Page:Chandrasekhar - On the decay of plane shock waves.djvu/9

$dP={\frac {\partial P}{\partial x}}\left[dx-(u+c)dt\right]+{\frac {1}{\gamma -1}}c^{2}{\frac {\partial \log \theta }{\partial x}}dt$,(4)

and

$dQ={\frac {\partial Q}{\partial x}}\left[dx-(u-c)dt\right]+{\frac {1}{\gamma -1}}c^{2}{\frac {\partial \log \theta }{\partial x}}dt$,(5)

where

(6)$P{=}{\frac {2}{\gamma -1}}c+u$;$Q{=}{\frac {2}{\gamma -1}}c-u$

and $\theta$ denotes the potential temperature (i.e., the temperature which the element of gas under consideration would have if reduced adiabatically to a certain standard pressure). For shocks of moderate Intensities the term in $\theta$ which, incorporates the changes in entropy can be ignored and we have

(7)$dP={\frac {\partial P}{\partial x}}\left[dx-(u+c)dt\right]$,

and

(8)$dQ={\frac {\partial Q}{\partial x}}\left[dx-(u-c)dt\right]$,

The foregoing equations are equivalent to

(9)${\frac {\partial P}{\partial t}}=-(u+c){\frac {\partial P}{\partial x}}$,

and

(10)${\frac {\partial Q}{\partial t}}=-(u-c){\frac {\partial Q}{\partial x}}$,

The particular significance of the case

(11)$Q{=}{\frac {2}{\gamma -1}}c-u{=}$ constant ${=}{\frac {2}{\gamma -1}}$

is now apparent: Q has the value 5 outside the shock pulse, it retains its value (to within 1% for y ≤ 2.5) as we cross the shock. Moreover, according to equation (8) since

—4—