The general thin aerofoil section
In general, the camber line can be any function of x (or 9) provided that yc = 0 at x = 0 and c (i. e. at 9 = 0 and 7г). When trigonometric functions are involved a convenient way to express an arbitrary function is to use a Fourier series. Accordingly, the slope of the camber line appearing in Eqn (4.22) can be expressed in terms of a Fourier cosine series
Sine terms are not used here because practical camber lines must go to zero at the leading and trailing edges. Thus yc is an odd function which implies that its derivative is an even function.
Equation (4.22) now becomes
thin aerofoil is constant, depending on the camber parameters only, and the quarter chord point is therefore the aerodynamic centre.
It is apparent from this analysis that no matter what the camber-line shape, only the first three terms of the cosine series describing the camber-line slope have any influence on the usual aerodynamic characteristics. This is indeed the case, but the terms corresponding to n > 2 contribute to the pressure distribution over the chord.
Owing to the quality of the basic approximations used in the theory it is found that the theoretical chordwise pressure distribution p does not agree closely with
experimental data, especially near the leading edge and near stagnation points where the small perturbation theory, for example, breaks down. Any local inaccuracies tend to vanish in the overall integration processes, however, and the aerofoil coefficients are found to be reliable theoretical predictions.