AcademySound
Academy
Shock Waves
Level 1 - Physics topic page in Sound.
Principle
Shock waves form when a source moves faster than sound in the medium.
Notation
\(v_S\)
source speed
\(\mathrm{m\,s^{-1}}\)
\(v\)
sound speed
\(\mathrm{m\,s^{-1}}\)
\(M\)
Mach number
1
\(\theta\)
Mach cone half-angle
\(\mathrm{rad}\)
\(t\)
elapsed time after emission
\(\mathrm{s}\)
Method
A subsonic source leaves wavefronts ahead of itself. A supersonic source outruns those wavefronts, so they pile up into a cone.
Mach number
\[M=\frac{v_S}{v}\]
Supersonic condition
\[M>1\]
Wavefront radius
\[R=vt\]
Source distance
\[d=v_St\]
Cone angle
\[\sin\theta=\frac{R}{d}=\frac{v}{v_S}=\frac{1}{M}\]
The shock is a large pressure change concentrated near the cone surface, not a separate object moving through the air.
Rules
These are the compact shock-wave relations.
Mach number
\[M=\frac{v_S}{v}\]
Supersonic condition
\[M>1\]
Mach angle
\[\sin\theta=\frac{1}{M}\]
Examples
Question
An aircraft travels at
\[680\,\mathrm{m\,s^{-1}}\]
where the sound speed is \[340\,\mathrm{m\,s^{-1}}\]
Find its Mach number and shock angle.Answer
\[M=\frac{680}{340}=2.0\]
Then \[\sin\theta=\frac{1}{2.0}\]
so \[\theta=30^\circ\]
Checks
- No Mach cone forms unless the source is faster than sound in that medium.
- Larger Mach number gives a smaller cone angle.
- The sound speed depends on the medium and its state.
- A sonic boom is the listener crossing the shock front.