 Solution

ID: past-exam-of-the-theoretical-mechanics-course-of-the-university-of-oxford/2009/b/6/a/solution

Integrate each steady one-dimensional conservation law across a thin region enclosing the shock. Mass conservation gives constant mass flux

j = ρ u

p + ρ u^{2}

is constant. The total energy flux is

u (ρ e + p) = j (e + p / ρ)

, where

e = c_{v} T + u^{2} /2

. For a genuine shock

j \neq = 0

c_{v} T = p / [(γ - 1) ρ]

give the conserved stagnation enthalpy

e + p / ρ = \frac{u ^{2}}{2} + \frac{γ p}{( γ - 1 ) ρ} .

[ρ u] = 0, [p + ρ u^{2}] = 0, [\frac{u ^{2}}{2} + \frac{γ p}{( γ - 1 ) ρ}] = 0.

The velocities here are measured in the stationary shock frame; a moving-shock problem must first be transformed to that frame.

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