Landau's equations for the two-fluid model of liquid helium IIare us ed as the basis for an investigation of the properties of thermalwave propagation. A number of assumptions are made which reducethe four original equations to a system of two non-linear partial differential equations valid to first order in the relative velocity of thetwo components. These equations are analogous to Riemann's equationswhich describe pressure waves in a classical fluid.
This system of equations, when reduced to just one spacedimension is shown to be hyperbolic and a set of characteristics andinvariants is found. A particularly simple, one-dimensional problemis then formulated and an explicit solution is given. This solution isthen studied in detail to show the distortion of a temperature pulse asit propagates and also to show effects such as non-linear breaking.
Subsequently, the restrictive assumptions are eliminatedindividually and the equations are then valid to second order in therelative velocity; the effects of including thermal expansion and usingthe relative velocity as a thermodynamic variable are given. Also,some effects due to the interaction of first and second sound areinvestigated. In all cases, the results are compared with otherresults based on equations differing from the Landau equations andwith results found by using perturbation techniques.
Finally, equations based on the same Landau equations arederived and discussed which describe steady state shock (discontinuous)solutions.
Suggestions for further theoretical and experimental work aremade.
