Shock wave angle, deflection angle: Foundations — Physics deep dive

Before you can read the parent note, you need to own every symbol it throws at you. Below, each idea is built from nothing: plain words → a picture → why the topic needs it. Read top to bottom; each rung stands on the one below it.

1. Speed of sound and the Mach number $M$

The subscripts you will see everywhere:

$1$ = state of the air before (upstream of) the shock.
$2$ = state of the air after (downstream of) the shock.

So $M_{1}$ = incoming Mach number, $V_{1}$ = incoming speed, and $ρ_{1}$ , $ρ_{2}$ below follow the same rule.

2. What a shock actually is

Figure — Shock wave angle, deflection angle

Two flavours appear in the parent note:

Normal shock — the shock line sits straight across the flow (perpendicular). See Normal Shock Waves.
Oblique shock — the shock line sits at a slant to the flow. This whole topic is about the oblique one.

The magic sentence of the parent note is: "an oblique shock is just a normal shock for the part of the velocity aimed straight into it." To understand that, we need to learn how to split a velocity — Section 4.

3. Density $ρ$ and the idea of compression

The topic needs $ρ$ because mass is never destroyed: whatever mass flows into the shock must flow out. That bookkeeping rule (called continuity) is what links density to velocity in Step 2 of the derivation.

4. Splitting a velocity into two arrows

This is the single most important tool on the page. A velocity is an arrow: it has a size and a direction. Any slanted arrow can be rebuilt from two arrows at right angles — like giving directions as "3 blocks east and 4 blocks north" instead of one diagonal walk.

To get $u$ and $w$ from the full speed $V$ and the shock's lean angle $β$ , we need the trig ratios — next.

5. Sine, cosine, tangent — earned on a right triangle

Everything angular in this topic reduces to a right triangle (a triangle with one $9 0^{\circ}$ corner). Put the angle you care about at one corner. Then:

6. The two star angles: $β$ and $θ$

Now that we can split arrows, the two headline angles are just labels on a picture.

7. The Mach angle $μ$ — the weakest possible shock

8. $γ$ — the gas's "springiness" number

How the foundations feed the topic

Read it bottom-up in one breath: sound sets $M$ ; $M > 1$ makes shocks; velocity-splitting on a right triangle turns speeds into the angles $β$ and $θ$ ; add the gas number $γ$ and mass bookkeeping, and you land on the $θ$ – $β$ – $M$ relation of the parent topic.

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note when each is instant.

What does the Mach number

M

compare?

The flow speed

V

to the local speed of sound

a

M = V / a

What does

M > 1

physically mean?

The air moves faster than its own warning signals, so obstacles arrive without warning and force abrupt shocks.

What do the subscripts

1

and

2

mean?

State

1

= upstream (before the shock); state

2

= downstream (after the shock).

What is a shock, in one line?

A hair-thin line across which pressure and density jump up and speed drops.

What is density

ρ

Mass packed per unit volume — how crowded the molecules are; a shock always makes

ρ_{2} > ρ_{1}

Why split velocity into normal and tangential parts?

A shock pushes only perpendicular to its face, so only the normal part gets squashed; the tangential part glides through unchanged.

Write

u_{1}

and

w_{1}

in terms of

V_{1}

and

β

u_{1} = V_{1} sin β

(into the shock),

w_{1} = V_{1} cos β

(along it).

Why does

tan

appear in the final relation?

Dividing normal by tangential (

u / w = tan β

) cancels the speed and leaves a pure angle.

Difference between

β

and

θ

β

= flow-to-shock lean angle;

θ

= how much the streamline bends. Always

θ < β

What is the Mach angle

μ

and its formula?

The lean of the weakest possible shock,

μ = sin^{- 1} (1/ M_{1})

— the lower limit of

β

What does

sin^{- 1}

(arcsine) do?

Undoes sine — answers "which angle has this sine value?"

What is

γ

and its value for air?

Ratio of specific heats, the gas's springiness;

γ = 1.4

for air.

Recall One-sentence summary

Sound sets a speed limit for warnings; beat it and the air must turn abruptly across a slanted shock, whose lean ( $β$ ) and the path's bend ( $θ$ ) are read off a right triangle built by splitting the velocity into "into-the-shock" and "along-the-shock" parts.