Civil Engineering Reference
In-Depth Information
cos
θ
1
=
1
−
1
/n
2
,and
V(s
0
)
is given with the same approximation by
jn(Z/φZ
0
)
cos
θ
0
cot
(k
0
l
√
n
2
(n
2
1
)
1
/
2
=
−
−
−
−
1
)
V(s
0
)
jn(Z/φZ
0
)
cos
θ
0
cot
(k
0
l
√
n
2
(7.B.20)
(n
2
−
1
)
1
/
2
−
−
1
)
If
θ
p
is also close to
π
/2, Equation (7.B..20) can be used to calculate cos
θ
p
which is
given by
nZ
n
2
1tan
k
0
l
n
2
j
φZ
0
cos
θ
p
=−
−
−
1
(7.B.21)
(see Equation 7.28 for the case of thin layers). The approximate reflection coefficient
V
has the same expression as
V
L
cos
θ
0
+
cos
θ
p
cos
θ
0
−
cos
θ
p
V(s
0
)
=
(7.B.22)
s
p
=
cos
2
θ
p
in the denominator at the
right-hand side of Equation (7.B.17), Equation (7.B.17) and (7.B.19) become identical.
Neglecting
the
term
multiplied
by
1
−
Appendix 7.C From the pole subtraction to the passage
path: locally reacting surface
,the reflected pressure evaluated with
the pole subtraction method has the same expression as that obtained with the passage
path method. The pole contribution, for the pole subtraction method, corresponds to the
In the case of a locally reacting surface, for large
|
u
|
term 2
√
πu
exp
(u
2
)
of Equation (7.48), and can be written
1
s
p
√
2
k
0
R
1
exp
−
3
jπ
2
√
π
exp
(u
2
)
−
exp
(
−
jk
0
R
1
)
R
1
4
SW
P
(s
p
)
=−
(7.C.1)
√
s
0
s
p
where
u
2
θ
0
)
]. This contribution is the same as
SW(s
p
)
given by
Equation (7.43) in the context of the passage path method. This contribution exists under
the same conditions. It may be shown that the condition Re
u>
0 in Equation (7.48)
corresponds to the fact that the pole has been crossed when the initial path of integration
is deformed into the passage path. Moreover, with
θ
0
and
θ
p
close to
π
/2 and
u
given by
Equation (7.50), the contribution of the term 1/2
u
2
in Equation (7.48) to
p
r
corresponds
to the contribution of
N
in Equation (7.20),
N
for an impedance plane being given in
Brekhovskikh and Godin (1992) by
=
jk
0
R
1
[1
−
cos
(θ
p
−
2cos
θ
p
(
1
−
cos
θ
0
cos
θ
p
)
(
cos
θ
0
−
cos
θ
p
)
3
N
=
(7.C.2)
and the product cos
θ
0
cos
θ
p
being neglected at the numerator. The expression of the
reflected pressure obtained with the pole subtraction method remains valid for large
numerical distances, and can replace the expression obtained with the passage path
method. A medium with a high flow resistivity can be replaced by an impedance plane
