Pressure-density relations across the shock

Eqn (6.38) may be rewritten (from e. g. Eqn (6.27)) as

Подпись: и? = Св Го =-£ + Подпись:7 Pi

2 7 – 1 pi

Pressure-density relations across the shock Подпись: (6.39)

which on rearrangement gives

From the continuity equation (6.36):

. , 1 1

М2 + Ml = /И I—— 1—

P2 Pi

and from the momentum equation (6.37):

m2 – мі =4-(рі -/>2)
m

Substituting for both of these in the rearranged energy equation (6.39)

Подпись:JLr(a_e)=i(f,,_„)(l+±)

7 -1 VPi Р2/ Pl /°2/

Pressure-density relations across the shock Подпись: (6.41)

and this, rearranged by isolating the pressure and density ratios respectively, gives the Rankine-Hugoniot relations:

l-H*

Ml

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