For real liquids, the relationships in Eq. (3.20) do not provide a complete description. The sphere model must be refined to account for the internal degrees of freedom of fluids and to include phenomena such as intermolecular interactions, conformational changes and chemical processes. The contributions arising from these processes are collectively referred to as excess attenuation, αex. It is obtained from the total attenuation coefficient, αtot:
Even in this general form, excess attenuation reveals the broad range of possible investigations offered by ultrasonic spectroscopy. To draw explicit conclusions about processes at the molecular level, however, the initially formal concept of excess attenuation must be given a physical interpretation.
To this end, the liquid is again regarded as a thermodynamic system whose state is determined by thermodynamic parameters such as pressure P, temperature T and volume V, or the associated density ρ. Initially, the system is in thermal equilibrium. When a sound wave enters the liquid, these parameters undergo spatial and temporal fluctuations that propagate through the system and disturb thermodynamic equilibrium.
According to Le Chatelier’s principle [25], the system responds to the external constraint by moving towards a new thermodynamic equilibrium. If an activation energy is involved, this process requires a certain time τ, which depends on the system and the particular process and is called the relaxation time. For small perturbations, the relaxation process can be described by a first-order differential equation. Denoting the new equilibrium state by Ξ̃(P,T,t) and the instantaneous state by Ξ(P,T,t), small deviations from equilibrium satisfy:
Since the perturbation considered consists of periodic pressure fluctuations in the form of a sinusoidal wave, the following complex trial solution is appropriate:
The solution consists of a real and an imaginary part:
Original formula typography retained. .