39C (Waves)

The three jumps express conservation of mass flux, normal and tangential momentum flux, and total enthalpy (energy) flux across the stationary shock.

b

c^{2} = γ p / ρ

eliminate the density and pressure ratios. The result is

M_{2}^{2} = \frac{2 + ( γ - 1 ) M _{1}^{2}}{2 γ M _{1}^{2} - ( γ - 1 )},

so a supersonic upstream flow becomes subsonic.

Using

T \propto p / ρ

together with the supplied density ratio and

p_{2} / p_{1} = [2 γ M_{1}^{2} - (γ - 1)] / (γ + 1)

gives

\frac{T _{2}}{T _{1}} = \frac{[ 2 γ M _{1}^{2} - ( γ - 1 )] [ 2 + ( γ - 1 ) M _{1}^{2} ]}{( γ + 1 ) ^{2} M _{1}^{2}} .

Apply the normal-shock relations to

M_{1 n} = M_{1} sin θ_{1}

. Tangential velocity continuity and mass conservation give

tan θ_{2} = tan θ_{1} \frac{2 + ( γ - 1 ) M _{1}^{2} s i n ^{2} θ _{1}}{( γ + 1 ) M _{1}^{2} s i n ^{2} θ _{1}} .

M_{1} \to \infty

, maximize

θ_{1} - θ_{2}

; with

k = (γ - 1) / (γ + 1)

the optimum has

tan^{2} θ_{1} = 1/ k

, and the maximum deflection is

arcsin (1/ γ)

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