Page:Chandrasekhar - On the decay of plane shock waves.djvu/29

Rewriting this in the form

(4′)$d\Delta _{i}={\frac {\partial \Delta _{i}}{\partial x}}\left[dx-\left(u\pm c\right)dt\right]+\left[{\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}\right]dt$

(the + or - sign going respectively with i = 1 or 2) we shall consider the second term in square brackets occurring as the coefficient of dt. We have (cf. eq. (3))

${\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}$

(5)$=\pm c{\frac {\partial \Delta _{i}}{\partial x}}+\left({\frac {\partial f}{\partial \rho }}{\frac {\partial \rho }{\partial x}}+{\frac {\partial f}{\partial \theta }}{\frac {\partial \theta }{\partial x}}\right)u+\left({\frac {\partial f}{\partial \rho }}{\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial \theta }}{\frac {\partial \theta }{\partial t}}\right)$

$\pm \left(u{\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial t}}\right)$.

Remembering that θ remains constant during the motion apd using the equations (1) we find that

(6)${\frac {\partial \Delta _{i}}{\partial x}}\left(u\pm c\right)+{\frac {\partial \Delta _{i}}{\partial t}}=\left(\pm c{\frac {\partial f}{\partial \theta }}\mp {\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}$.

Thus,

(7)$dP={\frac {\partial P}{\partial x}}\left[dx-\left(u+c\right)dt\right]+\left(c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}dt$.

and

(7)$dQ={\frac {\partial Q}{\partial x}}\left[dx-\left(u-c\right)dt\right]-\left(c{\frac {\partial f}{\partial \theta }}-{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}\right){\frac {\partial \theta }{\partial x}}dt$.

Now for a perfect gas (with a ratio of specific heats γ) we have

(9)$p=A^{2}\theta ^{\gamma }\rho ^{\gamma }$.

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