English translation

3.1 Ultrasound Spectroscopy

Two theoretical approaches are discussed: mode-coupling theory (MCT), and the phenomenological theory of Bhattacharjee and Ferrell based on dynamic scaling. Both seek to describe ultrasound absorption in binary fluids and other systems exhibiting critical behaviour through a scaling function.

1. Mode-coupling theory (MCT)

Mode-coupling theory originated in the work of Kawasaki [9] and was subsequently developed, in particular, by Fixman [8], Zwanzig and Mori, and Kadanoff [26]. Its basic idea comes from the classical hydrodynamic description of liquids: differential equations are constructed for all variable parameters. Nonlinear terms are particularly important, for example the heat density q, which couples to the convective part of heat transport in fluids:

This relation yields an equation of motion that must then be solved. The refinement introduced by Zwanzig and Mori was the use of the Hamiltonian formalism, in contrast to classical mechanics. Folk and Moser developed the theory further in [11–13]. Since the resulting scaling function lacks a closed-form representation, this approach is not pursued in the present work.

2. Scaling hypothesis of Bhattacharjee and Ferrell

The following section presents the principal elements of the theory published by Bhattacharjee and Ferrell [15–19] for ultrasound absorption in three-dimensionally fluctuating binary liquids. This theory plays a central role in the present investigation.

The approach to dynamic scaling begins with thermodynamic considerations of low-frequency, long-wavelength sound velocity near the critical point. According to [18], these lead to expressions for the frequency-dependent sound velocity and adiabatic compressibility, continued on the next page.

Original mathematical notation preserved from the thesis.