Yes, this is a common "urban legend", promoted by old pictures in high school
textbooks of placer mining using water cannons. The pictures plainly showed
the water line coming off the mountain, getting smaller as it came down.
However this was due strictly to cost savings in using a smaller high-pressure
(more expensive) pipe at the bottom, and a cheaper low pressure pipe at the
top. The pressure, as you pointed out, is strictly dependent on the amount of
head, and is only indirectly related to the pipe size (through friction loss).
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