Why does the Beech Staggerwing have its low wing in ahead of the high wing? What are the aerodynamic ideas behind this? Other biplanes of the era had the opposite, low wing behind the high one.
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Notice the retractable landing gear? With a tail-wheel design, the main gear work best at or ahead of the center of gravity. The Staggerwing has a neat and efficient retractable main gear, that folds onto the lower wing. But the lower wing has to be farther forward than usual for this to work.
The two wings on a biplane interfere with each other, but the biplane structure is very strong and stiff for its weight. So the Staggerwing's design gives up some wing efficiency for strength and light weight. Then it gives-up a little more for the reverse stagger that puts the downwash of the lower wing right under what should be the high pressure under the upper wing.
However, the design gets good speed from the retractable gear reducing drag. And both wings, upper and lower, join the fuselage at 90 degrees, with no struts or other sources of drag. Aerodynamically, a very clean design. the Beech Bonanza looked more modern, but the Staggerwing had the same cruising speed, 176 knots, 202 mph. It also had a 450 hp P&W Wasp Junior, while the Bonanza got along with a 300 hp Continental.
Wikipedia (certainly not a proper source) suggests:
The Model 17's unusual negative stagger wing configuration (the upper wing staggered behind the lower) and unique shape maximized pilot visibility while negligibly reducing interference between the wings.
which itself is sourced from "The Beechcraft Biplanes". Sport Avaition. January 1961.
Normally as you state the choice is for the lower wing to be further back due to the negative pressure on top which you do not want interfering and reducing the effectiveness of the upper wing.
Ted Wells did this just for visibility. It turned out that is was also good for flight characteristics as the bottom wing would stall first causing the airplane to pitch forward thus preventing the top wing from stalling. It also provided the ability to retract the gear into the wing.