  The second Stokes problem about behaviour of the rarefied gas filling
half-space is formulated. A plane, limiting half-space, makes harmonious
oscillations in the plane. The kinetic equation with model integral of
collisions in the form $ \tau $ - model is used. The case of diffusive
reflection of molecules of gas from a wall is considered. There are eigen
solutions (continuous modes) the initial kinetic equation, corresponding to the
continuous spectrum. Properties of dispersion function are studied. The
discrete spectrum of this problem consisting of zeroes of dispersion function
in complex plane is investigated. It is shown, that number of zero of
dispersion function to equally doubled index of coefficient of the problem. The
problem coefficient is understood as the relation of boundary values of
dispersion function from above and from below on the real axis. Further there
are eigen solutions (discrete modes) the initial kinetic equation,
corresponding to the discrete spectrum.
  The general solution of the kinetic equation in the form of expansion by
eigen solutions with the unknown coefficients corresponding to discrete and
continuous spectra is worked out. In the present work the analytical solution
of second Stokes problem is constructed. On the basis of the analytical
solution velocity of gas in half-space is calculated. Also velocity of gas
directly at oscillating boundary is calculated precisely. The force of a
friction operating from gas on the oscillating plate is found, and also
dissipation of energy of a plate is obtained.
