Geoscience Reference
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Let us consider periodic two-dimensional deep-water waves whose dynamics is described
by principal potential equations. Because of the periodicity condition, the conformal map-
ping for infinite depth can be represented by the Fourier series (see details in
Chalikov &
Sheinin
,
1998
;
Chalikov & Sheinin
,
2005
):
x
=
ξ
+
0
η
−
k
(τ )
exp
(
k
ζ)ϑ
k
(ξ),
(4.1)
−
M
≤
k
<
M
,
k
=
z
=
ζ
+
0
η
k
(τ )
exp
(
k
ζ)ϑ
k
(ξ)
;
(4.2)
−
M
≤
k
<
M
,
k
=
where
x
and
z
are Cartesian coordinates,
ξ
and
ζ
conformal surface-following coordinates,
τ
is time,
η
k
are coefficients of the Fourier expansion of a free surface
η(ζ, τ)
with respect
to the new horizontal coordinate
ζ
:
η(ζ, τ)
=
h
(
x
(ζ, ξ
=
0
,τ),
t
=
τ)
=
M
η
k
(τ )ϑ
k
(ζ ),
(4.3)
−
M
≤
k
≤
ϑ
k
denotes the functions
cos
k
ξ
for
k
≥
0,
ϑ
k
(ξ)
=
(4.4)
sin
k
ξ
for
k
<
0
and
M
is the truncation number.
Non-traditional presentation of the Fourier transform with definition
(4.4)
is, in fact,
more convenient
for computations with real numbers, such as
(ϑ
k
)
ξ
=
k
ϑ
−
k
and
(
A
k
ϑ
k
)
ξ
=−
kA
−
k
ϑ
k
. So, the Fourier coefficients
A
k
form a real array
A
,
thus making possible a compact programming in Fortran90. Such a presentation can be
generalised for the three-dimensional case.
Note that the definition of both coordinates
(
−
M
:
M
)
is based on Fourier coefficients for
surface elevation. It then follows from
(4.1)
and
(4.2)
that time derivatives
z
τ
ξ
and
ζ
and
x
τ
for
Fourier components are connected by a simple relation:
−
(
z
τ
)
−
k
for
k
>
0,
(
x
τ
)
k
=
(4.5)
(
z
τ
)
k
for
k
<
0.
coordinates. It
is shown in
Chalikov & Sheinin
(
1998
) and
Chalikov & Sheinin
(
2005
) that the potential
wave equations can be represented in the new coordinates as follows:
As a result of conformity, the Laplace equation retains its form in
(ξ, ζ )
ξξ
+
ζζ
=
0
,
(4.6)
z
τ
=
x
ξ
G
+
z
ζ
F
,
(4.7)
2
J
−
1
−
1
2
2
ζ
τ
=
F
ξ
−
ξ
−
z
,
(4.8)
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