Question

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

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

Answer 1

Answer: (C)

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