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).  Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected.  Such a system can be described by the following system of conservation laws, known as the 1D Euler equations, that in conservation form is:
where
* fluid mass density, 
* fluid velocity, 
* specific internal energy of the fluid, 
* fluid pressure, and
* is the total energy density of the fluid, [J/m], while e is its specific internal energy
Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form
is valid, where  is the constant ratio of specific heats .  This quantity also appears as the polytropic exponent of the polytropic process described by
For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).
Note: For a calorically ideal gas  is a constant and for a thermally ideal gas  is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation ().