taut string of length $3m$ along the x axis is fixed at $x$ $=$ $0$. The speed of the waves in the string is $100m{s}^{-1}$. 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 waves is (are)

• A. $y\left(t\right)=A\sin \frac{\pi x}{6}\cos \frac{50\pi t}{3}$
• B. $y\left(t\right)=A\sin \frac{\pi x}{3}\cos \frac{100\pi t}{3}$
• C. $y\left(t\right)=A\sin \frac{5\pi x}{6}\cos \frac{250\pi t}{3}$
• D. $y\left(t\right)=A\sin \frac{5\pi x}{2}\cos 250\pi t$

Answer 1

Answer: (A), (C)

$y\left(t\right)=A\sin \frac{\pi x}{6}\cos \frac{50\pi t}{3}$
$y\left(t\right)=A\sin \frac{5\pi x}{6}\cos \frac{250\pi t}{3}$

$y=A\sin kx.\cos \omega x$
$x=3,l=3$
at $x=0$ node
at $x=3$ antinode
$v=f\lambda$
$\lambda =4l=12$
$\therefore {\omega }_{fundamental}=\frac{50\pi }{3}$
$Thus,\omega =\frac{50\pi }{3},\frac{150\pi }{3},\frac{250\pi }{3},\frac{350\pi }{3},\frac{450\pi }{3},\frac{450\pi }{3},\frac{650\pi }{3},\frac{750\pi }{3}$