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# Three problems about of the shock waves. The second problem.

Three problems about of the shock waves for the curious students (hydrodynamics)

The second problem

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For it was possible to make any mathematical operations with any physical parameter of the continuous matter, those must be of everywhere determined simple function on the coordinates and time. In the one – dimentional plain stationary runing wave along axis X all this functions are: e=e(X), p=p(X), V=V(X) , etc. If on the section [Xs(t), (Xs(t)+h)] the function V(X) is monotonous one and dV/dX does not equals to nought almost some points on this section, we can resolve the equation V=V(X) relatively X and obtain X=X(V); by substituting it in other formulas we will obtain e=e(V) , p=p(V) , etc. The energy conservation law for adiabatical processes in differential form is here de= - p(V)dV . (1)

For p(V)=BV , B=Const. (2)

the integration (1) upon the section [Xs(t), (Xs(t)+h)] gives e – e0 = (1/2)(p + p0)(V0 – V) (3)

that is the Hugoniot equation; here V=V(Xs(t)), V0=V(Xs(t)+h), and as well for e and p . The result does not depend from the quantity of h and therefore is kept on the limit h=0: if shock wave is an adiabatic one, - and therefore the energy conservation law take form of the Hugoniot equation – then the p is a linear function of V. As it is easy to verify, any other physically justified form of function p(V) does not leads to (3) from (1). Meanwhile all experimentaly obtained with shock waves dependences of p from V in the conditions interesting for the techique are not linear. Why it is so?

- Leonid G. Philippenko's blog
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