Supersonic Speeds, Shock Waves — Question 5

Question 5

A warmer medium widens a cone at fixed source speed

A source maintains speed $v=680\,\mathrm{m/s}$. The medium changes uniformly from sound speed $c_1=340$ to $c_2=374\,\mathrm{m/s}$. Compare the corresponding steady weak Mach cones.

1. Calculate Mach number and cone angle in both states.

2. Derive the angle sensitivity $d\mu/dc$ at fixed $v$ and explain why the angle becomes especially sensitive near the sonic threshold.

Question 5 - Solution

A warmer medium widens a cone at fixed source speed

(a) Method and calculation

The Mach numbers are$$M_1=2.00,\qquad M_2=680/374=1.81818.$$$$\mu_1=\sin^{-1}(0.50)=30^\circ,\qquad\mu_2=\sin^{-1}(0.55)=33.367^\circ.$$The faster sound fronts make the cone wider even though the source speed is unchanged.

(b) Interpretation and further calculation

Differentiate $\sin\mu=c/v$ at fixed $v$:$$\cos\mu\,\frac{d\mu}{dc}=\frac1v,\qquad\frac{d\mu}{dc}=\frac1{v\sqrt{1-(c/v)^2}}.$$The derivative is in radians per unit speed and grows without bound as $c\to v^-$ in the ideal geometric model. If $c>v$, the source is subsonic and no steady Mach cone of this type exists.

Physical check

Mach number is a speed ratio relative to the medium; constant source speed does not imply constant Mach number when the medium changes.