Speeds and Mach Number - ATPL 081 — Principles of Flight

Every speed in the subsonic module was measured against the air or the ground. This module needs one more reference: a speed that belongs to the air itself: the speed of sound. How close an aeroplane flies to it decides whether the subsonic module's picture still holds, bends, or breaks. This lesson builds that reference, defines the Mach number, follows both through climbs and descents, where three separate ways of getting into trouble live, and ends with the map of flow regimes the rest of the module is organised around.

The speed of sound

Sound is a pressure disturbance, a tiny compression travelling through the air, passed from molecule to molecule. The speed of sound is the speed at which such small pressure disturbances propagate, and it matters far beyond hearing: it is the speed at which the air ahead of an aeroplane is warned that something is coming. All the smooth, anticipating streamline patterns of the subsonic module (the upwash reaching ahead of the wing, the flow parting before the nose arrives) exist because pressure signals run ahead of the aircraft at this speed. Its value follows from how fast the molecules themselves move, which is what temperature measures:

Read the formula for what is not in it: no pressure, no density, no altitude. For air, the speed of sound depends on the temperature alone, growing with its square root. At the standard sea-level temperature of +15 °C (288 K) it is 340 m/s (661 kt); in warmer air it is higher, in colder air lower, always as $\sqrt{T}$ with the temperature in kelvin.

The speed of sound and altitude

Altitude enters only through the temperature it brings. Climbing through the troposphere the air grows steadily colder (the first lesson's 2 °C per 1,000 ft), so the speed of sound falls with altitude: from 661 kt at sea level to about 573 kt at the tropopause near 36,000 ft. Above the tropopause the standard atmosphere holds its temperature constant, and the speed of sound holds constant with it. The operational consequence sits one step away: an aeroplane holding the same TAS while climbing is flying at an ever-larger fraction of an ever-smaller speed of sound. High altitude drags every jet toward its Mach limits without the aircraft accelerating at all.

Mach number

With the reference in hand, the module's central quantity is a ratio:

The Mach number says how fast the aircraft flies compared with how fast the air can pass warnings forward — which is why it, and not any airspeed, decides when compressibility trouble starts. Two aircraft at the same TAS can be at very different Mach numbers if one flies in cold air; the one in the colder air, with its lower speed of sound, is the one closer to trouble. This is also why high-altitude cruise is flown on the Mach indicator rather than the airspeed indicator: at cruise levels the Mach number is the limit that bites first.

Why temperature cancels

One relationship surprises almost everyone once: at a constant flight level and constant CAS (a flight level is a pressure altitude: flying one means flying a surface of constant static pressure), the Mach number does not change with temperature. A warmer or colder day at FL350 with the same indicated speed is the same Mach number. The reason is a clean cancellation. The flight level fixes the static pressure; the CAS fixes the pressure difference the pitot–static probes feel (the impact pressure); between them the ratio that IS the Mach number is already pinned down, and temperature never gets a vote. Watch the two halves move together: on a warmer day the air is thinner at the same pressure, so the same CAS corresponds to a higher TAS, but the warmth raises the speed of sound by exactly the same square-root-of-temperature factor. Numerator and denominator grow in step, and the ratio stands still.

Climb and descent: the three speeds together

In a climb the three speed scales drift apart, and the way they drift is exam territory. Climb at constant IAS and the true airspeed rises (the same dynamic pressure needs more true speed in thinning air) while the speed of sound falls: the Mach number rises from both ends of its ratio. Because the dynamic pressure is constant, the wing's demands do not change: the lift coefficient and the angle of attack stay essentially constant, and the pitch attitude slowly falls only because the climb itself flattens as the engines' surplus thrust fades with altitude. Climb far enough, though, and the rising Mach number approaches the aircraft's Mach limit, so at the crossover altitude the climb technique switches: hold constant Mach instead. Now the falling speed of sound pulls the TAS down with it, the IAS falls faster still, and the shrinking dynamic pressure must be paid for: the lift coefficient and the angle of attack rise steadily to keep carrying the weight. The rising angle of attack offsets the shallowing flight path, so the pitch attitude changes far less than in the constant-IAS climb: the nose is held up even as the climb flattens. The descent runs the same logic backwards: constant Mach down from cruise (IAS rising, lift coefficient and angle of attack falling, the nose lowering), then constant IAS below the crossover, where, as in the constant-IAS climb, the lift coefficient and angle of attack hold essentially constant.

Three risks live in that machinery, one per boundary of the high-altitude speed corridor:

Descending at constant Mach — the IAS rises the whole way down, climbing toward the maximum operating speed $V_{MO}$ : hold the cruise Mach too deep into the descent and the aircraft slides into an overspeed without accelerating in any way the pilot feels. The changeover to a constant-IAS descent exists to stop exactly this.
Climbing at constant IAS — the Mach number rises the whole way up, toward the maximum operating Mach number $M_{MO}$ : hold the climb IAS too long and the aircraft exceeds its Mach limit while the airspeed needle shows nothing unusual. The crossover to constant Mach exists to stop this one.
Climbing at too low a Mach number — the opposite corner: at high altitude the IAS behind a small Mach number is low, and the margin above the stalling speed (itself raised by compressibility, as the next section shows) can quietly vanish. The corridor between the stall and the Mach limit narrows with every foot of climb.

Compressibility

Now for what actually changes in the air. Compressibility means that the air's density is no longer constant along a streamline: where the flow accelerates and the pressure drops, the air now visibly expands; where it slows, it packs together. At low speed these density changes are so small that the subsonic module rightly ignored them; from about Mach 0.4 they grow large enough to matter, and they shape everything in the high subsonic, transonic and supersonic regimes. The measure of how compressible the flow behaves is not the airspeed but the Mach number — one more reason it is this module's ruling quantity.

What does it cost? The density changes distort the pressure field the wing has so carefully arranged: the accelerations over the upper surface produce deeper suction peaks and steeper pressure gradients than incompressible flow would, and the boundary layer must climb a harsher recovery behind them. The net effect on the wing's lifting ability is negative: the distorted, steepened gradients cost the top of the lift curve more than they add lower down (the exam's summary is the net line: an overall reduction of lift): $C_{L\,max}$ falls, and the critical angle of attack falls with it. A falling maximum lift coefficient means, through the stalling-speed formula, that the low-speed stalling IAS rises as altitude and Mach number climb. That rising stall floor is the left-hand wall of the corridor in the figure above, and the full story of lift above the critical Mach number is the effects lesson's subject.

The four flow regimes

The module's working map divides flight by Mach number into four regimes:

Subsonic, incompressible (below about M 0.4) — the whole subsonic module lives here: density treated as constant, and the error of doing so negligible.
Subsonic, compressible (about M 0.4 up to the critical Mach number) — all flow around the aircraft is still slower than sound, but density changes now matter: pressure fields deepen and the compressibility corrections begin. No shock waves yet.
Transonic (from the critical Mach number to about M 1.2) — the mixed regime: the aircraft flies below the speed of sound while some of the air over it, accelerated past the sound speed locally, is supersonic. Wherever supersonic flow must slow back down, shock waves stand, the next lesson's whole subject. Jet transport cruise sits at the bottom edge of this regime.
Supersonic (beyond about M 1.2) — the flow is supersonic everywhere around the aircraft; the air ahead receives no warning of its approach, and the flow pattern is rebuilt around shock waves and expansions.

One definition inside that list carries the rest of the module: the critical Mach number, the flight Mach number at which the airflow somewhere on the aircraft first reaches the local speed of sound. And one fact about it explains modern aviation: transport aeroplanes cruise above it. Crossing the critical Mach number is not where the penalty lies (the serious drag rise waits some hundredths of a Mach higher, at the drag-divergence boundary), so a stretch of cheap extra speed exists just beyond it, and supercritical wing design (the design lesson at the end of this module) exists precisely to widen that stretch. Flying there buys more speed, better fuel mileage per kilometre, and a higher optimum cruise altitude; the effects lesson quantifies all three.

What the exam asks

This lesson is definitions plus the climb/descent machinery, and the three-risk item is a bank favourite. The overview, following the learning objectives:

Define the speed of sound, and explain its variation with altitude and the influence of temperature

Define the Mach number as TAS over the local speed of sound

Explain why the Mach number does not change with temperature at constant flight level and CAS

Explain the relationships between Mach number, TAS and IAS in climbs and descents at constant Mach or constant IAS, including the variation of $C_L$ , α, pitch and flight-path angle

Explain the three risks: $V_{MO}$ in a constant-Mach descent, $M_{MO}$ in a constant-IAS climb, and the low-speed stall when climbing at too low a Mach number

State what compressibility means, from which Mach number it matters, and that Mach number is its measure

Describe compressibility's effect on lift: pressure gradients distorted, $C_{L\,max}$ reduced, critical angle reduced, low-speed stalling IAS increased

List the four flow regimes and describe their characteristics

Explain why transport aeroplanes cruise above the critical Mach number