Three problems about of the shock waves
for the curious students
(hydrodynamics)
The third problem
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The fundamental conclusions for shock waves lead from the second principle of thermodynamics. In the yours text-books such condlusions are made on the base of Zemplen theorem:
s - s0 = - (1/12T0)(d2p/dV2)s(V - V0)3 (1)
where s is the specific entropy, T - temperature. The main conclusion from (1) is formulated so: so far as in the adiabatic processes (shock waves are described by Hugoniot equation!) (s - s0) must be not smaller from the nought , then the stretching shock waves are impossible.
Meanwhile you satisfy himself in "second problem" (the formula (2)) that if the energy conservation law on shock wave have the form of the Hugoniot equation, then the function p(V) must be the linear function: p=Const.V. Therefore in such waves d2p/dV2 equels to nought and from (1) it leads to s = s0 independently of the character of deforming - the compression or the stretching .
Why it is so?