Civil Engineering Reference
In-Depth Information
Π
d
1
d
2
d
3
x
M
6
M
5
M
4
M
3
M
2
M
1
Figure 2.3
Three layers of fluid backed by an impedance plane II.
layer of fluid 1 is obtained from Equation (2.16) with
Z(M
1
)
infinite:
Z(M
2
)
=−
jZ
c
cotg
kd
(2.17)
where
Z
c
is the characteristic impedance and
k
the wave number in fluid 1.
The pressure and the velocity are continuous at the boundary. The impedance at
M
3
is equal to the impedance at
M
2
, the velocities and pressures being the same on either
side of the boundary:
Z(M
3
)
=
Z(M
2
)
(2.18)
2.3.3 Impedance at normal incidence of a multilayered fluid
A multilayered fluid is represented in Figure 2.3. If the impedance
Z(M
1
)
is known, the
impedance
Z(M
2
)
inside fluid 1 can be obtained from Equation (2.16). The impedance
Z(M
3
)
is equal to the impedance at
M
2
. The impedance at
M
4
,M
5
and
M
6
can be
obtained successively in the same way.
2.4
Reflection coefficient and absorption coefficient
at normal incidence
2.4.1 Reflection coefficient
The reflection coefficient
R
at the surface of a layer is the ratio of the pressures
p
and
p
created by the outgoing and the ingoing waves at the surface of the layer. For instance,
at
M
3
, in Figure 2.2, the reflection coefficient
R(M
3
)
is equal to
p
(M
3
,t)/p(M
3
,t)
R(M
3
)
=
(2.19)
This coefficient does not depend on
t
because the numerator and the denominator
have the same dependence on
t
. Using Equation (2.15), the reflection coefficient at
M
3
can be written as
Z
c
)/(Z(M
3
)
Z
c
)
R(M
3
)
=
(Z(M
3
))
−
+
(2.20)
where
Z
c
is the characteristic impedance in fluid 2. The ingoing and outgoing waves at
M
3
have the same amplitude if
|
R(M
3
)
|=
1
(2.21)
