Exercises

1. Consider the integral
   $$I_n(\phi) = \int_0^\pi \frac{\cos(n\,\phi')\,d\phi'}{\cos\phi'-\cos\phi},$$
   where $n$ is a non-negative integer. This integral is defined by its Cauchy principal value
   $$I_n(\phi)=\lim_{\epsilon\rightarrow 0}\left[ \int_0^{\phi-\epsilo... ..._{\phi+\epsilon}^\pi \frac{\cos(n\,\phi')\,d\phi'}{\cos\phi'-\cos\phi}\right].$$
   As was demonstrated in Section 9.9,
   $$I_0 = 0.$$
   Show that
   $$I_1=\pi,$$
   and
   $$I_{n+1} + I_{n-1} = 2\,\cos\phi\,I_n,$$
   and, hence, that
   $$I_n = \pi\,\frac{\sin(n\,\phi)}{\sin\phi}.$$

2. Suppose that an airfoil of negligible thickness, and wingspan $b$ , has a width whose $z$ variation is expressed parametrically as
   $$c(\phi) = \sum_{\nu=1,3,5,\cdots}c_\nu\,\sin(\nu\,\phi),$$
   for $0\leq \phi\leq \pi$ , where
   $$z = -\frac{b}{2}\,\cos\phi.$$
   Show that the air circulation about the airfoil takes the form
   $${\mit\Gamma}(\phi) = \sum_{\nu=1,3,5,\cdots} {\mit\Gamma}_\nu\,\sin(\nu\,\phi),$$
   where ${\mit\Gamma}_\nu = \pi\,V\,\alpha\,c_\nu$ . Here, $\alpha$ is the angle of attack (which is assumed to be small). Demonstrate that the downwash velocity at the trailing edge of the airfoil is
   $$w(\phi)= \sum_{\nu=1,3,5,\cdots} \frac{\nu\,{\mit\Gamma}_\nu}{2\,b}\,\frac{\sin(\nu\,\phi)}{\sin\phi}.$$
   Hence, show that the lift and induced drag acting on the airfoil take the values

   $$L = \frac{\pi}{4}\,\rho\,V\,b\,{\mit\Gamma}_1,$$

   $$D =\frac{\pi}{8}\,\rho\,\sum_{\nu=1,3,5,\cdots} \nu\,{\mit\Gamma}_\nu^{\,2},$$

   
   
   respectively. Demonstrate that the drag to lift ratio can be written
   $$\frac{D}{L} = \frac{2}{A}\,\alpha\left(1+\sum_{\nu=3,5,7,\cdots}\frac{\nu\,c_\nu^{\,2}}{c_1^{\,2}}\right),$$
   where $A$ is the aspect ratio. Hence, deduce that the airfoil shape (in the $x$ -$y$ ) plane that minimizes this ratio (at fixed aspect ratio) is an ellipse (i.e., such that $c_\nu=0$ for $\nu>1$ ).

3. Consider a plane that flies with a constant angle of attack, and whose thrust is adjusted such that it cancels the induced drag. The plane is effectively subject to two forces. First, its weight, ${\bf W} = -W\,{\bf e}_y$ , and second its lift ${\bf L} = -k\,v\,v_y\,{\bf e}_x + k\,v\,v_x\,{\bf e}_y$ . Here, $x$ and $y$ are horizontal and vertical coordinates, respectively, ${\bf v}$ is the plane's instantaneous velocity, and $k$ is a positive constant. Note that the lift is directed at right angles to the plane's instantaneous direction of motion, and has a magnitude proportional to the square of its airspeed. Demonstrate that the plane's equations of motion can be written

   $$\frac{dv_x}{dt} = - \frac{v\,v_y}{h},$$

   $$\frac{dv_y}{dt} = \frac{v\,v_x}{h} - g,$$

   
   
   where $h=k\,g/W$ is a positive constant with the dimensions of length. Show that
   $$\frac{1}{2}\,v^{\,2} + g\,y = {\cal E},$$
   where ${\cal E}$ is a constant. Suppose that $v_x=\sqrt{g\,h}\,(1+u)$ and $v_y=\sqrt{g\,h}\,w$ , where $\vert u\vert$ , $\vert w\vert\ll 1$ . Demonstrate that, to first order in perturbed quantities,

   $$\frac{du}{dt} \simeq - \sqrt{\frac{g}{h}}\,w,$$

   $$\frac{dw}{dt} \simeq 2\sqrt{\frac{g}{h}}\,u.$$

   
   
   Hence, deduce that if the plane is flying horizontally at some speed $v_0$ , and is subject to a small perturbation, then its altitude oscillates sinusoidally at the angular frequency $\omega= \sqrt{2}\,g/v_0$ . This type of oscillation is known as a phugoid oscillation.