Civil Engineering Reference
In-Depth Information
Appendix 8.C Rayleigh pole contribution
Equation (8.51) can be rewritten (see Equations 7.6, 7.7)
∞
F
4
π
U
s
ξr)U
z
(ξ)ξ
d
ξ
H
0
(
z
(r)
=−
−
(8.C.1)
−∞
where
U
z
is given by Equation (8.38). There is a simple zero of
D
2
given by Equation
(8.41) at
ξ
=
ξ
R
.Let
a
be the parameter defined by
∂D
2
∂ξ
/
ξ
=
ξ
R
a
=
(8.C.2)
The Rayleigh wave contribution is given in the frequency domain by
4
π
(
−
2
πj)H
0
(
−
ξ
R
r)
T
11
(T
32
+
φs
3
)
−
T
12
(T
31
+
φs
2
)/
ξ
=
ξ
R
F
U
s
Rz
(r)
=−
(8.C.3)
a
The variables in the derivation of
D
2
are
α
1
,α
2
,α
3
and cos
θ
. The derivatives of these
parameters are
∂α
i
/∂ξ
ξ/k
0
cos
θ
.
=−
ξ/α
i
and
∂
cos
θ/∂ξ
=−
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