A well-posed initial-boundary value problem is stated for the wave equation with variable propagation speed in a half-plane on a semi-infinite time interval. Certain conditions on the boundary data are indicated that ensure the existence and uniqueness of the solution as well as its stability under small changes of the boundary data in a certain function class. The problem statement is motivated by the need to justify the Poincaré wavelet-based integral representation of the wave equation solution in terms of localized solutions, in particular, quasiphotons.