Characteristic Equations for Gas Dynamics

Away from discontinuities, the 1D Euler equations take the form

ρt + (ρu)x = ρt + uρx + ρux = 0 (1)

(ρu)t + (ρu2 + P )x = ρut + ρuux + Px = 0 (2)

st + usx = 0 (3)

where s is the entropy. There are three families of characteristics: entropycharacteristics with characteristic speed λ0 = u and ds = 0 (which followsdirectly from ds/dt = 0 along λ0 from Eq. (3)), and pressure (sound wave)characteristics with characteristic speeds λ± = u± c and

dP ± ρc du = 0.

To derive this last relation, multiply Eq. (1) by c2 and Eq. (2) by ±c, usingthe fact that dP = c2dρ to obtain:

Pt + (u± c)Px ± ρc(ut + (u± c)ux) = 0.

The Riemann invariant for the entropy characteristics is

Γ0 = s, λ0 = u.

The Riemann invariants for the pressure characteristics are

Γ± = u±∫ c(ρ)

ρdρ, λ± = u± c.

For a γ-law gas, P = Aργ away from discontinuities, and

Γ± = u± 2c

γ − 1.

Note on Entropy Advection Equation

One can derive the entropy advection equation st + usx = 0 from the Eulerequations and the expression for the entropy s = cV ln(P/ργ) of a polytropicgas. This equation is valid only away from discontinuities. Hint: Start withthe 1D Euler equations with third equation st + usx = 0 and then derivethe energy conservation equation Et + (u(E + P ))x = 0. Note: The entropydensity S = ρs/m satisfies the conservation law St + (uS)x = 0.

Note on Soundspeed in Air

Newton assumed that sound wave propagation in air was isothermal and

predicted that the soundspeed c =√P/ρ, but it is actually a reversible

adiabatic process (isentropic), which implies P = const ργ with c =√γP/ρ.

Adiabatic means no gain or loss of heat, so reversible adiabatic implies thatthe change in heat dQ = Tds = 0, where s is the entropy. (An irreversibleprocess would have dQ < Tds.)

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