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I've heard of people talking of wingspan and effective wingspan.

Considering:

L = Lift Coefficient . 1/2 Rho . Velocity ^ 2 . Surface Area,

I understand that tip vortices increase drag, but do they reduce the effective wingspan of a wing?

How can this be calculated?

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I think the term "effective wingspan" should better be replaced by "effective aspect ratio". There is no change in wingspan when the wing flies closer to the ground, but it behaves like an equivalent one with a higher aspect ratio. The basic formulas can stay unchanged, only aspect ratio should be adjusted.

Think of it this way: If there was an increase in span, would wing area grow with it? Would that mean that wing loading goes down? No, it is better to assume that the wing near to the ground behaves like one of the same area, but of a higher aspect ratio with an increased span and a proportionally shortened chord.

Sighard Hörner gives an approximation on page 7-13 in his book Fluid Dynamic Lift for the increment by which the effective aspect ratio is increased over the geometric aspect ratio: $$\frac{∆AR}{AR} = 0.09\cdot\frac{h}{b}$$

b is wingspan and h is the height of the aerodynamic center over ground.

He also has graphs for the change in effective aspect ratios of vertical tails with the height at which a horizontal tail is attached, or the effective aspect ratio of a biplane wing: Plot of effective aspect ratio over geometric aspect ratio of a vertical tail depending on the vertical location of the horizontal tail
Plot of effective aspect ratio over geometric aspect ratio of a vertical tail depending on the vertical location of the horizontal tail (picture source)

The case where an effective span makes sense is in the calculation of the induced drag and vortex location of a rounded wingtip. Here is Hörner's list of wingtip shapes from the same book:

Wing-tip shape and tip-vortex location for a family of wings
Wing-tip shape and tip-vortex location for a family of wings (picture source)

And lastly, no, wingtip vortices don't cause induced drag. It doesn't become true just because it is repeated so often. Please look here or here for a better explanation.

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