$\frac{M_1}{C}=$:

A supersonic airfoil has equilateral triangular shape with tip angles of $5.$ It is placed at an angle of attack of $5$ against an airstream with $M_1=3.$

$($a$)$ Sketch the wave pattern around the airfoil.

$($b$)$ Use the exact $($shock$-$expansion$)$ theory to find the static pressure ratio $\frac{p}{p_1}$ in regions $2,3,$ and $4.$

$($c$)$ Using linear theory, find an approximate value for static pressure ratio $\frac{p}{p_1}$ in regions $2,3,$ and $4.$

$($d$)$ To compute the lift and drag of the airfoil, we first calculate the force $($per unit span$)$ acting on each airfoil surface $($since these are pressure forces, they act normal to the surface$).$ Summing the force components normal to flow direction gives the lift $L.$

Summing the force components parallel to flow direction gives the drag $D.$ The non$-$dimensional lift and drag coefficients are

$C_l=\frac{L}{\frac{1}{2}{}_1{U}_1^{2}c}=\frac{L}{\frac{1}{2}{p}_1{M}_1^{2}c}$

$C_d=\frac{D}{\frac{1}{2}{}_1{U}_1^{2}c}=\frac{D}{\frac{1}{2}{p}_1{M}_1^{2}c}$

where $c$ is the chord length. Compute $C_l$ and $C_d$ based on $($i$)$ the exact theory and $($ii$)$ the linear theory.