Sound absorbers 199
We have then determined the unknown total volume V of air in the chamber, V= V 0 + Vf.
Since the volume V 0 is easily measured, we have determined the sought after volume Vf
of the pores.
This is an outline of the basic principle of the measurement. The practical
implementation certainly involves problems such as ensuring an isothermal change of
state, the determination of small pressure changes etc. In the set-up mentioned, one is
using a piston of diameter 4 mm, positioned to an accuracy of 1 micrometer. A
differential pressure transducer detects pressure changes within 10-6 mm of mercury. The
material samples have a maximum volume of 1.5 litres. The general accuracy is, as
mentioned above, better than 1%.
5.6.3 Tortuosity, characteristic viscous and thermal lengths
There are several methods for the determination of tortuosity. For materials having a
non-conducting frame one may compare the conductivity when saturating the material
with an electrical conducting fluid with the conductivity of the fluid itself (see e.g. Allard
(1993:73)).
Later developments apply more efficient methods based on high frequency
measuring techniques. The principle is based on measuring the sound transmission
through the material, utilizing the high frequency asymptotic behaviour to determine the
tortuosity and well as the characteristic viscous and thermal lengths. At sufficiently high
frequencies, a practical frequency range for these measurements is 100–800 kHz, one
may assume that the frame is motionless. The complex wave number may then be
approximated to read
(^) s
00
11 2
1(1 j) where.(^2) Pr
kk
c
ω δ γμ
δ
ρω
⎡⎤⎛⎞−
′≈+−+⎢⎥⎜⎟ =
⎢⎥⎜⎟Λ Λ′
⎣⎦⎝⎠
(5.71)
The quantity δ is denoted the viscous skin depth. Allard et al. (1994) used this expression
to determine the tortuosity ks, utilizing the fact that the viscous skin depth approaches
zero at sufficiently high frequencies, i.e. that the following apply:
(^210)
0
sef
eff
where.
c f
kc
ck
f
⎛⎞ → ω
⎜⎟⎯⎯⎯⎯→=
⎝⎠ ′
(5.72)
The quantity ceff is thereby the effective speed of sound through the material. The
measurements are relatively easy to perform by placing a disc of the material between an
ultrasound source and receiver. One then compares the transit time of a broadband pulse
of ultrasound with and without the disc between source and receiver. A Fourier
transformation into the frequency domain then gives the sound speed as a function of
frequency, which enables one to use the extrapolation given in Equation (5.72). A
practical problem is to find suitable ultrasound transducers for air, both having sufficient
power and bandwidth. The range of ultrasound propagation in air is generally short in
addition to a normally large attenuation through the material sample. This implies that
the sample may have to be just a few millimetres in thickness, thus not being
representative of the material as such.
Leclaire et al. (1996) took the use of Equation (5.71) a step further by utilizing the
differences in the physical properties of air and helium to obtain an independent