Using Maslen method, find the approximate value of pressure and density at the junction of the sphere and the cone of Problem 7.29 at Mach number 8.

Problem 7.29

An empirical way to determine shock shapes based on experiments is given by Billig. A shape of shock created by a blunt body in \(x-y\) coordinate system is given as a hyperbola \(x=R+\delta-R_{c} \cot ^{2} \beta\left[\left(1+y^{2} \tan ^{2} \beta / R_{c} 2\right) 1 / 2-1\right]\). Here, the blunt body is taken as sphere-cone junction with \(\delta\) : the shock distance, \(\mathrm{R}\) : the radius of the sphere and \(\mathrm{R}_{\mathrm{c} \text { : the }}\) the shock curvature to give

\[ \delta / \mathrm{R}=0.143 \exp \left(3.24 / \mathrm{M}^{2}\right), \quad \mathrm{R}_{\mathrm{c}} / \mathrm{R}=1.143 \exp \left[0.54 /\left(\mathrm{M}^{2}-1\right)^{1.2}\right] \]

Here, \(\beta\) is the attached shock angle of the cone alone. Plot th shape of the shock generated by the solid surface shown above for Mach numbers of \(M=4\) and 8.