Intuition The one core ideaWhen a fast-moving stream of air (faster than sound) rounds a corner that bends away from it, the air fans out through a spray of ultra-thin ripples, speeding up and thinning out without any waste. This whole topic is just bookkeeping that spray : giving each ripple an angle, adding up the tiny turns, and reading off the new speed.
Before any of the parent note's formulas make sense, you need a small toolbox of ideas. Below, every symbol and word the parent uses is built from zero : plain meaning → the picture → why the topic needs it. Read top to bottom; each rung stands on the one below.
Definition Three gas constants named up frontThe sound-speed formula below uses three letters. Here is a one-line placeholder for each; §6 gives the full story:
T absolute temperature (kelvin, always ≥ 0 R gas constant for the gas (air: 287 J/(kg⋅K) γ ratio of specific heats (air: 1.4
Definition Speed of sounda speed of sound a 340 a = γ R T
Figure s01 — read it like this. The amber dot is the flow's source now ; each cyan circle is a sound ripple emitted at an earlier moment, grown to radius a t V t V > a M = 2 Mach lines . Look at how the ripples never reach ahead of the dot — that visual is what "supersonic" means.
Intuition Why the topic lives entirely atM > 1 ahead of its source if the flow is slower than the ripple. Once M > 1 Mach waves the whole topic is made of. No supersonic flow, no fan. That is why you will see M 2 − 1 M > 1
Definition Mach wave and Mach angleμ Mach wave is the weakest possible disturbance a supersonic flow can carry — one of those straight white lines in Figure s01. The angle it makes with the local flow direction is the Mach angle , written μ μ = arcsin ( M 1 ) , μ M
θ angle measures "how much you have rotated." The parent uses two units:
degrees (∘ 36 0 ∘ radians : a full circle is 2 π ≈ 57. 3 ∘
The Prandtl–Meyer function (defined in §5) is derived in radians (because calculus of angles is clean in radians) but quoted in degrees (because engineers eyeball corners in degrees). Always check which one a formula wants.
Definition Deflection angleθ wall bends away from the oncoming stream — the size of the corner. The flow copies the wall, so θ flow direction rotates. In this topic θ Sign convention: count θ positive for an expansion — the wall bends away from the flow (convex corner), which adds to ν negative θ would be a compression — the wall bends into the flow (concave corner), which subtracts from ν oblique shock rather than a smooth fan. This page and the parent only use θ > 0
Common mistake Feeding degrees into a radian formulaWhy it feels right: a number is a number.
Fix: arctan arcsin add a corner angle written in 1 0 ∘ θ
Every angle in this topic is read off a right triangle (one 9 0 ∘ μ is one of them.
Figure s02 — what to look for. This is the very triangle hiding inside Figure s01, drawn cleanly. The amber slanted side is the hypotenuse (the source's travel ∝ M ∝ 1 ∝ M 2 − 1 9 0 ∘ μ
sin μ = 1/ M a t V t sin μ = V t a t = V a = M 1 . 1/ M M 1/ M μ
Figure s03 — watch μ As you slide M down : the Mach angle keeps shrinking toward zero. That downhill curve is exactly why the last wave in a fan (high M M
Definition Inverse trig functions and their rangessin , cos , tan angle and give a ratio . Their inverses go backwards:
arcsin ( x ) which angle has this sine?" Its input x [ − 1 , 1 ] principal value ) lies in [ − 2 π , 2 π ] arctan ( x ) which angle has this tangent?" Its input x ( − 2 π , 2 π )
These principal-value ranges are the "one answer" the calculator returns. In this topic every argument is ≥ 0 1/ M > 0 M 2 − 1 ≥ 0 [ 0 , 2 π ) μ = arcsin ( 1/ M ) the (positive, acute) angle whose sine is 1/ M .
Definition The Prandtl–Meyer functionν ( M ) ν ν ( M ) the total angle a flow has turned to accelerate from sonic (M = 1 M — a running "turn score." Its formula (derived in the parent, not memorised here) is
ν ( M ) = γ − 1 γ + 1 arctan γ + 1 γ − 1 ( M 2 − 1 ) − arctan M 2 − 1 , ν ( 1 ) = 0 domain is M ≥ 1 ν increases steadily with M ν M arctan integrating the tiny turns (§8) produces exactly those "which-angle" shapes.
Intuition Why the topic needs the inverseYou measure a ratio (like 1/ M want an angle (the wave's slant μ ν
Definition The gas constants (full story)
T absolute temperature (in kelvin, never below zero). Sets the sound speed.R specific gas constant , a fixed number for a given gas (air: 287 J/(kg⋅K) γ ratio of specific heats , telling how "springy" the gas is when squeezed. For ordinary air γ = 1.4
These three feed a = γ R T 2 γ − 1 γ − 1 γ + 1 γ
Definition Static pressurep ρ T static quantities are what a tiny sensor drifting along with the flow would read:
p static pressure , the everyday push of the gas per unit area.ρ density , mass packed into each cubic metre.T static temperature , the local hotness.
These are the local values; they change through the fan (all three drop as the flow speeds up).
Definition Stagnation ("total") quantitiesT 0 , p 0 , ρ 0 complete stop with no losses. The temperature it would then have is the stagnation temperature T 0 p 0 ρ 0
\frac{p_0}{p}=\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}},\qquad
\frac{\rho_0}{\rho}=\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma-1}}.$$
Note each ratio pairs a total (subscript $0$) with a static value at the same point.
Definition Upstream and downstream: subscripts 1 and 2When the flow crosses the whole fan, we label the state before the corner with subscript 1 and the state after with subscript 2 . So p 1 , T 1 , M 1 p 2 , T 2 , M 2 p 2 / p 1
Intuition Why stagnation values matter for an expansion fanBecause the fan is isentropic (lossless), T 0 p 0 ρ 0 do not change as the gas speeds up. Only the local static T , p , ρ p 2 / p 1 p 0 p 0 ν ( M 2 ) = ν ( M 1 ) + θ
s Entropy s second law says s
isentropic = "s
A shock is a single violent jump: it raises s Mach waves , each raising s That is why the parent keeps insisting the fan is isentropic and the shock is not.
d d M M d θ d V tiny angle across one Mach wave, then adds up all the tiny turns to get the full turn.
Definition The integral sign∫ ∫ "add up infinitely many infinitely small pieces." In this derivation, ν ( M ) = ∫ 1 M ( tiny turn per bit of M ) d M d θ M = 1 ν ( M ) arctan ν the answer to that sum.
Intuition Why the topic can't avoid calculusA single Mach wave turns the flow by only a whisper. A real corner is thousands of these stacked in a fan. "Add up infinitely many infinitely small pieces" is exactly what ∫ ν ( M ) arctan
Right triangle sin cos tan
Mach angle mu equals arcsin one over M
Mach number M equals V over a
Only M greater than 1 supports Mach waves
Angle in degrees and radians
Deflection angle theta with sign
arctan arcsin which angle
Prandtl Meyer function nu of M turn score
Stagnation values T0 p0 rho0
Isentropic means s constant
Differentials and integral sign
Working rule nu M2 equals nu M1 plus theta
Pressure and temperature ratios
Test yourself — you are ready for the parent note if you can answer each without peeking.
What does the Mach number M The flow speed V a M = V / a
Why must the whole topic live at M > 1 Only when the flow outruns its own ripples do they pile into slanted Mach lines (the fan); also
M 2 − 1 is real only then.
What is a Mach wave and what is μ The weakest disturbance in supersonic flow; μ μ = arcsin ( 1/ M )
On a right triangle, what is tan μ opposite ÷ adjacent.
What question does arcsin ( x ) "Which angle has sine equal to x [ − π /2 , π /2 ]
In plain words, what is the Prandtl–Meyer function ν ( M ) The total angle a flow has turned to accelerate from M = 1 M M ≥ 1 ν M
Where does sin μ = 1/ M Ripple spreads a t V t sin μ = a / V = 1/ M
What is γ The ratio of specific heats (gas "springiness"); γ = 1.4
State the three stagnation-to-static ratios. T 0 / T = 1 + 2 γ − 1 M 2 p 0 / p = ( ⋅ ) γ / ( γ − 1 ) ρ 0 / ρ = ( ⋅ ) 1/ ( γ − 1 )
Distinguish static p , ρ , T p 0 , ρ 0 , T 0 Static = local values a co-moving sensor reads; stagnation = values if the flow were brought to rest losslessly (frozen through an isentropic fan).
What is the sign convention for θ Positive for expansion (wall bends away, ν
What do subscripts 1 and 2 mean? State 1 is upstream (before the corner), state 2 is downstream (after the fan).
What does "isentropic" mean and why does the fan qualify? Entropy s Δ s → 0
What does the integral sign ∫ It sums the infinitesimal turns d θ ν ( M )
Why take logarithms before differentiating V = M a ln d ( ln x ) = d x / x
