Exact Electric Analogy to the Vernotte Hypothesis

Abstract

The Fourier equation of heat conduction predicts the paradoxical result that the effect of a thermal impulse in an infinite medium will be felt instantaneously in all parts of the medium. In other words, a thermal impulse is propagated at infinite velocity. The result is paradoxical because it is incompatible with a dynamic interpretation of the mechanism of heat transfer in solids. In order to avoid this apparent paradox, Vernotte has proposed a modification of the Fourier hypothesis. This modification results in the transformation of the equation of heat conduction from a parabolic to a hyperbolic differential equation predicting finite velocity of propagation of thermal impulses. Vernotte's proposal is shown here to have an exact electric analogy. The proposal is equivalent to postulating the existence of a heat transfer quantity which is analogous to the electric quantity inductance. The transformed equation of heat transfer is therefore analogous to the differential equation of telegraphy.