  In the present work the second Stokes problem about behaviour of the rarefied
gas filling half-space is formulated. A plane limiting half-space makes
harmonious fluctuations with variable amplitude in the plane. The amplitude
changes on the exponential law. The kinetic equation with model integral of
collisions in the form $\tau$-model is used. The case of diffusion reflexions
of gas molecules from a wall is considered. Eigen solutions (continuous modes)
of the initial kinetic equation corresponding to the continuous spectrum are
searched. Properties of dispersion function are studied. It is investigated the
discrete spectrum of the problem consisting of zero of the dispersion functions
in the complex plane. It is shown, that number of zero of dispersion function
to equally doubled index of problem coefficient. The problem coefficient is
understood as the relation of boundary values of dispersion function from above
and from below on the real axis. Further are eigen solutions (discrete modes)
of the initial kinetic equation corresponding to the discrete spectrum are
searched. In the end of work the general solution of the kinetic equation in
the form expansion under eigen solutions with unknown coefficients
corresponding to discrete and continuous spectra is constructed.
