 Solution

As in case (i), the initial curve is characteristic and its displacement data force $g^{\prime }=1$. The new derivative data would therefore requireThe right-hand side varies along the initial segment. A single number $h^{\prime }(0)$ cannot satisfy those conditions, so 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.