Geoscience Reference
In-Depth Information
We shall further restrict ourselves to dealing with the plane problem. We write
equations (3.42) and (3.43) for the separate components:
∂
u
1
ρ
∂
p
=
−
x
+
Φ
x
,
(3.46)
∂
t
∂
∂
w
1
ρ
∂
p
=
−
z
+
Φ
z
−
g
,
(3.47)
∂
t
∂
∂
u
x
+
∂
w
=
s
.
(3.48)
∂
∂
z
w
∂
Neglecting vertical acceleration
∂
t
, we integrate equation (3.47) over the verti-
cal coordinate within limits from
z
to
ξ
. The result for the pressure is the following:
ξ
Φ
z
d
z
∗
,
p
(
z
)=
p
atm
+
ρ
g
ξ
−
ρ
g
z
−
ρ
(3.49)
z
where
is the displacement of the free surface and
z
is the running vertical coordi-
nate, varying within the limits
ξ
. Considering the free surface to deviate
insignificantly from its equilibrium position (
−
H
z
ξ
H
), it is correct to perform in-
tegration over the vertical coordinate not up to
z
=
ξ
ξ
,butto
z
= 0. Substituting
expression (3.49) into equation (3.46), we find:
0
∂
u
g
∂ξ
∂
∂Φ
z
∂
d
z
∗
+
=
−
x
+
Φ
x
.
(3.50)
∂
t
x
z
Integration of formula (3.50) over d
z
within limits from
−
H
to 0 yields the following
equation:
0
0
0
H
∂
U
∂
g
H
∂ξ
∂
∂Φ
z
∂
d
z
∗
+
=
−
x
+
d
z
Φ
x
d
z
,
(3.51)
t
x
z
−
H
−
H
where
U
is the horizontal velocity value averaged along the vertical direction. We
further integrate the continuity equation (3.48) over d
z
within the same limits:
0
H
∂
U
∂
+
∂ξ
∂
=
s
d
z
.
(3.52)
x
t
−
H
In obtaining expression (3.52) account was taken of the no-flow condition on the
ocean bottom,
w
(
x
,
H
,
t
)=0 (the ocean bottom is considered motionless for
the mean movement), while the vertical velocity at the surface is expressed as
the partial time derivative of the displacement
−
.
Further, calculating the partial derivatives with respect to
x
and
t
of equations
(3.51) and (3.52), respectively, and excluding the mixed derivative
ξ
2
U
∂
∂
x
∂
t
we
arrive at the inhomogeneous wave equation
