In this work the propagation of sound waves in a degenerate quantum gas is considered. We modify the Wang Chang and Uhlenbeck method for description of the sounds in a Maxwell gas for the case of a degenerate quantum gas. Using this approach, we constructed a dispersion relation for sound waves in a condensed Bose gas at finite temperatures, and calculate the velocities of first and second sounds in the first approximation. The possibility of the theoretical investigation for sound damping is discussed.