Supersonic Speeds, Shock Waves — Question 1

Question 1

A Mach cone is an envelope of earlier sound fronts

A compact source moves uniformly at $v=680\,\mathrm{m/s}$ through still air with small-disturbance sound speed $c=340\,\mathrm{m/s}$. Assume steady motion extending sufficiently far into the past and the weak-disturbance cone construction.

1. Derive the relation between Mach number $M=v/c$ and cone half-angle $\mu$.

2. Find $M$ and $\mu$. Explain why the relevant angle uses a sine, not a tangent.

Question 1 - Solution

A Mach cone is an envelope of earlier sound fronts

(a) Method and calculation

A disturbance emitted a time $\tau$ earlier occupies a sphere of radius $c\tau$ centered at the old source position. The current source is $v\tau$ from that center. The cone is tangent to the sphere; the radius to the tangent point is perpendicular to the cone generator. The right triangle therefore gives$$\sin\mu=\frac{c\tau}{v\tau}=\frac c v=\frac1M.$$

(b) Interpretation and further calculation

Numerically, $M=680/340=2$ and $\mu=\sin^{-1}(1/2)=30^\circ$. The source-to-old-center segment is the hypotenuse, not a leg, in the tangency triangle. Using $\tan\mu=c/v$ would instead predict $26.565^\circ$ and the wrong envelope.

Physical check

A real Mach angle requires $M\ge1$; a finite cone with $0<\mu<90^\circ$ requires supersonic motion.