I wondered about a simple formula for this as
well. If you assume the earth is a perfect sphere,
then a tower of height k meters will have a line
of sight tangent to the ocean surface at a
Sqrt(R^2 - (R-k)^2) meters,
neglecting the tilt of the tower relative to the
tangent point. (i.e. we assume that this angle is
small) This can be further approximated by noting
that R (the earth Radius) is very much larger than
the typical tower height. We get:
above) = Sqrt(2*R*k) - Sqrt(k^3/8R) -
Dropping the very small
corrections, we get the neat
which is proportional
to the geometric mean of the tower height and the
earth radius. Plugging in the radius of the earth
(6360km) and a 2meter tall man, we get: 5 km until
the ocean is tangent to the man. So a 2 meter man
can see a 2 meter man's head (standing on the
surface) at about 10km.
From a 40meter
(133') bluff -- the tangent point is 22.5km out.
Thus two 40meter tall crow's nests can spot each
other at 45km...
Without the crow's nest, you
can only see the other ship at best at 22km...