A wave-type heat transfer in both linear and nonlinear medias is considered on the basis of a new heat conductivity law taking into account higher order time derivatives further to the first and second order ones. This approach allows reducing the hyperbolic differential equation of heat transfer to the parabolic one with the retarded time argument. Emerging discontinuities of the first kind of temperature profile in nonlinear media under certain conditions of heat transfer at free boundaries are transformed into wave-like heat fluxes. The obtained results validate the proposed hypotheses and may be used to compute thermal conditions when designing hypersonic flying vehicles subjected to high-intensity gas dynamics and radiant heating.