Abstract <p>Heat transfer by thermal waves in media with local disequilibria over short time intervals is investigated analytically and by modeling. The times considered are multiples of the relaxation time, defined as the time delay of the heat flux due to the temperature gradient in thermal shock. A new analytical solution is obtained for the third initial–boundary problem associated with the hyperbolic heat-transfer equation derived on the basis of the Vernotte–Cattaneo–Lykov heat conduction law. New effects are observed: in particular, thermal waves moving into the depth of the body that are characterized by discontinuities of the first kind at the temperature distribution fronts, with declining amplitude; and upward convexity of the temperature profile close to the boundary. That contradicts the fundamental classical theorem of a maximum. The results of analytical solution are analyzed.</p>