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



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

(1)

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

(2)

where

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