Page:Chandrasekhar - On the decay of plane shock waves.djvu/27

APPENDIX

THE EQUATIONS OF MOTION IN RIEMANN'S FORM ALLOWING FOR CHANGES IN ENTROPY

In the text equations (4) and (5) were quoted from Penney (R.C., 260). Since Penney's report may not be generally accessible, it has been thought worth while to include here a brief outline of his derivation of these equations.

The equations of motion of a linear pulse are

${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}{=}-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}$,

(1)

${\frac {\partial \rho }{\partial t}}+u{\frac {\partial \rho }{\partial x}}{=}-\rho {\frac {\partial u}{\partial x}}$,

where p denotes the pressure, ρ the density, and u the mass velocity at any point. Introduce the two functions

$P{=}\Delta _{1}{=}f(\rho ,\theta )+u$,

(2)

$Q{=}\Delta _{2}{=}f(\rho ,\theta )-u$,

where

(3)$f{=}\int _{0}^{\rho }{\sqrt {\left({\frac {\partial p}{\partial \rho }}\right)_{\theta }}}d\log \rho$;$c^{2}{=}\left({\frac {\partial p}{\partial \rho }}\right)_{\theta }$,

and c the local sound velocity. In equations (3), θ denotes the temperature which the element of gas under consideration would have when it is reduced to a standard pressure adiabatically. It is evident that θ remains constant during the motion of any element of gas (except when it crosses a discontinuity).

Consider the total differential

(4)$d\Delta _{i}{=}{\frac {\partial \Delta _{i}}{\partial t}}dt+{\frac {\partial \Delta _{i}}{\partial x}}dx$(i=1,2).

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