Civil Engineering Reference
In-Depth Information
The Hankel functions can be expanded to
4
2
i
x
1
0
Hx
() 1
(
J
ln )
2
!
S
(14.12)
2
i
ix
1
x x
1
1
Hx
()
J
SS
(
ln )
!
x
2
2
2
where
J (Euler constant) and here only the first terms of the expansion are
shown. A plot of the fundamental solution for
U
is shown in Figure 14.1.
0.57721
14.2.2
Boundary integral equations
As with the static system the reciprocal theorem of Betti can be used to obtain the
integral equation. In the absence of body sources (distributed over the domain) we obtain
³
³
(14.13)
ˆ
()
cu P
U P Q
(, , ) (, )
Z
q Q
Z
dS
T P Q
(, , ) (, )
Z
u Q
Z
dS
S
S
w
w
u
and
c
depends on the boundary shape.
where the pressure gradient is defined as
q
n
Figure 14.1
Plot of the fundamental solution
U
for real part and
k
Z
/
c
4
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