  We develop a microscopic theory of sound damping due to Landau mechanism in
dilute gas with Bose condensate. It is based on the coupled evolution equations
of the parameters describing the system. These equations have been derived in
earlier works within a microscopic approach which employs the
Peletminskii-Yatsenko reduced description method for quantum many-particle
systems and Bogoliubov model for a weakly nonideal Bose gas with a separated
condensate. The dispersion equations for sound oscillations were obtained by
linearization of the mentioned evolution equations in the collisionless
approximation. They were analyzed both analytically and numerically. The
expressions for sound speed and decrement rate were obtained in high and low
temperature limiting cases. We have shown that at low temperature the
dependence of the obtained quantities on temperature significantly differs from
those obtained by other authors in the semi-phenomenological approaches.
Possible effects connected with non-analytic temperature dependence of
dispersion characteristics of the system were also indicated.
