Civil Engineering Reference
In-Depth Information
Substituting Equation (9.70) for
p(x
1
,x
2
,
0
)
and using Equation (9.73) yields
∞
A
m,n
(L)
πR
2
U
0
(
1
Z(B
)
=
+
β
m,n
)
m
=
0
n
=
0
,
±
1
,
±
2
···
2
1
/
2
2
πR
m
2
n
D
−
(9.76)
k
0
sin
θ
2
π
R
×
J
1
D
2
+
m
2
D
2
+
2
1
/
2
n
D
−
k
0
sin
θ
2
π
By the use of Equations (9.74) and (9.40), this equation can be rewritten
2
1
/
2
2
πR
m
2
n
D
−
k
0
sin
θ
2
π
J
1
D
2
+
∞
2
πφ(L)
Z(B
)
ν
m
Z
m,n
(B)
=
m
2
2
n
k
0
D
sin
θ
2
π
m
=
0
n
=
0
,
±
1
,
±
2
···
+
−
(9.77)
This equation is similar to Equation (9.48). The (0, 0) mode contribution to
Z(B
)
in
Equation (9.77) is
J
1
(Rk
0
sin
θ)
πφ(L)
k
0
D
sin
θ
2
π
Z
0
,
0
(B
)
=
Z
0
,
0
(B)
(9.78)
2
To a first-order approximation, this equation can be rewritten
s
φ(L)
Z
0
,
0
(B)
Z
0
,
0
(B
)
=
(9.79)
The components
k
1
and
k
2
for this mode are
k
1
=
k
sin
θ
. The impedance
Z
0
,
0
(B)/φ(L)
would be the impedance at
B
if there were no facing.
0and
k
2
=
9.4.2 Evaluation of the external added length at oblique incidence
The added length
ε
e
at oblique incidence can be evaluated by the following equation
2
1
/
2
2
πR
m
2
n
D
−
k
0
sin
θ
2
π
J
1
D
2
+
π
m,n
2
ν
m
Z
c
k
k
m,n
m
2
2
j
ωρ
0
ε
e
=
(9.80)
n
k
0
D
sin
θ
2
π
+
−
The double prime over the symbol
means that the term
m
=
0
,n
=
0 is excluded
∞
=
±
2
...
+
(9.81)
m,n
m
=
1
n
=
0
,
±
1
,
m
=
0
,n
=±
1
,
±
2
...
