Question

The equation for a standing wave generated by a string, with both ends fixed, is$y=(0.4cm)\sin (0.314c{m}^{-1})x]\cos [\left(600\pi s^{-1}\right)t].$. Figure out the smallest length of the string.

Answer 1

Step 1. Given data:

The standing wave equation $\mathrm{y}=(0.4\mathrm{cm})\sin (0.314{\mathrm{cm}}^{-1})\mathrm{x}]\cos [(600\mathrm{\pi }{\mathrm{s}}^{-1})\mathrm{t}]$

Step 2. Calculating the smallest length of the string.

The general equation of standing wave equation,

$\mathrm{y}=A\sin \left(k\mathrm{x}\right).\cos \left(w\mathrm{t}\right)$

On comparing the given equation and general equation of standing wave

$K=0.314$

Since,

$\begin{array}{rcl}\mathrm{\lambda }& =& \frac{2\mathrm{\pi }}{\mathrm{K}}\\ & =& \frac{2\times 3.14}{0.314}\\ & =& 20cm\end{array}$ (where $\lambda =$The wavelength of the standing wave)

The smallest length is,

$d=\frac{n\lambda }{2}$

Where $n=1$,

then,

$\begin{array}{rcl}d& =& \frac{20}{2}\\ & =& 10cm.\end{array}$

The smallest length is$10cm$.