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equation it should satisfy:
D
Dt
(
h
(
x, z
)
−
y
)=0
∂h
∂t
+
u
∂h
v
+
w
∂h
∂z
⇒
∂x
−
=0
∂h
∂x
,
=
v.
∂h
∂t
+(
u, w
)
∂h
∂z
⇔
·
Just as with shallow water, this looks like a two-dimensional material
derivative of height, with vertical velocity
v
as an additional term.
We'll also make the assumption that the ocean floor is flat, at depth
y
=
H
for some suitably large
H
. While this is almost certainly false,
the effect of variations in the depth will not be apparent for the depths and
the wavelengths we're considering.
2
The solid “wall” boundary condition
at the bottom, where the normal is now (0
,
1
,
0), becomes
∂φ/∂y
=0.
We can now write down the exact set of differential equations we want
to solve:
−
∇·∇
φ
=0
for
−
H
≤
y
≤
h
(
x, z
)
,
∂φ
∂y
=0
at
y
=
−
H,
∂t
+
2
∇
2
+
gh
(
x, z
)=0 at
y
=
h
(
x, z
)
,
∂φ
φ
∂h
∂x
,
=
v.
∂h
∂t
+(
u, w
)
∂h
∂z
·
This is still hard to deal with, thanks to the non-linear terms at the free
surface. We will thus use the clever mathematical trick of ignoring them—
in effect, assuming that
u
is small enough and
h
is smooth enough that all
the quadratic terms are negligible compared to the others. This cuts them
down to
∂φ
∂t
=
−gh
(
x, z
)
t
y
=
h
(
x, z
)
,
∂h
∂t
=
v.
2
Once you get to truly big waves, tsunamis, variation in ocean depth becomes im-
portant: for a tsunami the ocean looks shallow, and so the previous chapter actually
provides a better model. However, in the deep ocean these waves are practically invisi-
ble, since they tend to have wavelengths of tens or hundreds of kilometers but very small
heights on the order of a meter.
