Civil Engineering Reference
In-Depth Information
and
Z(M
2
)
is equal to
j
Z
c
Z(M
2
)
=−
φ
cot g
kd
(4.137)
where in both equations
d
is the thickness of the material.
4.8.2 Oblique incidence - locally reacting materials
Equations (4.130) - (4.137) are still valid for the case of oblique incidence. The impedance
Z(M
2
)
is given by Equation (4.137) and does not depend on the angle of incidence.
Sound propagation in each pore depends only on the pressure of air above the pore, and
the material is a locally reacting material. If the pores were interconnected, the surface
impedance could depend on the angle of incidence, because of the interference between
waves out of phase inside the material. An isotropic porous material, where pores are
isotropically distributed and connected, may be represented by an equivalent isotropic
fluid. It has been pointed out previously, in Section 3.4.2, that the impedance of a layer
of fluid depends weakly on the angle of incidence if the velocity in the porous material
is much smaller than that in air, and the transmitted wave propagates perpendicular to
the surface. In such a case, the material is also called a locally reacting material.
4.9
Tortuosity and flow resistivity in a simple anisotropic
material
In this section, the concept of tortuosity will be presented via a simple example. A porous
layer having pores of radius R lying in two directions symmetrical with respect to the
normal of the surface is represented in Figure 4.12. The thickness of the layer is
d
,the
length of the pores is
l
, and the differential pressure is
p
2
−
p
1
. The velocities in the two
pores, averaged over the cross-sections, are
υ
1
and
υ
2
.
With
n
pores per unit area of surface, the porosity
φ
is given by
nπR
2
cos
ϕ
φ
=
(4.138)
where
ϕ
is the angle between the axes of the pores and the surface normal. The pressure
gradient in the pores is
p
2
−
p
1
l
=
p
2
−
p
1
cos
ϕ
(4.139)
d
u
2
u
1
l
d
ϕ
x
Figure 4.12
A porous material with pores of constant and equal radius, and oriented in
two directions, symmetrical with respect to the normal of the surface.
