Displacement cancellation without slope cancellation in crossing pulses

For two-sided string data with zero slope, take $f(t)=\sin (\pi t/a)$ on $[-a,a]$ and zero outside. At $x=ca/2$, the displacement vanishes throughout $|t|<a/2$, whereas the spatial derivative is $y_x=-\pi \sin (\pi t/a)/(ca)$. Cancellation along a line in spacetime need not cancel the derivative transverse to that line. At $x=ca,t=0$, however, oddness gives $y(x,0)=0$ for every $x$, so the spatial derivative is zero even though its one-sided temporal limits there differ.