Mixed Problem for the Two-Velocity Wave Equation with Characteristic Oblique Derivative at the Endpoint of a Semibounded String

We substantiate a formula for a unique and stable smooth solution to the mixed problem for the two-velocity wave equation under a boundary mode with a nonstationary characteristic oblique derivative at the end of a semibounded string. Smooth solutions of order $$m$$ to this problem are two or more integer $$m$$ times continuously differentiable solutions to this mixed problem. The characteristic property of the oblique derivative at the end of a semibounded string means that at any moment of time it is directed along the critical characteristic of the wave equation. We derive a criterion for Hadamard correctness of the characteristic mixed problem, i.e., necessary and sufficient smoothness requirements for the initial data of this problem and conditions for matching the boundary mode with the initial conditions and the equation. These smoothness requirements and matching conditions ensure the existence, uniqueness, and stability of an $$m$$ times continuously differentiable solution outside and on the critical characteristic of the equation, respectively.