Let's figure this one out. Usually a stream flows along for a while
carrying about the same amount of water. Water doesn't normally keep
piling up at on place, overflowing the banks, and it's also not usually
evaporating very fast. Unless we look somewhere where there's a
tributary joining up or a big rainstorm, new water isn't joining in
That means that if you look at two slices across the stream, the
same amount of water flows past each slice in each second. The volume
of water going through the slice per second is the speed of the flow
times the area of the slice. So in places where the cross-sectional
area is small (that is, the stream is narrow and shallow) the flow must
be fast and where it's large (broad and deep) the flow must be slow.
Of course this whole argument breaks down if water is entering or leaving the stream.
Another disclaimer: real streams often have more complicated
structures -- they may have a deep channel in the middle with sandy or
rocky shallow flat sections of the bottom near the sides. In this case,
the water will flow more quickly in the deep center section than in the
shallow parts because of friction with the rocky or sandy bottom. If a
stream is making a tight bend, water will tend to flow on the outside
of the bend and leave silt on the inside. But the total amount of water
flowing across one slice of a stream has to be the same as any other
slice unless water is entering or leaving.
(republished on 07/31/06)