generalised thermoelasticity, a wave guide, a wave number, a harmonious wave, a thermal wave, an azimuthal number, a frequency equation.

In the given article within the limits of the generalised linear theory of GNIIIthermoelasticity, on the basis of the connected system of movement and heat conductivity equations the analysis of the harmonious thermoelastic waves extending along an axis of a free cylindrical wave guide of circular cross-section section is given. The lateral surface of a wave guide is supposed impenetrable for heat. The research of the frequency equation and forms of harmonious waves of movings and temperature is conducted in an infinite cylindrical wave guide in conditions both in the axially symmetric district wave profile, and in case of district harmonics of azimuthal order as high as possible. The dependence of wave number on frequency is numerically defined and the profiles of movings and temperature forms in the connected GNIII-thermoelastic wave of the third azimuthal order are constructed. It is established that to each set frequency answers a denumerable number of complex wave numbers. The basic research is preceded by a studying of distribution of the flat harmonious connected GNIII-thermoelastic waves.