Question

A string of length ${}^{\prime }{l}^{\prime }$ is fixed at both ends. It is vibrating in its $3^{rd}$ overtone with maximum amplitude ${}^{\prime }{a}^{\prime }$. The amplitude at a distance $\frac{l}{3}$ from one end is:

• A. $a$
• B. $0$
• C. $\frac{\sqrt{3}a}{2}$
• D. $\frac{a}{2}$

Answer 1

Answer: (C)

$\frac{\sqrt{3}a}{2}$

For a string vibrating in its $n^{th}$ overtone $(n+1{)}^{tj}$ harmonic)
$y=2A\sin \left(\frac{(n+1)\pi x}{L}\right)\cos \omega t$
For $x=\frac{l}{3}.2A=a$ and $n=3$;
$y=\left[a\sin (\frac{4\pi }{l}\cdot \frac{l}{3})\right]\cos \omega t$
$=a\sin \frac{4\pi }{3}\cos \omega t=-a\left(\frac{\sqrt{3}}{2}\right)\cos \omega t$
i.e. at $x=\frac{l}{3}$, the amplitude is $\frac{\sqrt{3}a}{2}$.