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

• A

  a
• B

  0
• Correct answer
  C

  $\frac{\sqrt{3} \mathrm{a}}{2}$
• D

  $\frac{a}{2}$

Solution

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

Hence, the correct answer is option: 'C'.