Page:Chandrasekhar - On the decay of plane shock waves.djvu/17

(29)${\frac {d}{dt}}\left(U-{\frac {1}{U}}\right)={\frac {q}{1+qt}}\left({\frac {1}{U}}-1\right)$.

Thus the differential equation governing the dependence of U on time is

(30)$\left(1+{\frac {1}{U^{2}}}\right){\frac {dU}{dt}}=-{\frac {q}{1+qt}}\left({\frac {U-1}{U}}\right)$.

This equation can be integrated to give

(31)$1+qt={\frac {\left(U_{0}-1\right)^{2}}{U_{0}}}e^{\left(U_{0}-U\right)}$

where U₀ denotes the shock velocity at time t = 0. The dependence of the ratio of pressures on the two sides of the shock front on time can be readily written down from equation (31). We have (for γ = 1.4)

(32)$1+qt={\frac {\left(U_{0}-1\right)^{2}}{U_{0}}}{\frac {\left[\left(6y+1\right)/7\right]^{1/2}}{\left[\left\{\left(6y+1\right)/7\right\}^{1/2}-1\right]^{2}}}e^{U_{0}-\left[\left(6y+1\right)/7\right]^{1/2}}$.

Equations (20), (21), (23), (24), (31) and (32) together describe completely the behavior of a linear shock pulse. The only limitation on this solution is that U₀ ≤ 1.5 for γ = 1.4 if an accuracy of the order of 1% is demanded.

In Fig. 2 we have illustrated the dependence of U on t for the case U₀ = 1.24 and γ = 1.4. Similarly in Fig. 3 the velocity field in the positive phase of the shock pulse at various instants is illustrated for the same case. And finally in Table II we have tabulated x_max, u_max, and U as functions of time also for the case U₀ = 1.24 and γ = 1.4.

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