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S. Braun:
"Über den Einfluß des thermischen Expansionskoeffizienten auf die nichtlineare Schallausbreitung in superfluidem Helium";
Supervisor, Reviewer: A. Kluwick, W. Schneider; Institut für Strömungslehre und Wärmeübertragung, 1997; oral examination: 11-03-1997.

The effect of the thermal expansion coefficient beta on the nonlinear propagation of first and second sound waves
in superfluid helium is commonly neglected. The reason for this usual simplification is the small numerical value of
beta.
If this quantity is taken into account consequently, it turns out that beta appears in the form of two dimensionless
groups. One of them is in fact very small. The other one, the socalled Grüneisenparameter, is typically of order
one. Consequently, beta cannot be neglected in general. The influence of beta on the propagation of sound from
this point of view is the main objective of the present study.

The linear wave speeds, the disturbances of the field quantities and the quadratic steepening parameters in the case
of inviscid flow are derived by means of a special multiple scale method, which was introduced by Taniuti and Wei.
This perturbation method reduces the basic hydrodynamic equations to two transport equations which are of the type
of inviscid Burgers equations. As shown, the restriction to weakly nonlinear waves leads to the possibility of considering the propagation process as locally one-dimensional and planar.

Taking beta into account, one finds the surprising result that first sound carries temperature perturbations in
leading order contrary to the current picture, which assumes first sound waves to be isothermal. The exact computation
of the nonlinearity parameter of first sound also leads to significant deviations from the beta=0 result. These
differences are small at low pressures but increase with pressure and as the phase transition line is approached.
In contrast, the commonly employed beta=0 approximation yields accurate results with respect to the disturbances
of the field quantities and the steepening parameter for second sound, if the temperature ranges near absolute zero and
the phase transition are excluded. In order to determine the correct asymptotic behaviour of the nonlinearity parameters
in the limit of vanishing temperature and close to the phase transition temperature, it was necessary to derive
sets of self-consistent asymptotic expansions for all thermodynamic quantities involved in the problem. The application of these expansions to the exact quadratic steepening parameters leads to a fundamentally different asymptotic
behaviour compared to the behaviour expected from the beta=0 theory for both first and second sound. The
reexamination of the asymptotic behaviour of nonlinear fourth sound shows clearly the shortcomings of the expansions used so far.

Nonlinear wave propagation requires the treatment of shock discontinuities. Therefore jump relations from the basic
conservation laws were derived. The shock speeds of weak shock waves were computed through use of the perturbation
method which was already mentioned. As can be shown by means of the second law of thermodynamics, the entropy
flux always has to increase across shocks. In contrast, the entropy may also decrease across shocks under certain
circumstances. The comparison of the present results with Khalanikov's theory of weak shocks and the numerical
solution of the exact jump relations shows the significance of beta, especially for first sound shock waves.

Finally an example of a boundary value problem clearly points out the role of beta as a coupling parameter
between first and second sound.