Standing wave node spacing ($\Delta x=\lambda /2$)

A node of a standing wave must have $y(x,t)=0$ for every time, not merely at one snapshot. For $y=2a\sin (kx)\cos (\omega t)$, with $a,k,\omega >0$, set $t=0$ to see that this requires $\sin (kx)=0$. ThusAn antinode instead maximizes the oscillation amplitude $2a|\sin (kx)|$ and lies halfway between successive nodes of a standing wave, at $x=(2j+1)\lambda /4$. Neighbouring antinodes are also separated by $\lambda /2$; the distance from a node of a standing wave to its nearest antinode is $\lambda /4$. These distances refer to the string's longitudinal coordinate, not its curved arc length.

Within one interval between nodes of a standing wave, all moving points have the same oscillation phase. Adjacent intervals have opposite signs of $\sin (kx)$, so their phase difference is $\pi$: one loop rises while the next falls. The oscillation phase at an exact node of a standing wave is undefined because its amplitude is zero. A ruler measures the spacing of the fixed pattern; it does not track a moving wave crest.