Q$.$ From the governing equations for the control volume around a

normal shock, Derive the followings:

$1)$ ratio of total temperature to static temperature, $\frac{T_0}{T}=1+\frac{-1}{2}{M}^2$

$2)$ Sonic condition, $\frac{T^{*}}{T_0}=\frac{2}{+1}$

$3)$ Characteristic Mach number, $M^{*}=\frac{u}{a^{*}}=\sqrt[\phantom{2}]{\frac{(+1)M^2}{2+(-1)M^2}}$