= Standing wave node spacing
{title2=$\Delta x=\lambda/2$}
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 $\sin(kx)=0$. Thus
$$
x_j=\frac{j\pi}{k}=\frac{j\lambda}{2},\qquad x_{j+1}-x_j=\frac{\lambda}{2},\qquad j\in\mathbb Z.
$$
An <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>.
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