39A (Waves)

In the shock frame, conservation of mass, normal momentum, and energy gives

ρ_{1} U = ρ_{2} (U - u_{2}),

p_{1} + ρ_{1} U^{2} = p_{2} + ρ_{2} (U - u_{2})^{2},

h_{1} + U^{2} /2 = h_{2} + (U - u_{2})^{2} /2.

b

Insert

h = γ p / [ρ (γ - 1)]

, eliminate

p_{2}

and the downstream normal speed using mass conservation, and solve for the density ratio. This gives

\frac{ρ _{1}}{ρ _{2}} = \frac{2 c _{1}^{2} + ( γ - 1 ) U ^{2}}{( γ + 1 ) U ^{2}}, c_{1}^{2} = \frac{γ p _{1}}{ρ _{1}} .

Tangential velocity is unchanged, while the downstream normal velocity is

U ρ_{1} / ρ_{2}

. Thus the incident and outgoing angles to the shock have tangents

1

and

ρ_{1} / ρ_{2}

. Using the tangent-difference identity, the ratio from part (b), and

M^{2} = 2 U^{2} / c_{1}^{2}

gives

tan δ = \frac{M ^{2} - 2}{γ M ^{2} + 2} .

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