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While studying the derivation of the lifting line theory using different text books like Fundamentals of Aerodynamics by John D. Anderson, Precisely when deriving the general lift distribution over the wing, it assumed the following equation \begin{equation} \Gamma(\theta) = 2bV_{\infty} \sum_{n=1}^{N} A_{n} \sin(n\theta) \end{equation}

How did he just assumed that multiplying the Fourier series expansion by $2bV$ is a good assumption, I mean there must be a mathematical motive to do so, and I could not find anyone explaining the equation, so I hope someone can help me with that.

Thanks in advance

P.S

I suspected that the value $2bV$ represents the the lift distribution at the origin as in the equation

\begin{equation} \Gamma(\theta) = \Gamma_{0} \sin \theta \end{equation}

But still I can't relate both quantities. I thought that I can derive it from the equation

\begin{equation} \alpha_{i} = - \frac{w}{V_{\infty}} = \frac{\Gamma_{0}}{2bV_{\infty}} \end{equation}

but to do so I think we should assume that the induced angle of attack is equal to one and I don't understand that physically. Thank you so much for your help!

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  • $\begingroup$ 2bV is Gamma zero (eq 5.47 in my version) $\endgroup$
    – Jim
    Commented Jun 19, 2022 at 2:04
  • $\begingroup$ When I compared both equations I deduced that the quantity is actually gamma zero but I still don't how he got from equation 5.47 to the Fourier expansion series equation. $\endgroup$
    – Mostafa Zewail
    Commented Jun 19, 2022 at 22:50
  • $\begingroup$ Your question concerns the lift over span (3D) while my answer was for the 2D case. $\endgroup$
    – Peter Kämpf
    Commented Jun 20, 2022 at 17:32

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