Question

The equation of a standing wave, produced on a string fixed at both ends, is

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

What could be the smallest length of the string?

Answer 1

Given:
Equation of the standing wave:
$$\begin{gathered} y=\left(0.4\mathrm{cm}\right)\sin \left[\left(0.314{\mathrm{cm}}^{-1}\right)x\right]\cos \left[\left(600\pi {\mathrm{s}}^{-1}\right)t\right] \\ \Rightarrow k=0.314=\frac{\pi }{10} \\ \mathrm{Also},k=\frac{2\mathrm{\pi }}{\lambda } \\ \Rightarrow \lambda =20\mathrm{cm} \end{gathered}$$

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