Page:Chandrasekhar - On the decay of plane shock waves.djvu/31

where A is a constant. From this equation we readily derive the formulae

(10)$c=A\gamma ^{1/2}\theta ^{\gamma /2}\rho ^{(\gamma -1)/2}$,

and

(11)$f={\frac {2}{\gamma -1}}A\gamma ^{1/2}\theta ^{\gamma /2}$$\rho ^{(\gamma -1)/2}={\frac {2}{\gamma -1}}c$.

Thus for the case under consideration

(12)$P={\frac {2}{\gamma -1}}c+u$ and $Q={\frac {2}{\gamma -1}}c-u$.

Moreover we readily verify from the foregoing that

(13)$c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}={\frac {1}{\gamma -1}}{\frac {c^{2}}{\theta }}$.

Combining equations (7), (12) and (13) we obtain the equations quoted in the text.

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