Civil Engineering Reference
In-Depth Information
θ
air
X
Porous layer
Z
Figure 10.4
A plane wave in air impinges on a porous layer bonded on a rigid impervi-
ous backing with an angle of incidence
θ
. The axis of symmetry is parallel to the normal
Z
of the surface of the layer.
the results obtained for the plane wave can be used with the Sommerfeld representation
as in Chapter 8.
10.5.2 Plane field in air
The layer has a finite thickness and is bonded onto a rigid impervious backing (see
Figure 10.4). The boundary conditions are
u
s
x
=
u
s
z
=
w
z
=
0
(10.57)
The summation is performed on all the waves at the bonded face. In what follows a
superscript
+
is used for the waves propagating toward the backing and a superscript - for
the waves propagating toward the free surface. At the free surface of the layer,
z
=
0
(see Figure 10.4) and the displacement components for a normalization factor
N
=
1of
the different waves are denoted as
a
1
(i),a
3
(i),b
1
(i),b
3
(i),i
=
1
,
2
,
3. The related
z
components of the slownesses are
q
±
(i)
q(i)
. As in Chapter 8, to avoid large systems
of equations, each wave propagating from the free surface to the backing is associated
with three waves propagating upward to satisfy the boundary conditions at the contact
surface with the backing. In a similar way as for isotropic media, the normalization factors
of the upward waves, related to a normalization factor
N
=±
1 of each downward wave,
are denoted as
r
ik
where
i
defines a downward wave and k one of the three associated
upward waves. The coefficients
r
ik
satisfy the following equations
=
a
3
(i)
r
ik
a
3
(k)
exp[
jωl(q(i)
+
+
q(k))
]
=
0
(10.58)
k
=
1
,
3
a
1
(i)
r
ik
a
1
(k)
exp[
jωl(q(i)
+
+
q(k))
]
=
0
(10.59)
k
=
1
,
3
b
3
(i)
+
r
ik
b
3
(k)
exp[
jωl(q(i)
+
q(k))
]
=
0
(10.60)
k
=
1
,
3
where
l
is the thickness of the layer. Any superposition of the three downward waves, each
one being associated with its three upward waves, satisfies the boundary conditions at the
bonded face. At the surface in contact with air the boundary conditions can be written
v
z
=
jω
[
u
s
z
+
w
z
]
(10.61)
σ
xz
=
0
(10.62)
