Graphics Reference
In-Depth Information
The
Fourier coecient
is
h
ij
(
t
); the
t
is included to emphasize that it
depends on time, but not on spatial variables. In general
h
ij
(
t
) will be
a complex number, even though in the end we'll construct a real-valued
height field—more on this in a minute. I use the notation
√
−
1instead
of the usual
i
(for mathematicians) or
j
(for engineers) since
i
and
j
are
reserved for integer indices. Speaking of which, the integers
i
and
j
are the
indices of this Fourier component—they may be negative or zero as well
as positive. The vector (
i, j
)
/L
gives the spatial frequency, and the vector
k
=2
π
(
i, j
)
/L
is called the
wave vector
. Define the
wave number
k
as
=
2
π
i
2
+
j
2
k
k
=
L
.
The
wavelength
is
λ
=2
π/k
=
L/
i
2
+
j
2
. As you can probably guess,
this corresponds precisely to what we would physically measure as the
length of a set of waves that this Fourier mode represents.
We'll now make the guess, which will prove to be correct, that when we
plug this in for the height field, the corresponding solution for
φ
(
x, y, z, t
)
will be in the following form:
φ
(
x, y, z, t
)=
φ
ij
(
t
)
e
√
−
12
π
(
ix
+
jz
)
/L
d
ij
(
y
)
.
We don't yet know what the depth function
d
ij
(
y
) should be. Let's first
try this guess in the interior of the domain, where
∇·∇
φ
= 0 should hold:
∇·∇φ
=0
∂
2
φ
∂x
2
+
∂
2
φ
∂y
2
+
∂
2
φ
∂z
2
⇔
=0
φ
+
d
ij
d
ij
4
π
2
i
2
L
2
4
π
2
j
2
L
2
−
φ
+
−
⇔
φ
=0
d
ij
d
ij
φ
=
k
2
φ
⇔
d
ij
=
k
2
d
ij
.
We now have an ordinary differential equation for
d
ij
, with the general
solution being a linear combination of
e
ky
and
e
−ky
. Note that the bottom
boundary condition,
∂φ/∂y
=0at
y
=
⇔
H
, reduces to
d
ij
(
H
) = 0. Since
our guess at
φ
already has a yet-to-be-determined factor
φ
ij
(
t
) built in, we
take
−
−
d
ij
(
y
)=
e
ky
+
e
−
2
kH
e
−ky
.
