Civil Engineering Reference
In-Depth Information
layers. Moreover, the pole contributions decrease exponentially when
k
0
R
1
increases and
become negligible. The asymptotic expression for
p
r
obtained to second-order approxi-
mationin1
/(k
0
R
1
)
by Brekhovskikh and Godin (1992), when the pole contributions are
neglected, can be written
V(
sin
θ
0
)
exp
(
−
jk
0
R
1
)
R
1
j
k
0
N
R
1
p
r
=
+
(7.20)
1
−
s
2
∂
2
V
∂s
2
1-2
s
2
2
s
∂V
∂s
N
=
+
(7.21)
2
s
=
sin
θ
0
This result is valid independently of the dependence of the reflection coefficient
V
on the angle of incidence. It has been shown in Brekhovskikh and Godin (1992) that
this result is also valid at small angles of incidence for large
k
0
R
1
. The coefficient
N
is given by Equations (7.A.3) -(7.A.4) for layers of finite thickness. For a semi-infinite
layer, when
l
→∞
,
N
is given by
n
2
)
[2
m(n
2
+
3
m
cos
2
θ
0
−
m
cos
4
θ
0
N
=
m(
1
−
−
1
)
n
2
n
2
−
sin
2
θ
0
cos
θ
0
(
2
n
2
+
sin
2
θ
0
)
]
(m
cos
θ
0
+
−
sin
2
θ
0
)
−
3
+
(7.22)
(n
2
−
sin
2
θ
0
)
−
3
/
2
×
where
m
ρ/(φρ
0
)
,
ρ
being the effective density of the air in the medium and
ρ
0
being
the free air density. This expression can be obtained by replacing the ratio of the densities
in Equation (1.2.10) in Brekhovskikh and Godin (1992), which gives the coefficient
N
at
a fluid - fluid interface, by
m
. In Brekhovskikh and Godin (1992), for a semi-infinite layer,
a supplementary term, the lateral wave, is added in Equation (7.20). This contribution is
neglected in the present work. For a layer of finite thickness, and for
θ
0
=
0
,N
is given by
=
1
+
M(
0
)
2
nk
0
l
(
1
−
n
2
)
sin
(
2
nk
0
l)
N
=
(7.23)
2
m
(
0
)(
1
−
n
2
)
M(
0
)
=
(m
(
0
)
+
n)
2
n
(7.24)
m
(
0
)
=−
jm
cot
(lk
0
n)
The coefficient
N
, for a given layer, only depends on the angle of specular reflection.
Predictions of the reflected pressure
p
r
obtained with the exact formulation, Equation
(7.6) and the passage path method are compared in Figures 7.3 and 7.4 and for a layer
of material 1 defined in Table 7.1. The source is the unit source of Equation (7.1).
Ta b l e 7 . 1
Acoustic parameters for different materials.
Materials
Tortuosity
Flow resistivity
Porosity
Viscous
Thermal
σ
(Nm
−
4
s)
α
∞
φ
dimension
dimension
(
(
µ
m)
µ
m
)
Material 1
1.1
20 000
0.96
100
300
Material 2
1.32
5500
0.98
120
500
