How does a supersonic jet engine differ from a subsonic jet engine?
Thrust is air mass flow multiplied by the speed difference between flight and nozzle speed of the engine. To increase thrust, subsonic engines try to maximize mass flow (by increasing the bypass ratio) while supersonic engines rely more on increasing the nozzle speed (by using afterburners).

The core engines do not differ much - after all, the intake will make sure that air reaches the engine at a speed of Mach 0.4 to 0.5, regardless of flight speed. The main difference is in their bypass ratio. The slower the design speed, the bigger the bypass ratio may become. At very low speed, the ungeared, shrouded fan is exchanged for a geared, free spinning propeller, in other words, the jet changes to a turboprop. The intake and nozzle, however, are very different indeed.

The optimum bypass ratio changes continuously, but since the drag coefficient drops after crossing Mach 1, airplanes are either designed for a maximum Mach number of 0.9 or less, or 1.6 and above. The corresponding bypass ratios today are up to 12 for subsonic engines, and less than 1 for supersonic engines. This produces a sharp boundary at the speed of sound, and many military engines designed for supersonic flight lost their afterburners and were fitted with a big fan to become the engines for subsonic transport aircraft. Examples are:


The Pratt &amp; Whitney J57 turbojet (installed in the F-100 and F-102 fighters, among others) became the JT-3D turbofan which powered the Boeing 707 or the DC-8.
The core of the General Electric F110 (installed in the F-15 and F-16 fighters, among others) became the CFM-56 turbofan which is used in the Boeing 737 or the Airbus A320.


The differences between sub- and supersonic engines grow bigger the more you move away from their core. High-pressure compressor, combustion chamber and high pressure turbine look and work the same, but the low pressure compressor of subsonic engines swallows a lot more air and has a much bigger diameter. Supersonic engines in turn mostly have an afterburner. The biggest difference, however, are the intakes (large pitot intake with blunt lips for subsonic aircraft versus adjustable spike or ramp intakes for supersonic flight) and the nozzle (fixed for subsonic flight versus a complex, adjustable convergent-divergent nozzle for supersonic flight). This is due to the very different air speeds and the much higher exit velocities required for supersonic flight.