= Solution
As in case (i), the initial curve is characteristic and its displacement data force $g'=1$. The new derivative data would therefore require
$$
1=\frac{\sqrt y}{2}\bigl(1-h'(0)\bigr),\qquad h'(0)=1-\frac2{\sqrt y}.
$$
The right-hand side varies along the initial segment. A single number $h'(0)$ cannot satisfy those conditions, so \b[no classical solution exists].
Characteristic <Cauchy data> impose a compatibility condition on transverse derivatives without determining the missing independent wave family. Compatible data can leave nonuniqueness, as in case (i); incompatible data have no solution, as here. The distinction follows from <compatibility of Cauchy data on a characteristic curve>, rather than a noncharacteristic existence-and-uniqueness theorem.
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