Cross-coupled type-III thermoelastic waves of a given azimuthal number in a waveguide under sidewall heat interchanging

The paper is devoted to a study of cross-coupled type-III generalized thermoelastic waves of a given azimuthal order propagating via a long cylindrical waveguide with circular cross-section. Sidewall of the waveguide is assumed free from tractions and permeable to heat. The study is carried out in the framework of coupled generalized theory of type-III thermoelasticity (GNIII) consistent with the fundamental principles of continuum thermomechanics.
The type-III theory combines the both possible mechanisms of heat transfer: thermodiffusion and wave. Type-III generalized thermoelasticity includes classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity (GNII) as limiting cases. The GNII-theory can be formulated as a field theory and differential field equations are of hyperbolic analytical type. Closed solution of the coupled linear GNIII-thermoelasticity partial differential equations satisfying the required boundary conditions on the surface of waveguide including convective heat interchanging condition is obtained by the separation of variables technique. For a given azimuthal number the frequency equation is derived. A numerical analysis of frequency equation is carried out by Mathematica. A scheme of frequency equation roots localization is proposed and wavenumbers of the coupled type-III thermoelastic waves of the first and seventh azimuthal numbers are computed. A numerical study of the coupled thermoelastic waves of the 70th azimutal number is also presented. Some aspects of numerical realization of the proposed approach are discussed.