Mach number M = V - a — subsonic ( - 1), transonic (~1), supersonic ( - 1), hypersonic ( - 5): Question bank

The traps cluster around four confusions: (1) treating Mach as a speed instead of a ratio, (2) confusing free-stream Mach with local Mach, (3) forgetting sound is adiabatic, and (4) mishandling the boundaries ( $M = 0$ , $M = 1$ , $M \to \infty$ ).

True or false — justify

Is "Mach 1" the same number of metres per second everywhere?

False.

a = γ R T

depends on temperature, so Mach 1 is ~340 m/s at sea level but only ~295 m/s in the cold stratosphere — Mach is a ratio

V / a

, not a fixed speed.

A plane flying at a constant true airspeed always has a constant Mach number.

False. If it climbs into colder air,

a

drops, so the same

V

gives a higher

M

— the plane becomes "more compressible" at altitude even though the speedometer reads the same.

If the whole aircraft is subsonic (

M < 1

), then every parcel of air around it is also subsonic.

False. Air accelerates over curved surfaces (wings); at free-stream

M \approx 0.85

the local flow can exceed

M = 1

and form a shock. That is exactly what makes transonic flight hard.

Mach number is dimensionless.

True. It is speed divided by speed, so all units cancel — this is why it's the universal knob for compressibility across gases and scales.

Sound speed in a gas increases if you raise the pressure at fixed temperature.

False. For an ideal gas

a = γ R T

depends only on

T

. Raising

p

at fixed

T

raises

ρ

proportionally, so

p / ρ

(and hence

a

) is unchanged.

M = 1

the Mach cone becomes a flat wall perpendicular to the flight path.

True.

sin μ = 1/ M = 1

gives

μ = 9 0^{\circ}

, so the "cone" opens completely flat — all the emitted wavefronts pile up into a single plane front.

Incompressible flow is a law of physics that holds below

M = 0.3

False. It is a modeling convenience: below

M \approx 0.3

the fractional density change (

\sim \frac{1}{2} M^{2} \approx 4.5%

) is small enough to ignore. Nothing switches off — we just choose to neglect a small effect.

Newton's isothermal

a = p / ρ

gives the correct speed of sound.

False. Sound oscillates too fast for heat to leak, so it is adiabatic, giving

d p / d ρ = γ p / ρ

. This adds a factor

γ \approx 1.18

— Laplace's correction that matches experiment.

Spot the error

"Since the jet flies at Mach 2, and sound is 343 m/s, it flies at 686 m/s."

The error is using sea-level

a

. At cruise altitude

T

is far lower, so

a \approx 295

m/s and

M = 2

means ~590 m/s. You must use the local speed of sound.

"The plane broke Mach 1 the moment it exceeded 340 m/s."

The threshold is

V = a

at the local temperature, not a fixed 340 m/s. In cold high-altitude air

a

can be ~295 m/s, so Mach 1 is crossed at a lower true airspeed.

"Density change scales like

\frac{d ρ}{ρ} \sim M

, so at

M = 0.3

we get 30% compression."

It scales as

\frac{1}{2} M^{2}

, not linearly. At

M = 0.3

that is

\frac{1}{2} (0.09) \approx 0.045

, i.e. ~5%, which is why 0.3 is the practical cutoff.

"At

M = 0.85

everything is subsonic, so I'll use incompressible Bernoulli for the wing pressures."

Two errors: local flow can go supersonic on the wing, and

M = 0.85 ≫ 0.3

so density varies strongly — incompressible Bernoulli gives wrong pressures. Use isentropic compressible relations.

"The Mach angle is

μ = arctan (1/ M)

It is

sin μ = 1/ M

, giving

μ = arcsin (1/ M)

. The geometry compares the wave radius

a t

to the hypotenuse

V t

(source travel), which is a sine, not a tangent.

"In the wave-fixed frame we keep the term

d ρ d a

to be accurate."

For a weak (infinitesimal) sound wave,

d ρ d a

is a product of two tiny quantities — second-order and negligible. Keeping it contradicts the linearization that defines a sound wave.

"Since

a^{2} = d p / d ρ

, and pressure equals

ρR T

, then

a^{2} = R T

You cannot just differentiate

p = ρR T

treating

T

constant — the process is adiabatic, so

T

changes with

ρ

too. Correctly

d p / d ρ = γ p / ρ = γ R T

, giving the extra

γ

Why questions

Why is Mach number, and not raw speed, "the single most important number" in compressible flow?

Because it decides whether the fluid has time to adjust: pressure signals travel at

a

, so

V / a

tells you if the fluid is warned (

M < 1

) or overrun (

M > 1

). The physics changes qualitatively at the ratio, not at any fixed speed.

Why does the speed of sound depend on

T

rather than on pressure?

Sound is a pressure pulse passed by molecular collisions; hotter gas means faster molecules, so collisions transmit the pulse sooner. Higher

T

raises molecular speed as

T

, and

a

tracks that.

Why do disturbances "pile up" into shocks only when

M > 1

The source outruns its own emitted wavefronts, so waves emitted at different times overlap and concentrate along the Mach cone envelope instead of spreading ahead — a steep pressure jump, i.e. a shock.

Why is the transonic regime (

M \approx 1

) the hardest to design for?

Subsonic and supersonic pockets coexist on the same body, producing local shocks that move with small speed changes. The mixed, shock-terminated flow is unstable and highly nonlinear — no single simple theory covers it.

Why must we treat air as more than a simple perfect gas above

M \approx 5

At hypersonic speeds the shock layer gets so hot that molecules vibrate, dissociate, and ionize, absorbing energy. The specific heats stop being constant, so the calorically-perfect-gas assumption behind

a = γ R T

fails.

Why did Newton's speed-of-sound prediction come out ~18% low?

He assumed the compressions were isothermal (heat has time to equalize). Sound is actually adiabatic, which makes the gas stiffer by a factor

γ

, and

γ \approx 1.18

is exactly the missing ~18%.

Why does the same true airspeed feel "more compressible" at high altitude?

Cold high-altitude air has a lower

a

, so the same

V

produces a higher

M

. Since compressibility effects scale with

M^{2}

, the higher Mach means larger fractional density changes for identical speed.

Why can we use steady conservation laws to derive

a

even though a sound wave is moving?

We jump into the wave-fixed frame so the wave is stationary and gas flows steadily through it. This swaps hard time-dependent equations for simple steady mass and momentum balances.

Edge cases

What is the Mach number of a stationary object in still air?

M = V / a = 0/ a = 0

. This is the extreme incompressible limit — no motion, no pressure disturbance, no compressibility to worry about.

M \to \infty

, what happens to the Mach angle

μ

sin μ = 1/ M \to 0

, so

μ \to 0^{\circ}

— the Mach cone collapses to a needle-thin sliver hugging the flight path, meaning disturbances stay tightly wrapped around the body.

Is the Mach cone formula

sin μ = 1/ M

valid at exactly

M = 1

It gives

sin μ = 1 \Rightarrow μ = 9 0^{\circ}

, a limiting flat front. For

M < 1

it has no solution (

1/ M > 1

), correctly signalling that no cone forms when the source is slower than its waves.

Can local Mach exceed 1 while the free-stream Mach is well below 1?

Yes — over a wing at free-stream

M \approx 0.7

–

0.85

, acceleration can push local flow past

M = 1

, creating a shock. The free-stream number is one value; the flow field has many.

At exactly

M = 0.3

, is the flow incompressible or compressible?

Neither — it is the conventional cutoff where density change reaches ~5%. It's a soft, human-chosen boundary; the physics varies smoothly, so treat 0.3 as a guideline, not a switch.

What does

M < 1

tell you about whether shocks can exist in the flow?

Purely subsonic flow (

M < 1

everywhere) has no shocks — every parcel is warned in advance and adjusts smoothly. Shocks require a supersonic pocket somewhere, even if the vehicle overall is subsonic.

Wrap-up recall

Recall One-line summary of the traps

Mach is a ratio $V / a$ , never a fixed speed.
$a$ depends on temperature only (for an ideal gas), via $T$ .
Free-stream Mach ≠ local Mach; wings make local flow faster.
Compressibility scales as $\frac{1}{2} M^{2}$ ; ~5% at $M = 0.3$ .
Sound is adiabatic → factor $γ$ , not isothermal.
$sin μ = 1/ M$ : no cone below $M = 1$ , needle-thin as $M \to \infty$ .

Related: Speed of Sound in Gases · Isentropic Flow Relations · Compressibility & Bernoulli's Limits · Prandtl–Glauert Correction · Reynolds Number