text: The gas motion behind the shock wave is governed by Euler equations. For an ideal polytropic gas with spherical symmetry, the equations for the fluid variables such as radial velocity , density  and pressure  are given by
  At , the solutions should approach the values given by the Rankine-Hugoniot conditions defined in the previous section.
  The variable pressure can be replaced by the sound speed  since pressure can be obtained from the formula . The following non-dimensional self-similar variables are introduced,
  The conditions at the shock front  becomes
  Substituting the self-similar variables into the governing equations will lead to three ordinary differential equations. Solving these differential equations analytically is laborious, as shown by Sedov in 1946 and von Neumann in 1947. G. I. Taylor integrated these equations numerically to obtain desired results.
  The relation between  and  can be deduced directly from energy conservation. Since the energy associated with the undisturbed gas is neglected by setting , the total energy of the gas within the shock sphere must be equal to . Due to self-similarity, it is clear that not only the total energy within a sphere of radius  is constant, but also the total energy within a sphere of any radius  (in dimensional form, it says that total energy within a sphere of radius  that moves outwards with a velocity  must be constant). The amount of energy that leaves the sphere of radius  in time  due to the gas velocity  is , where  is the specific enthalpy of the gas. In that time, the radius of the sphere increases with the velocity  and the energy of the gas in this extra increased volume is , where  is the specific energy of the gas. Equating these expressions and substituting  and  that is valid for ideal polytropic gas leads to
  The continuity and energy equation reduce to
  Expressing  and  as a function of  only using the relation obtained earlier and integrating once yields the solution in implicit form,
  where
  The constant  that determines the shock location can be determined from the conservation of energy
  to obtain
  For air,  and . The solution for  is shown in the figure by graphing the curves of , ,  and  where  is the temperature.