Dispersion relations of (sound-like) wave modes, which can exist in a helium II layer of arbitrary width, are calculated numerically. The basis of our considerations is the complete system of the linearized Landau-Khalatnikov equations, in which only the dissipative processes involved with I/and ~2 are taken into account. Apart from the linearization, no approximation or averaging is performed. The thermal expansion of helium II is taken into account. Symmetry properties of the velocities of flow, usually required, are dropped here. A hint is given as to how all the Khalatnikov coefficients may be measured by sound absorption experiments.

In liquid helium two wave modes are possible. Their properties may be analysed by solving the thermohydrodynamical equations under the condition that the tangential component of the normal fluid velocity is vanishing on the walls. In the present paper, these two types of wave propagation are determined for a plane-parallel capillary with the heat conduction and the thermal expansion being neglected and with the width of the capillary being much smaller than the penetration depth of a viscous wave. In particular, the dispersion relations of both, the so called fourth sound and an overdamped mode are calculated. (This overdamped mode may be called fifth wave mode.) The velocity fields can be determined, too, as well as the pressure fields and the temperature fields. It is shown, that pressure and temperature are coupled in both cases.11"

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