Civil Engineering Reference
In-Depth Information
ular to the direction of the fibres. The characteristic dimension
at normal incidence is
calculated in Appendix 5.C. The fibres are modelled as infinitely long cylinders having
a circular cross-section with radius
R
. In the case of materials with porosity close to 1,
is given by
1
2
π
LR
=
(5.29)
where
L
is the total length of fibres per unit volume of material. The characteristic thermal
dimension, evaluated from Equation (5.27) is given by
1
π
LR
=
2
=
(5.30)
5.3.5 Direct measurement of the high-frequency parameters, classical
tortuosity and characteristic lengths
In this present subsection, let
n
2
be the squared ratio of the velocity in a free fluid to the
velocity when it saturates a porous structure. In the high-frequency range, using Equation
(5.26) for the effective density and Equation (5.28) for the bulk modulus,
n
2
is given, to
first-order approximation in 1
/
√
ω
by
1
+
δ
1
γ
−
1
B
n
2
=
α
∞
+
(5.31)
A sketch of the experimental set-up for the measurement of
n
is represented in
Figure 5.4.
The phase velocity is obtained by comparing the phase spectra at the receiver with
and without the porous layer. In the domain of validity of Equation (5.31),
n
2
is linearly
dependent on 1
/
√
f
. In Figure 5.5 the squared velocity ratio is represented as a function
of the square root of the inverse of frequency.
The porous layer has been successively saturated with air and with helium. For helium,
(
γ
−
1)/B is close to 0.81 and for air it is close to 0.48. A comparison of both slopes in
transducer
transducer
Porous
layer
Amplifier
Amplifier
Function
generator
Oscilloscope
Figure 5.4
Measurement of the refraction index.
