Page:Chandrasekhar - On the decay of plane shock waves.djvu/23

It is of interest to compare these results with those of Penney and Dasgupta. Though these authors considered an initial shock pulse with a Mach number in the neighborhood of 3, it is seen that their curves illustrating the velocity in the pulse at various instants are very similar to those illustrated in Fig. 3.

We may further note that, according to equations (31) and (32), for γ = 1.4

$\left.{\begin{array}{lll}qt\sim {\dfrac {\left(U_{0}-1\right)^{2}}{U_{0}}}\ \mathrm {e} ^{U_{0}}\ {\dfrac {1}{\left(U-1\right)^{2}}}&\ &\left(t\rightarrow \infty ,U\rightarrow 1\right)\\\\qt\sim {\dfrac {49\left(U_{0}-1\right)^{2}}{9U_{0}}}\ \mathrm {e} ^{U_{0}}\ {\dfrac {1}{\left(y-1\right)^{2}}}&\ &\left(t\rightarrow \infty ,y\rightarrow 1\right)\end{array}}\right.$

(33)

It is possible that the laws

$\left(U-1\right)\propto {\frac {1}{\sqrt {t}}}\ \mathrm {and} \left(y-1\right)\propto {\frac {1}{\sqrt {t}}}\left(t\rightarrow \infty \right)$

(34)

are more general than their derivations from equation (31) and (32) would suggest.

Table II

qt	qxmax	U	umax	qt	qxmax	U	umax
0	0.434	1.24	0.361	004.878	006.000	1.10	0.159
0.195	0.673	1.22	0.334	008.199	009.616	1.08	0.128
0.450	0.982	1.20	0.306	015.375	017.28	1.06	0.097
0.796	1.394	1.18	0.277	035.88	038.77	1.04	0.065
1.280	1.960	1.16	0.248	146.6	152.5	1.02	0.033
1.986	2.771	1.14	0.219	∞	∞	1.00	0.000
3.074	3.999	1.12	0.189

 

3. On the General Solution of Shock-Pulses with Q = Constant. In §2 we have discussed the special case of linear shock pulses which are characterized by Q = Constant $\left(=2/\left(\gamma -1\right)\right)$ throughout. In this section we shall briefly indicate how the most general shock pulses under the circumstances Q = constant can be

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