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bug check is to make sure they all are zero or very close to zero. Some
FFT libraries allow you to specify that the result should be real-valued
and then allow you to define and pass in only half the
Y
coecients: this
can certainly be a worthwhile optimization, but the specifics of how to do
it vary from library to library.
13.4 Unsimplifying the Model
We made a lot of simplifying assumptions to get to an easily solved fully
linear PDE. Unfortunately, the resulting height field solution isn't terribly
convincing beyond very small amplitudes. In this section we'll try to boost
our solution to look better even at larger amplitudes, by compensating for
some of the terms we dropped earlier.
The first order of business is looking at the solution for the potential
φ
that accompanies our height field solution in Equation (13.8). We left
this hanging before, rushing on to the height field instead, but
φ
offers
some extremely useful information: in particular,
φ
gives us the implied
velocity field. As we'll see in a moment, the plain height field solution is
what you get when you ignore horizontal motion, letting the water bob
up and down but not side to side; the characteristic look of larger waves,
with wide flat troughs and sharper peaks, is largely due to this horizontal
motion so we will bring it back.
It's not hard to verify that the potential
φ
which matches the height
field in Equation (13.8) is as follows, building on our earlier incomplete
form and taking the limit as
H
∇
→∞
as we did for the wave speed
c
k
and
time frequency
ω
k
:
φ
(
x, y, z, t
)=
i,j
A
ij
ω
k
k
sin(
k
ω
k
t
+
θ
ij
)
e
ky
.
·
(
x, z
)
−
(13.15)
Taking the gradient gives us the complete velocity field, both at the surface
(
y
≈
0) and even far below:
∂x
=
i,j
u
(
x, y, z, t
)=
∂φ
A
ij
ω
k
2
πi
kL
cos(
k
ω
k
t
+
θ
ij
)
e
ky
,
·
(
x, z
)
−
=
i,j
v
(
x, y, z, t
)=
∂φ
∂y
A
ij
ω
k
sin(
k
ω
k
t
+
θ
ij
)
e
ky
,
·
(
x, z
)
−
(13.16)
=
i,j
w
(
x, y, z, t
)=
∂φ
∂z
A
ij
ω
k
2
πj
kL
cos(
k
ω
k
t
+
θ
ij
)
e
ky
.
·
(
x, z
)
−
