Civil Engineering Reference
In-Depth Information
10.5.3 Decoupling of the air wave
In a first approximation, the frame displacement induced by a source in air is negligible
compared with the air displacement. Equation (10.44) can be rewritten
B
8
c
2
B
8
q
c
b
1
b
3
−
+
C
1
−
=
0
(10.73)
B
8
q
c
B
8
q
2
−
−
+
C
3
where
b
1
and
b
3
can be replaced by
φd
1
and
φd
3
. A nontrivial solution exists if
C
3
B
8
−
1
c
2
C
3
C
1
q
2
=
(10.74)
There are two opposite
z
slowness components for a given
x
slowness component
1/
c
. Denoting the wave number for the wave propagating in the x direction by
k
P
,the
wave number for the waves propagating in the
z
direction by
k
N
,(
k
N
=
ω
2
C
3
/B
8
,k
P
=
ω
2
C
1
/B
8
)
the wave number components
ω/
cby
k
x
and
ωq
by
k
z
, Equation (10.74) can
be rewritten
k
N
1
k
x
k
P
k
z
=
−
(10.75)
Equation (10.27) can be rewritten
jB
8
φ
u
1
k
x
u
3
k
z
p
=
+
(10.76)
The system of Equation (10.58) - (10.60) is replaced by
r
11
=
exp
(
−
2
jk
z
l)
(10.77)
The surface impedance
Z
s
can be written
p
e
v
z
=
jφB
8
(d
1
k
x
+
d
3
k
z
)(
1
+
exp
(
−
2
jk
z
l))
jωφd
3
(
1
(10.78)
−
exp
(
−
2
jk
z
l))
From Equation (10.73), the ratio
d
1
k
x
/(d
3
k
z
)
is given by
B
8
/c
2
C
1
−
d
1
k
x
d
3
k
z
=
(10.79)
B
8
/c
2
which leads to
k
z
1
+
d
1
k
x
d
3
k
z
k
N
=
(10.80)
1
−
1
/
2
k
x
k
P
The surface impedance can be rewritten
(ρ
ef
K
f
)
1
/
2
jφ
[1
Z
s
=
(k
x
/k
P
)
1
/
2
]
cot
jlk
z
(10.81)
−
