The Memory Trick
💡 Mach Number = Source Speed ÷ Sound Speed
The Mach number (M) compares a source's speed to the local speed of sound: M = v_source/v_sound. M less than 1 is subsonic (slower than sound). M equal to 1 is transonic (exactly at the speed of sound — the 'sound barrier'). M greater than 1 is supersonic — the source is moving faster than the sound waves it's generating.
Why It Works
When a source moves faster than its own wavefronts can spread out ahead of it, those wavefronts can no longer 'get out of the way' — instead they pile up on top of each other, constructively interfering into an extremely high-pressure cone-shaped shock wave trailing behind the source.
Step by Step
Understanding the Mach Cone
1
The half-angle of the shock cone
The angle of the Mach cone is directly related to the Mach number: sin θ = v_sound/v_source = 1/M. A faster source (higher M) produces a narrower, sharper cone.
A jet flying at Mach 2 produces a narrower shock cone than one flying at Mach 1.2, since it's moving relatively faster compared to the speed of sound.
2
A sonic boom is continuous, not a single event
Despite how it's described, a sonic boom isn't a one-time event that happens only when a plane 'breaks' the sound barrier — it's the continuous shock wave cone trailing the aircraft the entire time it flies supersonically, which an observer hears as a boom only at the moment the cone sweeps past their location.
A supersonic aircraft produces a continuous trailing shock cone throughout its entire supersonic flight; observers on the ground each hear their own single boom only at the instant the cone passes over their specific location.
3
Cherenkov radiation — the same physics for light
An analogous phenomenon occurs when charged particles move faster than the speed of light IN A MEDIUM (which is slower than light's vacuum speed c) — producing a characteristic blue glow called Cherenkov radiation, most famously seen in nuclear reactor cooling pools.
The eerie blue glow seen in photos of nuclear reactor pools is Cherenkov radiation, caused by particles from the reactor traveling faster than light travels through that particular water.
🏥 Worked Example
A jet is flying at Mach 2 (twice the speed of sound). What is the half-angle of the Mach cone it produces?
1
Apply the Mach cone angle formula: sin θ = 1/M.
2
Plug in M = 2: sin θ = 1/2 = 0.5.
3
Solve for θ: θ = sin⁻¹(0.5) = 30° — the Mach cone's half-angle is 30° at Mach 2, and would narrow further at higher Mach numbers.
📌 Exam Application
Exams test correctly calculating Mach number and Mach cone angle, and correctly explaining that a sonic boom is a continuous trailing shock wave rather than a single instant of 'breaking' the sound barrier.
⚠️ Most Common Shock Waves and Sonic Boom Mistakes
The most common trap is thinking a sonic boom happens only once, at the exact moment a plane crosses Mach 1 — in reality, the shock cone trails continuously throughout the ENTIRE supersonic portion of flight; each ground observer just hears their own boom only when that cone happens to sweep past their location.
✓ Quick Self-Test
1) Write the formula for Mach number. M = v_source / v_sound. 2) What does M > 1 mean, and what forms as a result? Supersonic (faster than sound); a cone-shaped shock wave (Mach cone) forms. 3) Write the formula for the half-angle of the Mach cone. sin θ = v_sound/v_source = 1/M. 4) Is a sonic boom a single one-time event or a continuous phenomenon? Continuous — the shock cone trails the entire time the source flies supersonically; an observer hears a boom only when the cone passes their location. 5) What is Cherenkov radiation, and what causes it? A blue glow produced when charged particles travel faster than light travels through a particular medium (like water) — the light-wave analog of a sonic boom.