= Two-sided string data with zero slope
{title2=$y(x,t)=[f(t-x/c)+f(t+x/c)]/2$}
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 $P'=Q'$, 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.
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