Civil Engineering Reference
In-Depth Information
were
D
2
is given by
D
2
=
[
T
41
+
(
1
−
φ)s
2
][
T
32
+
φs
3
]
−
[
T
42
+
(
1
−
φ)s
3
][
T
31
+
φs
2
]
(8.40)
The denominators
D
1
and
D
2
are related by
D
2
=
T
41
T
32
−
T
31
T
42
−
D
1
Z
0
cos
θ
(8.41)
A comparison of Equations (8.30) and (8.41) shows that the singularities of the
velocity components
U
z
and
U
x
are located at the same
x
wave number component
ξ
as
the singularities of the reflection coefficient.
The frame velocity components created by a plane field in air are used at the end
of the chapter to predict the displacements induced by a point source in air. The frame
velocity components obtained for a mechanical excitation are used in what follows for
different geometries of the excitation.
8.2.2 Circular and line sources
Let g(
r
) be the radial spatial dependence of the circular source
τ
s
. Using the direct and
inverse Hankel transforms
∞
G
(ξ)
=
r
g
(r)
J
0
(ξr)
d
r
0
(8.42)
∞
g
(r)
=
ξ
G
(ξ)
J
0
(ξr)
d
ξ
0
the axisymmetric source can be replaced by a superposition of excitations with a spa-
tial dependence given by the Bessel function J
0
(
rξ
). The following equivalence (see
Equations (7.3) - (7.6)) and Appendix 8.B
∞
∞
∞
1
2
π
G
(ξ)
J
0
(rξ)ξ
d
ξ
=
G
(ξ)
exp
(
−
j(ξ
1
x
+
ξ
2
y)
d
ξ
1
d
ξ
2
(8.43)
−∞
−∞
0
ξ
1
+
ξ
2
ξ
=
shows that J
0
(ξr)
can be replaced by a superposition of unit fields with different orienta-
tions and the same radial wave number
ξ
. The total vertical velocity of the frame
s
z
(r)
U
can be written (see Appendix 8.B)
∞
1
2
π
U
s
ξG(ξ)J
0
(ξ) U
z
(ξ)
d
ξ
z
(r)
=
(8.44)
0
Using
2
π
j
2
π
J
1
(u)
=
exp
(
−
ju
cos
θ)
cos
θ
d
θ
0
