Question

The equation of a standing wave, produced on a string fixed at both ends, is
\[y = \left( 0 \cdot 4  cm \right)  \sin  \left[ \left( 0 \cdot 314  {cm}^{- 1} \right)  x \right]  \cos  \left[ \left( 600\pi  s^{- 1} \right)  t \right]\]
What could be the smallest length of the string?

Answer 1

Given:
Equation of the standing wave:

\[y = \left( 0 . 4  cm \right)  \sin  \left[ \left( 0 . 314  {cm}^{- 1} \right)  x \right]\cos  \left[ \left( 600  \pi s^{- 1} \right)  t \right]\] 

\[ \Rightarrow k = 0 . 314 = \frac{\pi}{10}\] 

\[Also,   k = \frac{2\pi}{\lambda}\] 

\[ \Rightarrow \lambda = 20  \text{ cm }\]
We know:
\[L = \frac{n\lambda}{2}\]
For the smallest length, putting n = 1:
\[\Rightarrow L = \frac{\lambda}{2} = \frac{20  \text{ cm}}{2} = 10  \text{ cm }\]
Therefore, the required length of the string is 10 cm.