Intuition The one core ideaA supersonic flow that is forced to turn compresses through a tilted shock, and the tilt is fixed entirely by three numbers: how fast the flow comes in, how much it must turn, and how steep the shock sits. To understand that we only need one trick — split the incoming velocity into a piece that crosses the shock (which does all the compressing) and a piece that slides along it (which does nothing).
This page assumes you know nothing and builds every symbol the parent Oblique shock waves — θ-β-M relation uses, one at a time, each earning the next.
Definition Flow and streamlineA flow is air (or any gas) moving. We draw its direction with an arrow. A streamline is the path a tiny dust speck would trace as it rides along — the picture of "where the air is going".
Everything below is about one incoming stream of air hitting an obstacle. Keep that single arrow in your mind; we are going to bend it.
Before we can say "supersonic", we need a yardstick for fast.
Definition Speed of sounda speed of sound a a ≈ 340 m/s
M M = a V V a pure number (no units — a speed divided by a speed).
M < 1 subsonic — the flow moves slower than its own messages.M = 1 sonic .M > 1 supersonic — the flow outruns its own pressure messages.
M V a ratio V / a M V Mach Angle and Mach Waves .
When M > 1
μ μ = arcsin ( M 1 )
arcsin ( 1/ M ) a V opposite side is a hypotenuse is V μ sin μ = hyp opposite = V a = M 1 recover the angle from its sine we use arcsin
The parent calls μ weakest possible oblique shock . That is why we build it here first: it is the limiting case of everything to come.
Now put a wedge (a pointed ramp) in the flow.
Definition Deflection angleθ physically bent as it follows the wedge surface. If the wedge surface tilts up by θ θ
Picture: incoming arrow horizontal, wedge surface sloping up at θ θ toward the oncoming stream is a compression (the flow gets squeezed into a smaller channel).
The turn cannot happen smoothly — the flow is supersonic and cannot be warned ahead. So a thin sheet of sudden compression appears: the shock . It does not lie along the wedge; it sits at its own steeper tilt.
β incoming flow direction and the shock sheet . Always β > θ
Intuition Two different angles, do not confuse themθ air turns (set by the wedge). β shock leans (nature chooses it). The whole topic is one equation linking θ β M
Here is the master trick. Take the incoming velocity arrow V 1 shock sheet as the reference line, break it into two perpendicular pieces.
Definition Normal and tangential components
Normal component u n 1 straight across (perpendicular to) the shock sheet.Tangential component w along the shock sheet.
From the right triangle formed by V 1 u n 1 = V 1 sin β , w = V 1 cos β
sin β cos β β V 1 along the shock is the adjacent side of that angle → cos β across the shock is the opposite side → sin β
Intuition Why bother splitting at all?Because the two pieces behave completely differently. The normal piece slams into the compression and obeys the ordinary Normal Shock Waves laws. The tangential piece slides along the sheet untouched (there is no sideways force to change it). One hard collision + one free slide = the whole oblique shock. This is the "Normal Normally, Tangent Tags Along" mnemonic.
Only the normal piece "does the shock". So the effective Mach number that enters the normal-shock physics uses only u n 1
Intuition Why not just useM 1 M 1 all of the speed crosses the shock — but the tangential slide never does. It would overstate the compression. Only M n 1 = M 1 sin β
A real shock needs M n 1 > 1 sin β > 1/ M 1 β > μ β = μ
Quantities after the shock get subscript 2 1
M 2 θ The downstream velocity V 2 θ ( β − θ ) u n 2 = V 2 sin ( β − θ )
( β − θ ) toward the shock by θ β β − θ
Definition The tools, in plain words
tan α = cos α sin α = adjacent opposite steepness of an angle. It appears because the final θ-β-M relation compares two "steepnesses" (normal vs tangential change).cot β = tan β 1 = sin β cos β cos 2 β double-angle ; it shows up after simplifying sin 2 β cos 2 β cos 2 β = 1 − 2 sin 2 β arcsin arctan
γ γ γ = 1.4 γ how much . It comes packaged with the Rankine-Hugoniot Relations that govern any shock.
ρ ρ compresses , so ρ 2 > ρ 1 ρ 1 u n 1 = ρ 2 u n 2
split V into normal and tangential
normal Mach Mn1 = M sin beta
Read it top to bottom: sound speed births M M θ β M n 1 γ ρ
Cover the right side and answer before revealing.
What does M The flow speed V a M = V / a
What does M > 1 The flow outruns its own pressure ripples — supersonic.
Define the Mach angle and its formula. The half-angle of the ripple pile-up cone, μ = arcsin ( 1/ M )
What is θ The angle the air is physically bent by the wedge.
What is β θ The tilt of the shock sheet vs the incoming flow; always β > θ
Into which two pieces do we split V 1 Normal and tangential, both measured relative to the shock sheet.
Write u n 1 w V 1 β u n 1 = V 1 sin β w = V 1 cos β
Why is sin β cos β β sin cos
What effective Mach number enters the normal-shock laws, and why? M n 1 = M 1 sin β
What condition on β sin β > 1/ M 1 β > μ M n 1 > 1
What angle does the downstream flow make with the shock, and why? ( β − θ ) θ
What is γ Ratio of specific heats, γ = 1.4
