Considering the limiting cases of ωτ yields the following results. When ωτ ≪ 1, the expression in Eq. (3.26) approaches unity. Thus Ξ = Ξ̃, and the system has sufficient time to attain the new equilibrium state. Conversely, when ωτ ≫ 1, the system can no longer follow the variation imposed by the sound wave. At ωτ = 1, the imaginary part reaches its maximum. This simple relaxation function, involving a single relaxation time, is called the Debye spectral function:

Here, c_s,∞ = c_s(ω → ∞), and A is the relaxation amplitude.

3.1.1.3 Critical Attenuation

Another type of attenuation is caused by the critical fluctuations near the critical point discussed in Section 2.2. These fluctuations can no longer be described by the relaxation dynamics represented by Eqs. (3.20) and (3.27). All thermodynamic parameters, particularly the density ρ, undergo very strong fluctuations, which manifest themselves as a reduction in the energy of the sound wave. In general, the total attenuation therefore consists of the following terms:

One of the thermodynamic parameters mentioned above plays a central role in theories of critical contributions to ultrasonic absorption: the isobaric heat capacity C_p. As noted in Section 2.2, it also obeys a power law:

Here, α̃ is a critical exponent (see Section 2.2.1). As already discussed, the static scaling hypothesis fails to describe dynamic critical behavior in systems. The following sections therefore introduce several important theories and hypotheses that account for critical fluctuations.

Two major approaches are those of Folk and Moser and of Bhattacharjee and Ferrell. As mentioned in the introduction, these theories represent two independent approaches to the problem of critical systems. Folk and Moser proceed from mode-coupling theory, whereas Bhattacharjee and Ferrell present a phenomenological theory based on dynamic scaling.

Original formula typography retained. .