A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $\text{y}\left(\text{x},\text{t}\right)=\left(\text{0.01 m}\right)sin\left[\left({62.8\; m}^{-1}\right)x\right]cos\left[\left({628\; s}^{-1}\right)t\right]$. Assuming $\pi =\text{3.14}$, the correct statement(s) is (are)

• A

  The number of nodes is 5.
• Correct answer
  B

  The length of the string is 0.25 m.
• Correct answer
  C

  The maximum displacement of the midpoint of the string, from its equilibrium position is 0.01 m.
• D

  The fundamental frequency is 100 Hz.

Solution

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$\text{Nodes}=\mathrm{6\; }$
$\lambda =\text{0.1}$
$\therefore 5^{\text{th}}harmonic, 5\frac{\lambda }{2}=\text{L}$
$\text{L}=\text{0.25}$
$\text{At}\frac{\text{L}}{2}$
${\text{y}}_{\text{max}}=\text{0.01 sin in }\left(\text{62.8}\times \frac{\text{0.25}}{2}\right)$
          $=\text{0.01}$
$\text{Interfering wave velocity}=\frac{\text{w}}{\text{k}}=\frac{628}{\text{62.8}}=10$
$\therefore \text{fundamental frequency}=\frac{\nu }{2\text{L}}=20$

Hence, the correct answer is options: 'B', 'C'.