Geoscience Reference
In-Depth Information
We now introduce dimensionless variables (the sign '*' will be further dropped)
m
∗
,
n
∗
}
x
∗
,
y
∗
,
z
∗
,
a
∗
,
b
∗
}
=
H
−
1
{
{
m
,
n
}
{
{
x
,
y
,
z
,
a
,
b
}
=
H
;
;
(2.108)
t
∗
,
τ
∗
}
g
1
/
2
H
−
1
/
2
;
{
ξ
∗
,
ζ
∗
}
η
−
1
0
{
=
{
t
,
τ
}
=
{
ξ
,
ζ
}
.
Part of the integrals, present in formula (2.39), can be calculated analytically. Zip-
ping intermediate calculations, we shall write out the formulae describing the per-
turbation of a free surface in the case of ocean bottom deformations of the form
(2.105)-(2.107):
•
Piston-like displacement
ξ
1
(
x
,
y
,
t
)=
θ
(
t
)
ζ
1
(
x
,
y
,
t
)
−
θ
(
t
−
τ
)
ζ
1
(
x
,
y
,
t
−
τ
)
,
(2.109)
•
Membrane-like displacement
ξ
2
(
x
,
y
,
t
)=2
θ
(
t
)
ζ
1
(
x
,
y
,
t
)
−
θ
−
τ
/
2)
ζ
1
(
x
,
y
,
t
−
τ
/
2)+2
θ
−
τ
ζ
1
(
x
,
y
,
t
−
τ
)
,
4
(
t
(
t
)
(2.110)
where
∞
∞
4
ζ
1
(
x
,
y
,
t
)=
d
m
d
n
2
π
τ
0
0
sin (
ma
) sin (
nb
) cos (
mx
) cos (
ny
) sin
(
k
tanh
k
)
1
/
2
t
mn
cosh
k
(
k
tanh
k
)
1
/
2
×
,
(2.111)
k
2
=
m
2
+
n
2
,
•
Running displacement
∞
+
∞
η
dm
exp
{
imx
sin (
nb
) cos (
ny
)
cosh (
k
)
n
}
0
ξ
3
(
x
,
y
,
t
)=
dn
2
π
2
i
−
∞
0
exp
i
2
a
m
+
(
k
tanh(
k
))
1
/
2
v
⎛
1
−
−
⎝
exp
i
(
k
tanh(
k
))
1
/
2
t
×
m
+
(
k
tanh(
k
))
1
/
2
v
exp
i
2
a
m
⎞
(
k
tanh(
k
))
1
/
2
v
1
−
−
−
⎠
exp
i
(
k
tanh(
k
))
1
/
2
t
+
−
.
(
k
tanh(
k
))
1
/
2
v
m
−
(2.112)
