The theory of Casimir (1938) for steady-state heat flow in the boundary-scattering limit for cylindrical samples is extended to include time-dependent effects. This shows that a term of the form (∂3T/∂x2∂t) is the first correction term which should be added to the usual thermal diffusion equation, the range of validity of the theory being restricted by the condition that the wavelength of the disturbance, [Formula: see text], where L is the mean free path of phonons in the bulk material and a is the radius of the cylinder. The dispersion relation is derived, and, as an example, dispersion curves are computed for liquid helium as the dielectric medium, contained in a tube of 1-cm radius. At high frequencies the velocity of the temperature wave is reduced below the value predicted by the diffusion equation.
