Civil Engineering Reference
In-Depth Information
X
′
n
ϕ
X
porous layer
ϕ
Z
Z
′
Figure 10.6
The axis of symmetry
Z
and the new axis
X
in a plane perpendicular to
the surface of the layer. The axes
Y
and
Y
are perpendicular to the figure.
the coordinates (
x,y,z)
must be replaced by (
x
,y,z
) and the formalism of Section 10.5
must be modified.
Theanglebetween
Z
and
Z
is
ϕ
. As in the previous sections, the waves are char-
acterized by a given slowness in the plane
XY
of the surface of the layer. The slowness
components in this plane are
q
x
=
1
/c
and
q
y
. The slowness vector components in both
sets of axes are related by
q
x
=
q
x
cos
ϕ
−
q
z
sin
ϕ
(10.92)
q
y
=
q
y
(10.93)
q
z
=
q
z
cos
ϕ
+
q
x
sin
ϕ
(10.94)
q
x
cos
ϕ
−
q
x
=
q
z
tan
ϕ
(10.95)
The components
q
x
and
q
y
are defined in the description of the source in air or on
the surface.
The slowness vector in the system
X
Y
Z
with
Y
identical to
Y
is defined by the
three components
q
x
,q
y
and
q
z
. The new meridian plane is defined by the axis
Z
,and
in the plane
x
y
by the vector
q
p
of components
q
x
and
q
y
(see Figure 10.7).
The square modulus of
q
p
is given by
q
x
cos
2
ϕ
−
2
q
x
sin
ϕ
q
P
=
q
y
+
q
z
tan
2
ϕ
cos
2
ϕ
q
z
+
(10.96)
Let
X
be the axis parallel to
q
p
and
Y
be the axis perpendicular to
X
and
Z
.
For the waves polarized in the new meridian plane
Z
,X
, Equation (10.42) is always
valid, 1
/c
=
q
x
being replaced by
q
x
=
q
p
,and
q
=
q
z
being replaced by
q
z
. With this
Y
′
Y
″
X
″
q
y
′
q
p
′
ψ
X
′
q
x
′
in the plane
x
y
.
Figure 10.7
The vector
q
p
