= Solution
Take $z$ positive upwards, so the source's sign convention makes $p$ a downward applied load. The small-<slope> <tension> contribution on a short interval is $Tw''$. Let $r\geq0$ be the upward contact reaction. Force balance gives $Tw''-p+r=0$. <Nonpenetration constraint>[Nonpenetration] is $w\geq f$, and reaction is absent where $w>f$. Therefore
$$
\boxed{w-f\geq0,\qquad Tw''-p=-r\leq0,\qquad(w-f)(Tw''-p)=0.}
$$
The contact boundary must have continuous $w$ to avoid a broken string, and continuous <slope> in this smooth-contact model with no concentrated point force. Integrating force balance over a vanishing interval around a contact endpoint gives $T[w']=0$ when the load and reaction are locally bounded. This is the one-dimensional <obstacle problem>, with its reaction acting as a nonnegative <constraint> force.
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