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The horizontal velocity 
can be found by substituting
, made up of its
sines and cosines, back into equation (3.1),
differentiating with respect to 
then integrating with respect to 
. Now,
differentiating and integrating
sines and cosines still gives sines and cosines: the
horizontal velocity of
the ocean also oscillates sinusoidally with time and
distance. If, when we first
wrote down the acceleration in the Newton's Second Law
equation (1.5), we had
considered the acceleration in the vertical
(
-direction) as well as in the
horizontal, we would have found that the vertical
velocity, too, oscillates
sinusoidally. This is what we experience when floating
well out from shore;
fluid particles in the ocean (and the swimmers and
surfboards and fish buoyed
up by them) trace out elliptical paths in vertical planes
- the result of a
combined sinusoidal velocity in the horizontal and
vertical.
Figure 4: Waves propagate but particles in the ocean bob up and down.
Using the standard identities for sums and products of sines and cosines, (3.14) and (3.15) can be re-arranged to give a better way of looking at the wave solution,
where 
and 
are constants. Look only at the first term (the
cosine) and draw a
graph of it at time 
, as in figure 4(a). (Things
turn out similarly for
the sine term.)
In figure 4(b) the graph has been redrawn for a time of
second. The wave
pattern has moved 
metres to the right,
and as more time goes
by it will move further to the right. But the actual particles in the
ocean (like the boat and the fish) do not move in the
same way as the wave
pattern; they just bob around on their elliptical paths.
This is consistent
with what we know happens in reality, for the ocean far
from shore.
Since the wave pattern moves metres in a
second, its wave speed 
is given by
(The sine term
in equation (4.1) represents a wave pattern going the
other way, with a wave
speed 
in reality, a
disturbance in the ocean makes
waves that travel both ways, away from the disturbance.)
A practical application of this wave solution can already
be made. Say there is
a cyclone making a giant disturbance out in the Capricorn
Channel 500 km north
of Brisbane - a physical situation for which our
equations are relevant. The
ocean in the Capricorn Channel has a depth 
of about
40 m (and by the way, this isn't
far off the scaling assumption we made in assuming that
the ocean was
`something like' 50 m deep). The acceleration due to
gravity, 
, is about
. Then from (4.2),
The big waves generated by the cyclone travel at
, and make it to
the coast near Brisbane in 25000 seconds - about 7 hours,
enough time for
everyone to be warned. (In practice, more sophisticated
equations would be
used, but the principles of the analysis are similar.)
However the linear wave solution (4.1) - the
one
obtained by scaling out
the nonlinear terms - does not always describe reality.
Physically, as the wave
pattern travels into shallower water - towards a reef or
beach, say, its speed
slows down, because the depth of the ocean is
getting less. The fluid
particles, however, have an enormous momentum behind them
as they oscillate to
and fro, and they do not slow down as much as the wave
pattern. As the water
gets shallower and shallower, the fluid particles start
to `catch up' with the
crests of the wave pattern; the front of the wave pattern
gets steeper.
Eventually the wave is slowed down so much that the fluid
particles actually
overshoot the crest, plunging out from the tip of the
crest in a curling
breaker.
Mathematically, the scaling assumptions, in particular
the assumption 
, begin to get unrealistic when the wave
gets into shallower
water. This is what we should expect; we chose the
scalings to represent the
conditions far from shore. As the water gets shallower,
gets smaller
(and 
often gets smaller too, for reefs and
beaches often have
horizontal scales less than 100 m). Because
and are the
denominators of the nonlinear terms, the nonlinear terms
get larger until they
can no longer be neglected. Eventually, the nonlinear
terms dominate the
acceleration in the Newton's Second Law equation, causing
wild variations from
the linear wave solution which has completely broken
down.
In summary, the mathematical equations of fluid dynamics are `nonlinear differential equations'. The nonlinear terms in the equations come from the acceleration in Newton's Second Law, and represent a fluid particle's special ability to change its velocity by changing its position within the flowing mass. Most of the time, as with the ocean far from shore, the nonlinear terms are unimportant - the behaviour is described by the solutions to the scaled or `linearized' equations. However, the ocean waves that are a fundamental feature of the linear solution eventually break as the conditions change and the nonlinear terms take control. Surf is the final, chaotic solution to the nonlinear equations. Solving nonlinear equations, analytically or by computer, is one of the most interesting areas of applied mathematics and is a skill increasingly demanded of scientists and engineers - especially those working with fluids!
R. Manasseh Papers
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