Graphics Reference
In-Depth Information
Equation (13.5) tells us that the value at a horizontal position (
x, z
)and
time
t
is the same as the initial wave form at time
t
and position (
x, z
)
c
k
tk
.
It's a lot like the advection equations we have seen over and over again,
only this time it's just the wave moving at that speed, not the individual
molecules of water. Equation (13.6) is called the
dispersion relation
, giving
the speed of a wave as a function of its wave number
k
. Remembering that
wavelength is inversely proportional to
k
, the dispersion relation shows
that waves of different sizes will travel at different speeds—in particular if
they all start off in the same patch of ocean, as time progresses they will
disperse, hence the name. This fact is probably the most crucial visual
element in a convincing ocean: it's what communicates to the audience
that there is significant depth below the water, whether or not they know
the physics underlying it.
In fact, what we have derived so far is just as valid for shallow water
(
H
small) as deep water (
H
big). If the waves are shallow, i.e.,
H
is
small compared to the wavelength so that
kH
is small, then asymptotically
c ∼
√
gH
. That is, the speed depends on gravity and depth but not
wavelength, which is exactly what we saw for shallow water in Chapter 12.
On the other hand, for deep water and moderate-sized waves, i.e., where
kH
is very large, to a very good approximation we have
−
g
k
,
gk,
c
k
≈
k
≈
(13.7)
which is in fact the simplified formula normally used in the ocean—beyond
a certain depth the dependence on
H
doesn't really matter. This form
makes it clear that longer waves (
k
small) move faster than short waves
(
k
large) in the ocean, which again is a very characteristic look: big swells
rushing past underneath slow moving ripples.
3
13.3
Evaluating the Height Field Solution
We derived the wave speed using only one component of the general solution
in time for the height field. You can double check that the other component
gives a wave moving at the same speed, but in the opposite direction
k
.
This leads to some redundancy with the Fourier mode associated with wave
vector
−
k
, which has the same wave number
k
, the same wave speed, and
−
3
Incidentally, the full spectrum of dispersion is also very easily seen in boat wakes:
if you look far enough away from the boat, the wake will have separated into big wave-
lengths at the leading edge and smaller scale stuff behind.
