We study shock waves propagating in a rigid heat conductor at low temperature using a generalized Maxwell-Cattaneo equation. The existence of a critical temperature 6, characteristic of the material, for which the structure of the shock changes is proved. When the unperturbed temperature ^o is less than 9 the temperature ^i behind the shock wave front is such that 9\ > 6o (hot shock), and, vice versa, if ao> ^then Oi < OQ {cold shock). We find ^=15.36 K for NaF and (9 = 3.38 K.for Bi. These temperatures are very close to the values at which the second sound was identified experimentally in pure NaF and Bi crystals.

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A continuum approach, based on the principles of modern extended thermodynamics, describing the model of a phonon gas is performed. The main difference with the ideal phonon gas theory consists in the presence of a thermal inertia. We apply the shock wave theory and discuss the selection rules for physical shocks (the Lax conditions and the entropy growth). In this way the existence of two new kinds of shocks (hot and cold shocks) in rigid heat conductors at low temperature is pointed out. In particular a critical temperature, characteristic of each material, changing the structure of the previous types of shocks is analytically deduced. This characteristic temperature permits also to explain the modification of the received second sound wave form with respect to the initial wave profile. Finally, the results are applied to the case of high-purity crystals (NaF, Bi, 3He and 4He) and compared with experimental results. PACS 44.10 -Heat conduction (models, phenomenological description). PACS 66.70 -Nonelectronic thermal conduction and heat-pulse propagation in nonmetallic solids.

In his monograph Thermodynamics, I. Müller proves that for incompressible media the volume does not change with the temperature. This Müller paradox yields an incompatibility between experimental evidence and the entropy principle. This result has generated much debate within the mathematical and thermodynamical communities as to the basis of Boussinesq approximation in fluid dynamics. The aim of this paper is to prove that for an appropriate definition of incompressibility, as a limiting case of quasi thermal-incompressible body, the entropy principle holds for pressures smaller than a critical pressure value. The main consequence of our result is the physically obvious one, that for very large pressures, no body can be perfectly incompressible. The result is first established in the fluid case. In the case of hyperelastic media subject to large deformations the approach is similar, but with a suitable definition of the pressure associated with convenient stress tensor decomposition.

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