Civil Engineering Reference
In-Depth Information
Π
M
2
M
1
n
θ
θ
X
3
X
2
n
′
d
X
1
Figure 3.3
The ingoing and the outgoing plane waves associated with an impedance
plane II.
Substitution of Equations (3.30) - (3.33) into Equation (3.34) yields
A
exp
(jk
3
x
3
)
A
exp
(
−
jk
3
x
3
)
+
Z(M
1
)
=
(3.35)
A
Z
c
A
Z
c
k
3
k
exp
(
k
3
k
exp
(jk
3
x
3
)
−
jk
3
x
3
)
−
This expression depends only on
x
3
.
Equation (3.35) can be obtained by the substitution of
Z
c
k/k
3
and
k
3
, instead of
Z
c
and
k
, respectively, into Equation (2.14). With the same substitution, the impedance
translation theorem, Equation (2.16), can be rewritten
−
jZ(M
1
)
cotg
k
3
d
+
Z
c
k
k
3
Z
c
k
k
3
Z(M
2
)
=
(3.36)
jZ
c
k
k
3
Z(M
1
)
−
cotg
k
3
d
where
d
is equal to
x(M
1
)
−
x(M
2
)
.
3.4.2 Impedance at oblique incidence for a layer of finite thickness
backed by an impervious rigid wall
The layer is represented in Figure 3.4 in the incidence plane with the incident and
the reflected waves. The angles are real or complex. For instance, fluid 2 can be a
nondissipative medium with a real wave number
k
and fluid 1 a dissipative medium
with a complex wave number
k
.If
θ
in fluid 2 is real,
θ
in fluid 1 is complex and
defined by Equation (3.22).
k
1
=
k
sin
θ
=
k
1
=
k
sin
θ
(3.37)
Equation (3.4) yields
(k
2
k
1
)
1
/
2
k
3
=
−
(3.38)
