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With the simplified Newton's Second Law equation, the two differential equations now look like this,
The first step is to
get from two equations
in two unknowns (
and 
) to one equation in one
unknown. Eliminate by
differentiating (3.1) just with respect to
and
(3.2) just with
respect to 
,
and
by multiplying (3.3) by
and adding the result to (3.4),
Re-arranging (3.5) gives a famous equation in mathematical physics,
Since
depends on both
time and distance , assume that
where 
is just a function
of time and 
is just a function of distance. Now, as far as the
differentiation just with
respect to (on the left-hand side of (3.6)) is
concerned, is just a
constant (it doesn't have any s in it to
differentiate). The reverse is
true for the 
on the
right-hand side of (3.6),
which now becomes,
Now there is something special about
(3.9). The left-hand side
is just a function of while the right-hand side is
just a function of .
The only way this could be possible is if both sides are
equal to a
constant (
, say),
The equation for
has been split into two equations; let's look
at the equation (3.10) for the time dependence of
,
which can be written,
A simple exponential would satisfy (3.12) but it
would not
be a
physically realistic
solution, since (the local ocean surface height above
sea level) cannot
change exponentially with time forever - it must be zero
(at sea level) on
average. It's easier to see the true solution by writing
where
is just another constant,
The
solution can now be seen to be a sine wave multiplied by
some constant 
,
(We have to
include a cosine since a cosine could satisfy (3.13)
just
as well as a sine).
Similarly the equation (3.11) for yields a
sinusoidal
solution,
In fact, equation (3.6) is famous because its realistic solutions are sinusoidal waves; it is a form of the wave equation that describes all sorts of waves, like radio waves and sound waves, as well as waves on the surface of the ocean. Since
it can now be seen that the natural state of the ocean surface, for our chosen conditions far from shore, is a motion where it oscillates sinusoidally with both time and distance - waves !
R. Manasseh Papers
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