Compressibility is an important factor in aerodynamics. At low speeds, the compressibility of air is not significant in relation to aircraft design, but as the airflow nears and exceeds the speed of sound, a host of new aerodynamic effects become important in the design of aircraft.  These effects, often several of them at a time, made it very difficult for World War II era aircraft to reach speeds much beyond .
Many effects are often mentioned in conjunction with the term "compressibility", but regularly have little to do with the compressible nature of air.  From a strictly aerodynamic point of view, the term should refer only to those side-effects arising as a result of the changes in airflow from an incompressible fluid (similar in effect to water) to a compressible fluid (acting as a gas) as the speed of sound is approached. There are two effects in particular, wave drag and critical mach.
One complication occurs in hypersonic aerodynamics, where dissociation causes an increase in the “notional” molar volume because a mole of oxygen, as O, becomes 2 moles of monatomic oxygen and N similarly dissociates to 2 N.  Since this occurs dynamically as air flows over the aerospace object, it is convenient to alter the compressibility factor , defined for an initial 30 gram moles of air, rather than track the varying mean molecular weight, millisecond by millisecond.  This pressure dependent transition occurs for atmospheric oxygen in the 2,500–4,000 K temperature range, and in the 5,000–10,000 K range for nitrogen.
In transition regions, where this pressure dependent dissociation is incomplete, both beta (the volume/pressure differential ratio) and the differential, constant pressure heat capacity greatly increases. For moderate pressures, above 10,000 K the gas further dissociates into free electrons and ions.  for the resulting plasma can similarly be computed for a mole of initial air, producing values between 2 and 4 for partially or singly ionized gas. Each dissociation absorbs a great deal of energy in a reversible process and this greatly reduces the thermodynamic temperature of hypersonic gas decelerated near the aerospace object. Ions or free radicals transported to the object surface by diffusion may release this extra (nonthermal) energy if the surface catalyzes the slower recombination process.