Two-sided string data with zero slope ($y(x,t)=[f(t-x/c)+f(t+x/c)]/2$)

For the wave equation on a string, prescribing $y(0,t)=f(t)$ and $y_x(0,t)=0$ selects equal incoming and outgoing profiles. Write D'Alembert's formula as $P(t-x/c)+Q(t+x/c)$. The two traces imply $P+Q=f$ and $P^{\prime }=Q^{\prime }$, proving the displayed expression. These data do not generally describe a disturbance launched from an everywhere resting string. For nonsmooth $f$, derivatives and boundary traces need their piecewise classical or weak derivative interpretation; an endpoint cusp may prevent a classical derivative at an isolated time.

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.