Page:Chandrasekhar - On the decay of plane shock waves.djvu/15

Fig. 1

Fig. 1

Let ABC denote the pulse at time t. At time t + Δt, A moves to A′ and C to C′ where

(25)$AA'=\Delta t$ and $CC'=U\Delta t$,

where U denotes the shock velocity at time t. During the same time the slope of AB also changes. For at t + Δt the velocity field will be governed by

(26)$u={\frac {2}{\gamma +1}}\left({\frac {1+qx}{1+q\left(t+\Delta t\right)}}-1\right)$

Accordingly the value of u at the new position B′C′ of the shock front is given by

(27)$u_{\max }\left(t+\Delta t\right)={\frac {2}{\gamma +1}}\left[{\frac {1+q\left(x+U\Delta t\right)}{1+q\left(t+\Delta t\right)}}-1\right]$.

The increment in u_max during the time Δt is therefore given by

(28)

$\Delta u_{\max }$	$={\frac {2}{\gamma +1}}\left[{\frac {1+q\left(x+U\Delta t\right)}{1+q\left(t+\Delta t\right)}}-1\right]-{\frac {2}{\gamma +1}}\left[{\frac {1+qx}{1+qt}}-1\right],$
	$={\frac {2}{\gamma +1}}{\frac {q}{1+qt}}\left[U-{\frac {\gamma +1}{2}}u_{\max }-1\right]\Delta t$.

Using equation (23) we can rewrite the foregoing equation in the form

—7—