Geoscience Reference
In-Depth Information
∞
(
x
,
t
)=
η
0
π
d
k
ω
cos(
kx
)(
ω
sin(
ω
t
)
−
p
0
sin(
p
0
t
))
X
i
(
k
)
ξ
,
(3.59)
p
0
)
cosh(
k
)(
ω
2
−
0
where
+
∞
p
0
=
k
tanh(
k
)
,
X
i
(
k
)=
d
x
exp
{
ikx
}
η
i
(
x
)
.
−
∞
Expressions (3.57)-(3.59) contain under the integral sign dimensionless variables
(the asterisk '*' is omitted)
t
∗
=
t
g
H
1
/
2
k
∗
=
Hk
,
,
H
g
1
/
2
(3.60)
(
x
∗
,
z
∗
,
a
∗
,
b
∗
,
c
∗
)=
1
ω
∗
=
ω
,
H
(
x
,
z
,
a
,
b
,
c
)
,
but the coefficients in front of the integrals are dimensional.
Numerical calculation of the flow velocity components has shown that in the fre-
quency range considered, immediately after oscillations of the ocean bottom are
'switched on', each point of the liquid starts performing harmonic oscillations with
an amplitude depending only on its coordinates,
u
(
x
,
z
,
t
)=
u
(
x
,
z
) cos(
w
(
x
,
z
,
t
)=
w
(
x
,
z
) cos(
ω
t
)
,
ω
t
)
.
(3.61)
Substituting formulae (3.61) into expression (3.55) and subsequently averaging
over the period of oscillations, we obtain formulae for calculating the horizontal and
vertical components of the force field,
Φ
x
and
Φ
z
, respectively,
u
(
x
,
z
)
∂
,
u
(
x
,
z
)
∂
u
(
x
,
z
)
∂
1
2
+
w
(
x
,
z
)
∂
Φ
x
(
x
,
z
)=
−
(3.62)
x
z
u
(
x
,
z
)
∂
.
w
(
x
,
z
)
∂
w
(
x
,
z
)
∂
1
2
+
w
(
x
,
z
)
∂
Φ
z
(
x
,
z
)=
−
(3.63)
x
z
Functions
u
(
x
,
z
) and
w
(
x
,
z
) can be calculated from formulae (3.57) and (3.58)
at
t
= 0:
∞
−
η
0
ω
π
d
k
sin(
kx
) sinh(
kz
)
X
i
(
k
)
cosh(
k
)
u
(
x
,
z
)=
,
(3.64)
0
∞
w
(
x
,
z
)=
η
ω
π
d
k
cos(
kx
) cosh(
kz
)
X
i
(
k
)
cosh(
k
)
0
.
(3.65)
0
As a result we arrive at the following expressions for the components of the force
field:
