Justification of Fourier method in a mixed problem for wave equation with non-zero velocity

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions. The problems are considered under minimal requirements on initial data. A. N. Krylov's idea of accelerating Fourier series convergence is essentially employed.