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. $y(t)=A\sin \frac{\pi x}{6}\cos \frac{50\pi t}{3}$
• 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. $y(t)=A\sin \frac{5\pi x}{2}\cos 250\pi t$

Answer 1

Answer: (A), (C), (D)

Taking $y(t)=Af(x)g(t)$ & Applying the conditions:
1 ; here $x=3m$ is antinode $\&x=0$ is node
2 ; possible frequencies are odd multiple of fundamental frequency.
where, $v_{\text{fiudamental }}=\frac{v}{4\ell }=\frac{25}{3}Hz$