Foundations — Shock waves — Mach number, Mach cone — - CRITICAL for rockets -

This page assumes you know nothing about the symbols on the parent note. We build each one — plain words, then a picture, then why the topic can't live without it — in an order where every new piece leans only on pieces already in your hands.

0. What is a "wavefront"? (the picture behind everything)

Before any symbol, you need one mental image. Drop a stone in a still pond. A ring spreads outward. That ring is a wavefront: the set of all points the disturbance has reached at one instant.

Prerequisite: this is the pond-ripple idea from Speed of sound in a medium and, more generally, wave motion.

1. Speed $v$ and time $t$ — how to read "metres per second"

Picture a snail crawling. If it covers $3$ metres in $2$ seconds, its speed is $3/2=1.5\text{m/s}$. Read $\text{m/s}$ as "so many metres for every one second".

The one algebra fact we will reuse constantly:

Reveal check:

2. $v_s$ — the speed of sound (a fixed number, that's the key)

The subscript "$s$" is just a label meaning "sound" — it does not multiply anything. Think of $v_s$ as one word.

$v_s$ changes with the air (colder/thinner air ⇒ different $v_s$) — that story lives in Speed of sound in a medium. For this page treat it as a known fixed number.

3. $v_{\text{obj}}$ — the speed of the object (the second racer)

Now we have the two racers standing side by side:

4. $M$ — the Mach number (turning two speeds into one verdict)

Read it like this: $M=2$ means "the object goes twice as fast as its own sound".

Reveal check:

5. The right triangle — where geometry enters

Now the two racers run for the same elapsed time $t$ (the seconds from section 1). Two distances appear — each is just $vt$:

• The oldest ripple has grown to radius $v_{s}t$ (sound speed $\times$ time).
• The source has travelled $v_{\text{obj}}t$ along its straight track.

When $v_{\text{obj}}>v_s$ the source ends up outside that sphere, and the sphere just touches the cone's slanted wall. That touch point, the source's start, and the source's now-position make a right-angled triangle.

In our picture:

• hypotenuse $=v_{\text{obj}}t$ (how far the source ran — always the longest side, since the source is fastest),
• opposite side to the apex angle $=v_{s}t$ (the ripple radius),
• adjacent side to the apex angle = the remaining side lying along the flight track,
• the small corner at the apex is the angle we want.

6. $\theta$ and $\sin$ — measuring the cone's steepness

Now we need a tool that connects an angle to the two sides of our triangle. That tool is the sine.

Feed our sides in: $\sin \theta =\frac{v_{s}t}{v_{\text{obj}}t}=\frac{v_s}{v_{\text{obj}}}=\frac{1}{M}.$ The $t$ cancels (top and bottom both have it), and $v_s/v_{\text{obj}}$ is the upside-down Mach number — that's where $1/M$ comes from. This single line is the whole parent formula.

Reveal check:

7. $\tan$ — the extra tool for the "boom on the ground" problem

Worked example 3 on the parent needs one more ratio, because there we know the height and want the ground lag — the two short sides of a triangle (opposite & adjacent, both defined back in section 5), not the hypotenuse.

Prerequisite map

Read the arrows as "feeds into": each box is one idea from this page, and following the arrows shows the exact order in which they stack up to build the shock-cone result. Start top-left (the wavefront picture and the speed formula) and finish at the bottom (the Mach cone).

Equipment checklist

Cover the right side and test yourself — if any line is shaky, re-read its section before the parent page.

Connections

• Speed of sound in a medium — supplies the fixed $v_s$ every symbol here leans on.
• Doppler effect — the same growing-wavefront picture, before the source outruns them.
• Superposition & constructive interference — why the touching wavefronts build a single strong wall.