Civil Engineering Reference
In-Depth Information
Π
(3)
(2)
(1)
θ″
M
6
M
5
M
4
M
3
M
2
M
1
X
3
X
1
d
2
d
1
Figure 3.5
Three layers of fluid backed by an impedance plane
. The incident and
reflected waves are represented in fluid 3.
The impedance
Z(M
2
)
can be calculated by Equation (3.36),
d
being replaced by
d
1
.
In the same way,
Z(M
4
)
and
Z(M
6
)
can be calculated successively. This approach can
be used for any number of layers.
3.5
Reflection coefficient and absorption coefficient
at oblique incidence
The reflection coefficient at the surface of a layer is the ratio of the pressures
p
and
p
created by the outgoing and the incoming wave, as in the case of normal incidence. More
generally, this coefficient can be defined everywhere in a fluid backed by an impedance
plane. At any point
M(x
1
,x
2
,x
3
)
in the fluid represented in Figure 3.3, the reflection
coefficient is given by
R(M)
=
p
(M)/p(M)
(3.42)
where the pressures
p(M)
and
p
(M)
are given by Equations (3.30) and (3.32). Substi-
tution of Equations (3.30) and (3.32) in Equation (3.42) yields
A
A
exp
(
2
jk
3
x
3
)
R(x
3
)
=
(3.43)
and, like the impedance,
R
is a function of
x
3
only. If
k
3
is real,
R(x
3
)
describes a circle
in the complex plane as
x
3
varies, while if
k
3
is complex,
R(x
3
)
describes a spiral. With
the aid of Equation (3.35), the reflection coefficient can be written as
Z
c
k
k
3
Z(x
3
)
−
R(x
3
)
=
(3.44)
Z
c
k
k
3
Z(x
3
)
+
where
k/k
3
can be replaced by 1/cos
θ
.
An absorption coefficient
α
at oblique incidence can also be defined by Equation
(2.23):
2
α(M)
=
1
−|
R(M)
|
(3.45)
