Oscillations, Waves, and Interactions - GWDG

Liquids: Formation of complexes and complex dynamics 369
Here α is the attenuation coefficient, β = 2π/λ is the wave number with λ = c/ν
denoting the wavelength, c is the sound velocity, ν the frequency, and ı 2 = −1. K is
the complex adiabatic compression modulus, given by
K = κ −1
S (0) + ı ωηv , (3)
where
�
κS(0) = lim −
ω→0
1
� � �
∂V
(4)
V ∂p S
is the static adiabatic compressibility at very small angular frequency ω = 2πν, and
ηv is the volume viscosity. At very small attenuation (α ≪ β) the imaginary part of
Eq. (2) yields
α = 2π2 ν 2
c 3 ϱ
�
4
3 ηs
�
+ ηv . (5)
Frequently it is assumed that, within the frequency range of measurement, roughly
10 kHz ≤ ν ≤ 10 GHz, the shear viscosity is independent of frequency but the volume
viscosity
ηv(ν) = ∆ηv(ν) + ηv(∞) (6)
may be composed of two parts, of which ∆ηv(ν) displays relaxation characteristics
whereas ηv(∞) does not depend upon frequency. The sonic attenuation coefficient
α(ν) = αexc(ν) + B ′ ν 2
thus contains a part with quadratic frequency dependence and coefficient
B ′ = 2π2
c3 �
4
ϱ 3 ηs
�
+ ηv(∞)
and an excess contribution
(7)
(8)
αexc(ν) = 2π2
c 3 ϱ ∆ηv(ν) , (9)
which is of primary interest in acoustical spectrometry. Because of the frequency
dependence of the asymptotic high-frequency “background” contribution (7) it is
convenient to display experimental spectra in the frequency normalized format
α αexc(ν)
=
ν2 ν2 + B′ , (10)
accentuating the low-frequency part of the spectrum. If interest is focussed on the
high-frequency regime and if, particularly, comparison with theoretical models and
their thermodynamic parameters is derived an (αλ)exc-versus-ν plot is appropriate.
It is obtained from subtracting the asymptotic high-frequency contribution
from the total attenuation per wavelength, αλ.
Bν = B ′ cν (11)