On the initial-boundary value problem for the wave equation on a semi-infinite time interval

We state a correct initial-boundary value problem for the wave equation with variable propagation speed in a half-plane on a semi-infinite time interval. We find out what conditions should be imposed on the boundary data so that the solution of the problem exists, is unique, and stable with small changes of the boundary data in some functional class. The problem statement is motivated by the need to justify of the Poincare wavelet-based integral representation of the wave equation solution in terms of localized solutions, in particular, quasi-photons.