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



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

This equation can be integrated to give

where U$0$ 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)

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 U$0$ ≤ 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 U$0$ = 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 U$0$ = 1.24 and γ = 1.4.