text: The following assumptions are made in order to simplify the Rankine–Hugoniot equations. The mixture is assumed to obey the ideal gas law, so that relation between the downstream and upstream equation of state can be written as
  where  is the universal gas constant and the mean molecular weight  is assumed to be constant (otherwise,  would depend on the mass fraction of the all species). If one assumes that the specific heat at constant pressure  is also constant across the wave, the change in enthalpies (calorific equation of state) can be simply written as
  where the first term in the above expression represents the amount of heat released per unit mass of the upstream mixture by the wave and the second term represents the sensible heating. Eliminating temperature using the equation of state and substituting the above expression for the change in enthalpies into the Hugoniot equation, one obtains an Hugoniot equation expressed only in terms of pressure and densities,
  where  is the specific heat ratio, which for ordinary room temperature air (298 KELVIN) = 1.40. An Hugoniot curve without heat release () is often called a "shock Hugoniot", or simply a(n) "Hugoniot". Along with the Rayleigh line equation, the above equation completely determines the state of the system. These two equations can be written compactly by introducing the following non-dimensional scales,
  The Rayleigh line equation and the Hugoniot equation then simplifies to
  Given the upstream conditions, the intersection of above two equations in the - plane determine the downstream conditions; in the - plane, the upstream condition correspond to the point . If no heat release occurs, for example, shock waves without chemical reaction, then . The Hugoniot curves asymptote to the lines  and , which are depicted as dashed lines in the figure. As mentioned in the figure, only the white region bounded by these two asymptotes are allowed so that  is positive. Shock waves and detonations correspond to the top-left white region wherein  and , that is to say, the pressure increases and the specific volume decreases across the wave (the Chapman–Jouguet condition for detonation is where Rayleigh line is tangent to the Hugoniot curve). Deflagrations, on the other hand, correspond to the bottom-right white region wherein  and , that is to say, the pressure decreases and the specific volume increases across the wave; the pressure decrease a flame is typically very small which is seldom considered when studying deflagrations.
  For shock waves and detonations, the pressure increase across the wave can take any values between ; the steeper the slope of the Rayleigh line, the stronger is the wave. On the contrary, here the specific volume ratio is restricted to the finite interval  (the upper bound is derived for the case  because pressure cannot take negative values). If  (diatomic gas without the vibrational mode excitation), the interval is , in other words, the shock wave can increase the density at most by a factor of 6. For monatomic gas, , the allowed interval is . For diatomic gases with vibrational mode excited, we have  leading to the interval . In reality, the specific heat ratio is not constant in the shock wave due to molecular dissociation and ionization, but even in these cases, density ratio in general do not exceed a factor of about .