If one assumes that the specific heat at constant pressure formula_17 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 a Hugoniot equation expressed only in terms of pressure and densities,  where formula_20 is the specific heat ratio. Hugoniot curve without heat release (formula_21) is often called as Shock 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 formula_24 plane determine the downstream conditions. If no heat release occurs, for example, shock waves without chemical reaction, then formula_25. The Hugoniot curves asymptote to the lines formula_26 and formula_27, i.e., the pressure jump across the wave can take any values between formula_28, but the specific volume ratio is restricted to the interval formula_29 (the upper bound is derived for the case formula_30 because pressure cannot take negative values). The Chapman–Jouguet condition is where Rayleigh line is tangent to the Hugoniot curve. If formula_31 (diatomic gas without the vibrational mode excitation), the interval is formula_32, in other words, the shock wave can increase the density at most by a factor of 6. For monoatomic gas, formula_33, therefore the density ratio is limited by the interval formula_34. For diatomic gases with vibrational mode excited, we have formula_35 leading to the interval formula_36. 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 the factor formula_37. Consider gas in a one-dimensional container (e.g., a long thin tube). Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls).