The physical foundations and mathematical formalism of the contemporary wave continuum thermomechanics are considered; and coupled thermoelastic waves travelling at finite velocities in long waveguides are studied. Lagrange field formalism sequent to the minimum action principle and a possibility of full variational description of a physical field in a space-time is developed. The theory of variational symmetries for a cross-coupled thermoelastic field and conservation laws (including their new forms) are presented. Field theoretic approach is then applied to mathematically modeling heat transport in solids by hyperbolic partial differential equations which ensure heat impulses to travel at finite speeds and admit undamped heat waves of the second sound in solids. By the wave thermomechanics methods employing GN-theory equations a problem of propagation of harmonic cross-coupled thermoelastic waves of an arbitrary azimuthal order via a long waveguide (sidewall of the waveguide is assumed free from tractions and permeable to heat or isolated from heat interchanging) is studied. As a limiting case of hyperbolic GN scheme the Pochhammer-Chree theory of elastic wave propagating in a rod is derived.
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