Civil Engineering Reference
In-Depth Information
p
′
p
x
M
2
M
1
d
Z(M
2
)
Z(M
1
)
Figure 2.1
Plane waves propagate both in the
x
direction and in the opposite direction.
The impedance at
M
1
is
Z(M
1
)
.
At
M
2
, the impedance
Z(M
2
)
is given by
A
exp[
jkx(M
2
)
]
A
exp[
−
jkx(M
2
)
]
−
A
exp[
jkx(M
2
)
]
Z
c
A
exp[
−
jkx(M
2
)
]
+
Z(M
2
)
=
(2.14)
From Equation (2.13) it follows that
A
A
=
Z(M
1
)
Z
c
Z(M
1
)
+
Z
c
exp[
−
2
jkx(M
1
)
]
−
(2.15)
By the use of Equations (2.14) and (2.15) we finally obtain
Z
c
−
jZ(M
1
)
cotg
kd
+
Z
c
Z(M
2
)
=
(2.16)
Z(M
1
)
−
jZ
c
cotg
kd
where
d
is equal to
x(M
1
)
−
x(M
2
)
. Equation (2.16) is known as the impedance trans-
lation theorem.
2.3.2 Impedance at normal incidence of a layer of fluid backed
by an impervious rigid wall
A layer of fluid 1 backed by a rigid impervious plane of infinite impedance at
x
=
0is
represented in Figure 2.2. Two points
M
2
and
M
3
are shown at the boundary of fluids 1
and 2,
M
3
beinginfluid2and
M
2
in fluid 1. The impedance at
M
2
at the surface of the
M
3
M
2
M
1
p
′
Fluid 2
p
Fluid 1
x
x=0
x=
−
d
Figure 2.2
A layer of fluid of finite thickness in contact with another fluid on its front
face and backed by a rigid impervious wall on its rear face.
