Geoscience Reference
In-Depth Information
Fig. 2.8 Mathematical formulation of the 3D problem
shall put the origin of the Cartesian reference frame, 0
xyz
, in the unperturbed free
surface and direct the 0
z
axis vertically upward (Fig. 2.8). The liquid is at rest until
the time moment
t
= 0. To find the wave perturbation
(
x
,
y
,
t
),formedonthesur-
face of the liquid, and the velocity field, v(
x
,
y
,
z
,
t
), throughout the thickness of
the layer in the case of motions of the basin floor, occurring in accordance with
the law
ξ
(
x
,
y
,
t
), we shall resolve the problem with respect to the velocity poten-
tial
F
(
x
,
y
,
z
,
t
) [Landau, Lifshits (1987)]:
η
2
F
2
F
2
F
∂
+
∂
+
∂
= 0
,
(2.29)
∂
x
2
∂
y
2
∂
z
2
2
F
∂
g
∂
F
−
∂
=
t
2
,
z
= 0
,
(2.30)
∂
z
∂
F
=
∂η
∂
t
,
z
=
−
H
.
(2.31)
∂
z
The physical meaning of the boundary condition (2.30) consists in the pressure
on the free surface of the liquid being constant. The boundary condition (2.31)
signifies equality of the vertical component of the flow velocity to the velocity of
motion of the basin floor (a no-flow condition). Displacement of the free surface and
the flow velocity vector are related to the potential of the flow velocity by the fol-
lowing known formulae:
z
=0
1
g
∂
F
ξ
(
x
,
y
,
t
)=
−
,
(2.32)
∂
t
