Civil Engineering Reference
In-Depth Information
M
4
,
1
=
A
1
(
1
+
r
11
)
+
A
2
r
12
(8.A.19)
M
4
,
2
=
A
2
(
1
+
r
22
)
+
A
1
r
21
(8.A.20)
M
4
,
3
=
A
1
r
31
+
A
2
r
32
(8.A.21)
M
5
,
1
=
B
1
(
1
+
r
11
)
+
B
2
r
12
−
C
3
r
13
(8.A.22)
M
5
,
2
=
B
2
(
1
+
r
22
)
+
B
1
r
21
−
C
3
r
23
(8.A.23)
M
5
,
3
=
C
3
(
1
−
r
33
)
+
B
1
r
31
+
B
2
r
32
(8.A.24)
M
6
,
1
=−
jξ(
1
+
r
11
+
r
12
)
−
jα
3
r
13
(8.A.25)
M
6
,
2
=−
jξ(
1
+
r
21
+
r
22
)
−
jα
3
r
23
(8.A.26)
M
6
,
3
=
jα
3
(
1
−
r
33
)
−
jξ(r
31
+
r
32
)
(8.A.27)
Appendix 8.B Double Fourier transform and Hankel
transform
Double Fourier transform
∞
∞
1
4
π
2
f
(x,y)
=
F
(u,v)
exp[
−
j(ux
+
vy)
]d
u
d
v
(8.B.1)
−∞
−∞
∞
∞
F
(u,v)
=
f
(x,y)
exp[
j(ux
+
vy)
]d
x
d
y
(8.B.2)
−∞
−∞
If f depends only on
r
=
x
2
+
y
2
, F depends only on
w
=
√
u
2
+
v
2
.
Hankel transform
∞
f
(r)
=
w
F
(w)
J
0
(rw)
d
w
(8.B.3)
0
∞
F
(w)
=
r
f
(r)
J
0
(rw)
d
r
(8.B.4)
0
Relationship between the two transforms
Equations (8A.1) and (8A.2) can be rewritten with the polar coordinates
r
,
θ
instead of
x,y
,and
w
,
ψ
instead of
u,w
∞
2
π
1
4
π
2
f
(r)
=
F
(w)w
d
w
exp[
−
j(rw
cos
(ψ
−
ϕ))
]d
ψ
(8.B.5)
0
0
