Civil Engineering Reference
In-Depth Information
or by
Nδ
3
−
ω
2
ρ
11
ω
2
ρ
22
µ
3
=
(6.85)
6.5.3 The three Biot waves in ordinary air-saturated porous materials
For the case where a strong coupling exists between the fluid and the frame, the two
compressional waves exhibit very different properties, and are identified as the slow wave
and the fast wave (Biot 1956, Johnson 1986). The ratio
µ
of the velocities of the fluid
and the frame is close to 1 for the fast wave, while these velocities are nearly opposite
for the slow wave. The damping due to viscosity is much stronger for the slow wave
which, in addition, propagates more slowly than the fast wave. With ordinary porous
materials saturated with air, it is more convenient to refer to the compressional waves as
a frame-borne wave and an airborne wave.
This new nomenclature is obviously fully justified if there is no coupling between the
frame and air. For such a case, one wave propagates in the air and the other in the frame.
For the case where a weak coupling exists, the partial decoupling previously predicted
by Zwikker and Kosten (1949) occurs. With the frame being heavier than air, the frame
vibrations will induce vibrations of the air in the porous material, yet the frame can be
almost motionless when the air circulates around it. More precisely, one of the two waves,
the airborne wave, propagates mostly in the air, whilst the frame-borne wave propagates in
both media. The wave number of the frame-borne wave and its characteristic impedance
corresponding to the propagation in the frame can be close to the wave number and the
characteristic impedance of the compressional wave in the frame when in vacuum. The
shear wave is also a frame-borne wave, and is very similar to the shear wave propagating
in the frame when in vacuum.
6.5.4 Example
The two compressional waves propagating in a fibrous material at normal incidence
are described in the context of the Biot theory. The material is a layer of glass wool
'Domisol Coffrage' manufactured by St Gobain-Isover (BP19 60290 Rantigny France).
The material is anisotropic, but the compressional waves in the normal direction are
identical in this material and in an equivalent isotropic material which presents the same
stiffness in the case of normal displacements, and whose other acoustical parameters are
the same as for the fibrous material in the normal direction. The parameters
α
∞
,
ρ
1
,
σ
,
φ
,
N
and
ν
are indicated in Table 6.1. The shear modulus
N
is evaluated from acoustic
measurements, as indicated in Section 6.6, and the Poisson coefficient is equal to zero
(Sides
et al
. 1971).
The diameter of the fibres, calculated by use of Equation (5.C.7) is
=
12
×
10
−
6
d
m
The characteristic dimensions
and
are obtained by the use of Equations (5.29)
and (5.30)
10
−
4
=
10
−
4
=
0
·
56
×
m
,
2
=
1
·
1
×
m
