Answer
$0.25\;m$
Work Step by Step
According the the give wave equation:
$k=4\pi\; rad/m$ and $\omega=400\pi\;rad/s$
Therefore, wave velocity is given by
$v=\frac{\omega}{k}=\frac{400\pi}{4\pi}\;m/s=100\;m/s$
and time period is given by
$T=\frac{2\pi}{\omega}=\frac{2\pi}{400\pi}\;s=0.005\;s$
We know that the oscillating point move from maximum upward displacement to maximum downward displacement in the time interval of $\frac{T}{2}$
Therefore, in this time interval, the displacement of each wave move along the string becomes
$s=v\times\frac{T}{2}$
or, $s=100\times\frac{0.005}{2}\;m$
or, $\boxed{s=0.25\;m}$
