Supersonic Speeds, Shock Waves — Question 3

Question 3

A boom delay determines Mach number only within its allowed range

For steady horizontal supersonic flight directly over an observer, the weak-cone model gives $t_b=(h/c)\sqrt{1-M^{-2}}$. A known altitude is $h=1000\,\mathrm m$, sound speed is $340\,\mathrm{m/s}$, and boom delay after overhead passage is $2.00\,\mathrm s$.

1. Infer Mach number and speed.

2. Determine the allowed range of positive delays under this model. Would a $3.20\,\mathrm s$ delay be compatible with the same altitude and sound speed?

Question 3 - Solution

A boom delay determines Mach number only within its allowed range

(a) Method and calculation

Let $q=ct_b/h=0.680$. Squaring and rearranging gives$$q^2=1-M^{-2},\quad M=\frac1{\sqrt{1-q^2}}=1.36386.$$$$v=Mc=463.71\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

For finite $M>1$,$$0<\sqrt{1-M^{-2}}<1\Rightarrow0<t_b<h/c=2.94118\,\mathrm s.$$A $3.20\,\mathrm s$ delay is incompatible with the stated steady, directly-overhead weak-cone model. It could not be repaired by choosing a larger Mach number; the delay approaches $h/c$ from below as $M\to\infty$.

Physical check

The inferred Mach number is greater than one. Near-sonic limiting results assume an increasingly long prior flight history, an important condition of the steady model.