The parabolic model for heat conduction discussed above shows that the Fourier equation (and the more general Fick's law of diffusion) is incompatible with the theory of relativity for at least one reason: it admits infinite speed of propagation of the continuum field (in this case: heat, or temperature gradients). To overcome this contradiction, workers such as Carlo Cattaneo, Vernotte, Chester, and others proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form, where the n, the temperature field  is governed by:
In this equation, C is called the speed of second sound (that is related to excitations and quasiparticles, like phonons). The equation is known as the "hyperbolic heat conduction" (HHC) equation. Mathematically, the above equation is called "telegraph equation", as it is formally equivalent to the telegrapher's equations, which can be derived from Maxwell’s equations of electrodynamics.  
For the HHC equation to remain compatible with the first law of thermodynamics, it is necessary to modify the definition of heat flux vector, q, to
where  is a relaxation time, such that  This equation for the heat flux is often referred to as "Maxwell-Cattaneo equation". The most important implication of the hyperbolic equation is that by switching from a parabolic (dissipative) to a hyperbolic (includes a conservative term) partial differential equation, there is the possibility of phenomena such as thermal resonance and thermal shock waves.