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