Three problems about of the shock waves

 Three problems about of the shock waves

                                      for the curious students

                                           (hydrodynamics)

                                          The first problem

As it informs in your text-books, for an ideal gas the Hugoniot equation has the form

p[(h+1)V-(h-1)V₀]=p₀[(h+1)V₀-(h-1)V] , h=c_p/c_v ;

here p- the pressure, V- the specific volume, c_p and c_V  - the specific heat capacities with constant p and V . Hence it follows

 [(h-1)V₀-(h+1)V](dp/dV)=[(h+1)p+(h-1)p₀]                                            (1)

As you know, the energy conserwation law is

                de=dQ-pdV                                    (2)

where e -  the specific inner energy, dQ - the summary contribution in de from all nonmechanical influences (so called "the heat flow"). The Hugoniot equation bases on the idea about of adiabatic deforming in a shock wave:dQ=0  , and from (2):

                                                                           de=-pdV                                             (3)

For the function e(p,V)  with any form,  de=(de/dp)_Vdp+(de/dV)_pdV_; the substitution it in (3) gives

                                        (dp/dV)=-[p+(de/dV)]/(de/dp)     ;                                      

for an ideal gas it will be

                                                                (dp/dV)=-hp/V                                       (4)

As you see, the formula (1) turns into (4) only at p=p₀ and V=V₀: in an ideal gas the shock waves can be described by Hugoniot equation only if its amplitudes near to nought.

Why it is so?