Page:Chandrasekhar - On the decay of plane shock waves.djvu/11

(12)$dQ=0$ for $dx=(u-c)dt$

it appears that after a sufficient length of time the shock pulse must be characterized by Q = 5 throughout. In any event it is clear that in considering the case Q = constant = 5 we are not limiting ourselves to too 'trivial' a class of shock pulses. Assuming then the validity of equation (11) we have

(13)$u={\frac {2}{\gamma -1}}\left(c-1\right)$;$P={\frac {2}{\gamma -1}}\left(2c-1\right)$;$u+c={\frac {\gamma +1}{\gamma -1}}c={\frac {2}{\gamma -1}}$,

and equation (9) reduces to

(14)${\frac {\partial c}{\partial t}}=-\left({\frac {\gamma +1}{\gamma -1}}c-{\frac {2}{\gamma -1}}\right){\frac {\partial c}{\partial x}}$.

Letting

(15)$\phi ={\frac {\gamma +1}{\gamma -1}}c-{\frac {2}{\gamma -1}}$

we have

(16)${\frac {\partial \phi }{\partial t}}+\phi {\frac {\partial \phi }{\partial x}}=0$.

A complete integral of the foregoing equation can be written down at once. We have

(17)$\phi ={\frac {1+qx}{b+qt}}$

where b and q are two arbitrary constants. In terms of this complete integral we can readily write down the general solution of equation (16). But postponing this discussion to §3 and limiting ourselves for the present to the solution (17), we have (cf. eqs. (13) and (15))

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

and

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

—5—