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

(20)$c={\frac {\gamma -1}{\gamma +1}}\left({\frac {1+qx}{1+qt}}+{\frac {2}{\gamma -1}}\right)$,

and

(21)$u={\frac {2}{\gamma +1}}\left({\frac {1+qx}{1+qt}}-1\right)$.

From the foregoing equations it follows that

(22)u = 0, c = 1 for x = t

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

(23)$u_{\max }={\frac {2}{\gamma +1}}\left(U-{\frac {1}{U}}\right)$.

Thus at any given instant the positive phase of the pulse extends from^[1]

(24)x = t to $qx_{\max }=(1+qt)\left({\frac {\gamma +1}{2}}u_{\max }+1\right)-1$.

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

1. In this paper we do not explicitly discuss the negative (or the suction) phase of the pulse. It is however clear that the discussion of the suction phase will proceed on lines exactly similar to that given for the positive phase.

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