Civil Engineering Reference
In-Depth Information
R
2
M
S
z
2
r
z
1
Z
θ
0
Reflecting layer
S
′
R
1
Rigid impervious backing
Figure 7.1
The source - receiver geometry, the monopole source at S, its image at S
,
and the receiver at M above the layer. The angle
θ
0
is the angle of specular reflection,
R
1
is the distance from the image of the source to the receiver, and
R
2
is the distance
from the source to the receiver.
and
p
at M can be rewritten
∞
∞
=
−
j
2
π
exp[
−
j(ξ
1
x
+
ξ
2
y
+
µ
|
z
2
−
z
1
|
)
]
p(R)
d
ξ
1
d
ξ
2
,
µ
−∞
−∞
(7.2)
k
0
−
ξ
1
−
ξ
2
,
Im
µ
µ
=
≤
0
,
Re
µ
≥
0
Let
V(ξ
1
,ξ
2
)
be the plane wave reflection coefficient of the layer. If the layer is
isotropic or transversely isotropic with the axis of symmetry Z,
V
only depends on
ξ
(ξ
1
+
ξ
2
)
1
/
2
, and the reflected pressure
p
r
at M can be written
=
∞
∞
=
−
j
2
π
V(ξ/k
0
)
exp[
−
j(ξ
1
x
+
ξ
2
y
−
µ(z
1
+
z
2
))
]
µ
p
r
d
ξ
1
d
ξ
2
(7.3)
−∞
−∞
The variables
ξ
and
µ
are related to an angle of incidence
θ
defined by
cos
θ
=
µ/k
0
(7.4)
sin
θ
=
ξ/k
0
and
V(ξ/k
0
)
is the reflection coefficient for the angle of incidence
θ
.For
ξ
≤
k
0
,
θ
is a
real angle, and for
π
2
+
jβ
ξ >k
0
,θ
=
where sinh
β
ξ/k
0
.
Using the polar coordinates (
ψ
,
ξ
)and(
r
,
ϕ
)definedby
ξ
1
=
jµ/k
0
,cosh
β
=
=
ξ
cos
ψ
,
ξ
2
=
ξ
sin
ψ
,
x
=
r
cos
ϕ
,
y
=
r
sin
ϕ
, Equation (7.3) can be rewritten
∞
2
π
=
−
j
2
π
z
2
)
]
V(ξ/k
0
)
µ
p
r
exp[
jµ(z
1
+
exp[
−
jrξ
cos
(ψ
−
ϕ)
]
ξ
d
ξ
d
ψ
(7.5)
0
0
Using
2
π
0
exp[
−
jrξ
cos
(ψ
−
ϕ)
]d
ψ
=
2
π
J
0
(rξ)
(Abramovitz and Stegun 1972),
Equation (7.5) becomes
p
r
=−
j
∞
0
V(ξ/k
0
)
µ
J
0
(rξ)
exp[
jµ(z
1
+
z
2
)
]
ξ
d
ξ
(7.6)
