A $2$D supersonic airfoil has a flat lower surface aligned with the flow direction and an upper surface described by $y=0\mathrm{.}5x(1-x),0x1.$ The freestream Mach number is $M_1=3$ and the freestream static pressure is $p_1=1$atm.

$($a$)$ Sketch all the wave phenomena on the airfoil.

$($b$)$ Find an expression for the distribution of the Prandtl Meyer angle

$u(x)$ along the upper surface.

$($c$)$ Calculate the pressure on the following points on the upper surface of the airfoil: leading edge; crest; and trailing edge.