For the flat plate in problem 9.4, the quarter-chord point is located, by definition, at a distance equal to \(c / 4\) from the leading edge. Using linearized theory, derive the following expression for the moment coefficient about the quarter-chord point for supersonic flow

\[C_{M_{c / 4}}=\frac{-\alpha}{\sqrt{M_{\infty}^{2}-1}}\]

where \(C_{M_{c / 4}} \equiv M_{c / 4} / \frac{1}{2} ho_{\infty} V_{\infty}^{2} S c\), and as usual in aeronautical practice, a positive moment by convention is in the direction of increasing angle of attack.

Data From Problem 9.4:

Consider a flat plate with chord length \(c\) at an angle of attack \(\alpha\) to a supersonic free stream of Mach number \(M_{\infty}\). Let \(L\) and \(D\) be the lift and drag per unit span, respectively, and \(S\) be the planform area of the plate per unit span, \(S=c(1)\).