Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/96

Chapter 10

The History of the Shock Wave Problem

The equation of the connection between pressure and density in a substance subjected to the action of a shock wave, which was derived from elementary considerations and from the study of the conservations laws, led to an unexpected result, namely, the increase in entropy with compression of the ideal gas in a shock wave. Entropy increase follows directly from juxtaposing the initial and final state of the substance, which are associated with one another by the conservation equations. We did not investigate the processes that occurred between the control surfaces A and B (Fig. 23b) which led to entropy increase. Formally, as already mentioned, only the conservation equations are symmetrical with respect to $\rho _{1}$, $p_{1}$ and $\rho _{2}$, $p_{2}$. We could also satisfy the conservation equations by investigating the inverse motion, viz., a expansion wave in which expansion occurs within a small interval AB (which we shall not investigate closer) in accordance with the Hugoniot equation. In actual fact, however, such a motion is impossible since entropy would drop in it (this is the so-called Zemplen theorem [99] mentioned earlier). This particular feature of the result of Chapter 9 where, without considering dissipation processes, we came to a change in entropy, creates specific difficulties in the understanding of the theory of shock waves which can be overcome only if we observe the processes inside the region of the change of state proper (between the control surfaces A and B, (Fig. 23b). This has held up considerably the evolution of the theory of shock waves.

It is remarkable that the first three most important works on the theory of shock waves were produced at different time periods but, apparently, completely independently from one another. We shall therefore investigate them not in their chronological order.

Riemann [81] set up for the first time two equations, one for the conservation of matter and one for the conservation of momentum. As a third equation he took Poisson's equation, i.e., he preassigns the conservation of entropy in a shock wave, similarly to the conservation of entropy in non-shock waves in which the effect of dissipation forces, viscosity and thermal conduction, is not considered. The relation between pressure and density obtained by him is