Typically jets cannot operate when intake airflow is supersonic relative to the engine. Why is this so? Also, why are scramjets able to use supersonic air?
To slow down the air to subsonic speeds, the air passes through a shockwave (if I understand correctly). How does this slow down the air?
In a nutshell
A compressor blade works best in subsonic flow. Supersonic flow introduces additional drag sources which should be avoided if efficiency is important. Thus, the intake has to slow down the air to a Mach number between 0.4 and 0.5. Note that the high circumferential speed of a large fan blade will still mean that its tips work at around Mach 1.5, but the subsequent compressor stages will operate in subsonic conditions.
A scramjet is possible with fuels with supersonic flame front speeds and rapid mixing of fuel and air. If the engine would burn regular kerosene, the flame would be blown out like a candle if the internal airspeed would be supersonic, and even if flame holders keep the flame in place, most combustion would take place only after the fuel-air mixture has left the engine due to the slow mixing of kerosene and air. By using hydrogen, a stable combustion can be achieved even in supersonic flow. Due to the high flight speeds, compression is possible by a cascade of shocks, so no moving turbomachinery is needed in ramjets and scramjets.
Background: Maximum heating of air
All jets decelerate air in their intake in order to increase air pressure. This compression heats the air, and in order to achieve a combustion which produces thrust, this heating must be restricted. If air is heated above approx. 6,000° K, adding more energy will result in dissociation of the gas with little further heat increase. Since thrust is produced by expanding air through heating, burning air that enters the combustion process already at 6,000° K will not achieve much thrust. If the air enters the intake at Mach 6, it must not be decelerated below approx. Mach 2 to still achieve combustion with a meaningful temperature increase - that is why scramjets are used in hypersonic vehicles.
Full disclosure: Oxygen starts to dissociate already between 2,000° and 4,000° K, depending on pressure, while Nitrogen will dissociate mainly above 8,000° K. The 6,000° K figure above is a rough compromise for the boundary where adding more energy starts to make less and less sense. Of course, even a 6,000° K flame temperature is a challenge for the materials of the combustion chamber, and ceramics with film cooling are mandatory.
The equation for the stagnation temperature $T_0$ of air shows how important the flight speed $v$ is: $$T_0 = T_{\infty} \cdot \frac{v^2}{c_p} = T_{\infty} \cdot \left(1 + \frac{\kappa - 1}{2}\cdot Ma^2 \right)$$
$T_{\infty}$ is the ambient temperature, $c_p$ the specific heat at constant pressure and $\kappa$ the ratio of specific heats. For two-atomic gases (like oxygen and nitrogen), $\kappa$ is 1.405. Temperature increases with the square of flight speed, so at Mach 2 the factor of heat increase over ambient is only 3.8, while at Mach 6 this becomes 26.3. Even at 220° K air temperature, the air will be heated to 5,800° K when it is ideally compressed in case of a hypersonic vehicle traveling at Mach 6. Note that real compression processes will heat air even more due to friction.
Compression with shocks
Supersonic flow is slowed down by a pressure rise along the flow path. Since no "advance warning" of what is coming is possible, this pressure rise is sudden: Pressure jumps from a fixed value ahead to a higher, fixed value past the jump. This is called a shock. The energy for the pressure rise is taken from the kinetic energy of the air, so past the shock all other parameters (speed, density and temperature) take on new values.
F-16 air intake (picture source)
The simplest shock is a straight shock. This can be found at the face of pitot intakes like the one of the F-16 (see the picture above) in supersonic flight. More common are oblique shocks which are tilted according to the Mach number of the free flow. They happen on leading and trailing edges, fuselage noses and contour changes in general: Whenever something bends the airflow due to its displacement effect, the mechanism for this bending of the flowpath is an oblique shock.
straight and oblique shock (own work)
The index 1 denotes conditions ahead of the shock, and 2 those downstream of the shock. For weak straight shocks the product of the speed ahead of the shock $v_1$ and the speed past the shock $v_2$ equals the square of the speed of sound: $$v_1\cdot v_2 = a^2$$ If $Ma_1 > 1$, then $Ma_2$ must be smaller than 1, so the flow is always decelerated to subsonic speed by a straight shock.
The same equation works for the normal speed component $v_n$ ahead and past a weak oblique shock: $$v_{1n}\cdot v_{2n} = a^2$$ Note that the tangential component $v_t$ is unaffected by the shock! Only the normal component is reduced. Now the speed $v_2$ is still supersonic, but lower than $v_1$, so a weak oblique shock produces a modest increase of pressure, density and temperature.
The angle of the oblique shock wave is determined by the Mach number ahead of the shock.
Supersonic intakes
Weak shocks are desired, because they produce only small losses due to friction. Pitot intakes with their single, straight shocks work well at low supersonic speeds, but incur higher losses at higher Mach numbers. As a rule of thumb, a pitot intake is the best compromise at speeds below Mach 1.6. If the design airspeed is higher, more complex and heavier intakes are needed to decelerate the air efficiently. This is done by a sequence of weak, oblique shocks and by means of a wedge intake. The picture below shows the intake of the supersonic Concorde airliner:
Concorde intake (picture source)
Gradually increasing the angle of the wedge is causing a cascade of ever steeper, oblique shocks which gradually decelerate the air. The design goal is to position this cascade of shocks caused by the wedge on top such that they hit the lower intake lip. This is done by a moveable contour of the upper intake geometry and/or the lip. The goal is to achieve a uniform speed over the intake cross section and not to waste any of the compressed air to the flow around the intake. See the picture of the Eurofighter intake below for an example of a moveable intake lip (which admittedly is mainly for increasing the capture area at low speed and for avoiding flow separation even with a small intake lip radius).
Eurofighter intake (picture source)
Once the air has entered the intake, it is only mildly supersonic and can be further decelerated by a final, straight shock at the narrowest point of the intake. After that point, the intake contour is gradually widened, such that the air decelerates further without separation. To achieve this, a very even flow across the intake area is mandatory, and even the slight disturbance caused by the boundary layer of anything which is ahead of the intake must be avoided. This is achieved by a splitter plate which is clearly visible in the pictures of the F-16 and Eurofighter intakes. The splitter plate of the Eurofighter intake is even perforated to suck away the early boundary layer there.
The deceleration of the intake flow results in a significant pressure rise: In case of the Concorde at Mach 2.02 cruise, the intake caused a pressure rise by a factor of more than 6, so the engine compressor had to add "only" a factor of 12, such that the pressure in the combustion chamber of the four Olympus 593 engines was 80 times that of the ambient pressure (admittedly, this ambient pressure was only 76 mbar in the cruise altitude of 18 km).
This pressure increase means that a supersonic intake must be built like a pressure vessel, and the rectangular face of the intake must quickly be changed to a round cross section downstream to keep the mass of the intake structure low.
Intakes at higher speed
Going faster means the intake pressure recovery increases with the square of flight speed: In case of the SR-71 intake at Mach 3.2, the pressure at the engine face was already almost 40 times higher than the ambient pressure. Now it becomes clear that going faster than Mach 3.5 does away with the need for a turbocompressor: At these speeds a properly designed intake can achieve enough compression by itself for the combustion to produce enough thrust, and going above Mach 5 will need restraint in slowing down the intake flow in order to have enough temperature margin for combustion, requiring supersonic flow in the combustion chamber.
To avoid shock waves on the compressor blades which would make the engine unusable both because of the very large pressure fluctuations that would cause fatigue and failure of blades and because of the high level of drag that is developed in supersonic flows that would have the effect of slowing the blades down as they rotated. In fact, the engine just wouldn't run with a supersonic flow going into it.
Also, the flow needs to be slowed down as much as possible to allow enough time in the combustion chamber for the fuel to burn completely.
So... a cone or ramp shape at the inlet is used to create a little shockwave in front of the engine slowing down the incoming air to subsonic speeds and allowing the jet engine to operate efficiently.
A ramjet is able to use the compressed air because it is designed to do so. An excellent case study is the SR-71 Blackbird, which had engine cones that moved forward and backward based on speed/altitude, to transition from a turbine to a ramjet mission profile. (Fun fact: That plane is so darned fast that the limit to its speed comes not from engine power, but from MELTING THE PLANE because it is going so fast.) The SR-71 had "Bypass doors" to close of the main turbine of the engine when operating on a ramjet profile.
A ramjet sometimes referred to as a flying stovepipe or an athodyd, is a form of airbreathing jet engine that uses the engine's forward motion to compress incoming air without a rotary compressor. Ramjets cannot produce thrust at zero airspeeds; they cannot move an aircraft from a standstill. A ramjet powered vehicle, therefore, requires an assisted take off like a JATO to accelerate it to a speed where it begins to produce thrust. Ramjets work most efficiently at supersonic speeds around Mach 3. This type of engine can operate up to speeds of Mach 6.
A scramjet is a variant of a ramjet airbreathing jet engine in which combustion takes place in supersonic airflow. As in ramjets, a scramjet relies on high vehicle speed to forcefully compress the incoming air before combustion, but a ramjet decelerates the air to subsonic velocities before combustion, while airflow in a scramjet is supersonic throughout the entire engine. This allows the scramjet to operate efficiently at extremely high speeds: theoretical projections place the top speed of a scramjet between Mach 12 and Mach 24.
Besides the fact that beyond those 6000K combustion does not provide much expansion is also the fact that decelerating the flow to subsonic increases the drag of the engine because shock waves are not reversible and therefore the pressure is not recovered at the back (imagine a shut off engine with inner subsonic flow travelling at that speed, it would have a high drag due to shock waves). At hypersonic speeds overcoming that drag on top of the airframe drag would be a no-no. That is why I doubt whether the solution for the SABRE engine (you can google it), which has an internal subsonic flow, may be feasible even if it achieves a high degree of cooling before reaching the compressor.
Why can't jet engines operate with supersonic air?
"Because there has been no business case to develop an engine with supersonic flow at entry to the compressor." The advantages would be the same that led to todays transonic (supersonic relative flow over part of the blade span) compressors, ie smaller and lighter. Compressors with supersonic relative flow over the whole blade span have been rig-tested at steady-state speeds, eg see Naca RM E55A27. Problems to be addressed (there are many) would include shock-induced boundary-layer thickening and separation in the compressor blade passages which causes unacceptably high loss of the potentially "useful" energy that the compressor rotor is putting into the air (there would be too much temperature rise and not enough density and pressure rise) However, they can and do operate with supersonic air, but only over the outer portion of fan and core compressor front stages. Note this air is only supersonic relative to the fast-spinning rotor blades and is self-generated within the engine, ie is not received as supersonic air from the intake (see reason that air leaving intake and entering engine is subsonic in following answer).
The compressor's job is to compress and so the rotor, having first grabbed the air and whirled it around at high speed, also has to slow it down within the passage between the spinning rotor blades (and also through the following stator vane passages), ie it has to compress it if it wants to be called a compressor (no slowing down would mean no increase in pressure). The compressor rotor blade profiles and the diverging area of the passages between them give rise to the type of shock waves that have subsonic flow behind them. The shock waves that are the natural mechanism to get from supersonic to subsonic flow interact with the blade boundary layers and bl thickening and separation means high losses and losses are what the compressor efficiency is a measure of. So everything has to be done with as much finesse as possible to minimise the effects of BL separation and that means limiting the Mach number of the air relative to the blades to low supersonic values and these occur where the blade speed is highest, ie at the tips.
How do jet engines slow supersonic air down?
The question asks how does the engine slow the air down. It is often said that the intake slows the air down. However, the air is going to slow down anyway, with or without an intake. The air flow through the engine, and hence subsonic velocity at entry to the compressor, is set in the first instance by the pilot's request, ie compressor speed/fuel flow. At supersonic speed, if there is no intake, the air slows down to the subsonic entry speed through a plane shockwave. To improve the 'overall pressure-ratio' part of the engine efficiency an intake is added which is a more efficient supersonic compressor than the free-stream, ie it has features which produce a higher ram rise at compressor entry and less spillage drag round the outside of the engine (see later when it doesn't) This requirement is extreme at high supersonic speeds and is the reason for ramps/cones/lip shaping before entry and more ramps/cones/boundary layer bleed/duct shaping inside the intake.
When the intake doesn't do its job. This occurred many times when flying YF12 and SR71 aircraft at high supersonic speeds. In a split second the intake would increase the total pressure loss of the air entering the compressor from its low design value of about 20% to about 70%. The intake had changed (ie unstarted) from being an efficient supersonic intake to being the most inefficient type possible, ie a pitot intake with slowing down of the air from Mach 3 to subsonic in one violent step instead of a number of gentler ones.
The air in the intake slows down "because the engine has controlling areas inside the engine which set the average axial speed of the air through the engine (which has to be low to keep pressure losses to an acceptably low level) and hence at entry to the engine and this speed is subsonic". High air speeds only occur where energy exchange is taking place, ie from the compressor rotors to the incoming air and from the outgoing combustion gases to the turbine, and where the low Mach number flow in the jetpipe ( it's low to keep the pressure losses to an acceptable value) accelerates to sonic speed at the nozzle throat.
Controlling areas are the throat areas of the turbine nozzle guide vanes and the exhaust nozzle where the gas Mach number is 1 and cannot go higher. As stated in a previous answer, the low air speed requirement through the combustor sets the air speed at entry to the compressor. From this subsonic flow the compressor can generate its own supersonic flow relative to its rotor blades if it is driven fast enough by its turbine.
