Yes, I'm pulling this thread back from the dead just to update; it does shows up 2nd in a google search for the subject of "calculating area rule", so this is for anyone wanting to answer the question as I've long since answered it.
Using the formulas in the picture from the previous reply #8 above as reference:
First set up a coordinate system; x-axis is chordwise, y-axis is spanwise, and z-axis is [chord] thickness-wise.
Yle is the y1(x) formula for the leading edge of the wing
Yte is the y2(x) formula for the trailing edge of the wing
b/2 is of course the half span of the wing.
I've set the y foil as the wing cross section to instead be z(x) = 0.4x - 0.4x^2, where the max thickness zmax = z(x at dz/dx=0) = 0.1
I assumed the chord length to vary with the span-wise distance y from leading to trailing edges, so there are three segments:
1) Yleading - 0
... y1(x) - 0 = x from x=0 to x=1,
2) Yleading - Y trailing
... y1(x) - y2(x) = x - (1.5x-1.5) = 1.5 - x from x=1 to x=2,
3) b/2 - Y trailing.
... b/2 - y2(x) = 2 - (1.5x-1.5) = 0.5 - 1.5x from x=2 to x=2.33
** It is easier to do this on a spreadsheet program like MS
Excel as the above segments aren't continuous, it makes it difficult on paper or on a calculator. **
Since throughout each segment, the area is following along the wing cross section, that is the integration of the foil shape, we must scale the shape of the wing with this variation, as y1(x) - y2(x). This scale not only affects z(y(x)) plane chord length but also the thickness, which is zmax/chord * new chord scale, z(y(x)).
First integrate z(x) as z'(x) = 0.2x^2 - 0.1333x^3.
Then change formula by scale formula (and change of variable since plane of reference is normal to flight direction, we're assuming Mach 1):
z'(y(x)) = [(zmax/c) * (y1(x) - y2(x)]*[0.2*(x/(y1(x) - y2(x))^2 - 0.13333*(x/(y1(x) - y2(x))^3]
in our case, which chord = 1,
z'1(x) = [(0.1/1) * (x)]*[0.2*(x/x)^2 - 0.13333*(x/x)^3] from x = (0,1)
z'2(x) = [(0.1/1) * (1.5-x)]*[0.2*(x/(1.5-x))^2 - 0.13333*(x/(1.5-x))^3] from x = (1,2)
z'3(x) = [(0.1/1) * (2-1.5x)]*[0.2*(x/(2-1.5x))^2 - 0.13333*(x/(2-1.5x))^3] from x = (2,2.33)
When finally ploted out, it should look like this:
If wanting the volume of this wing section, simply integrate it (each individual z' within their ranges). I'm sure you can guess, the more complex airfoil along with complex leading and trailing edges, the crazier this math can get. Use a computer.
Note, with increasing Mach numbers, the oblique wave is angled away from 90-deg so it is that new slanted plane that "sees" the airplane and wing cross section, it isn't as simple as the above math.
Hope this helps anyone, if there are any questions, send me an email at firstname.lastname@example.org
[Edited 2009-12-26 23:24:31]
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