Page:Chandrasekhar - On the decay of plane shock waves.djvu/13

 According to equations (18) and (19), at any given instant both c and u are linear functions of x behind the shock front. By a translation of the time axis we can rewrite equations (18) and (19) more conveniently in the forms

and

From the foregoing equations it follows that

In other words the point at which u = 0 in the pulse moves forward with a uniform velocity equal to the velocity of sound in the undisturbed regions. Starting at this point, x = t, u and c increase linearly with x till they attain their maximum values immediately behind the shock front. And moreover the shock velocity U is related to the mass velocity u$max$ behind the shock front by the Rankine-Hugoniot equation

Thus at any given instant the positive phase of the pulse extends from

It now remains to determine u$max$ (or equivalently U) as a function of time. We proceed now to establish this relation.