Show that for a steady supersonic flow the lifting pressure coefficient becomes

\[ C_{p a}(x, y)=-\frac{4}{\pi} \frac{\partial}{\partial x} \iint_{V} \frac{w(\xi, \eta)}{U} \frac{d \xi d \eta}{R} . \]

Find the lift line slope for the wing of Problem 5.27 using Mach box technique. Note that over each box the integral of the Kernel reads as

\(\left.\frac{\partial}{\partial x} \iint_{B o x} \frac{d \xi d \eta}{R}=\left[\left(-\arccos \frac{\bar{\eta}}{\xi}\right)_{\bar{\eta}_{l}}^{\bar{\eta}_{u}}\right]_{\bar{\xi}_{l}}\right]_{\bar{\xi}_{u}}, \quad\) where \(\quad \bar{\xi}=x-\xi, \quad \bar{\eta}=\beta(y-\eta)\), \(\beta=\sqrt{M^{2}-1}\)

(If \(\bar{\eta} / \bar{\xi}>1\) then take:arccos \(\frac{\bar{\eta}}{\xi}=0\) ) and \(l\) and \(u\) stand for lower and upper integral limits respectively.

Problem 5.27

The wing given by Problem 5.25 oscillates with the reduced frequency of \(k=0.2\). Obtain the lifting pressure curve for the spanwise change. Find the total lift coefficient.