Supersonic Speeds, Shock Waves — Question 6

Question 6

Finite emission history can remove the nominal cone arrival

A source moves along $(x,y)=(vt_e,h)$, with $v=680\,\mathrm{m/s}$, $h=1000\,\mathrm m$ and $c=340\,\mathrm{m/s}$. A ground observer is at the origin. A signal emitted at $t_e$ arrives at $t_r=t_e+\sqrt{v^2t_e^2+h^2}/c$.

1. Find the emission time that minimizes arrival time when past emission is unrestricted.

2. If the source begins emitting only at overhead passage $t_e=0$, find the first arrival. Explain the failure of the earlier steady-cone prediction in that case.

Question 6 - Solution

Finite emission history can remove the nominal cone arrival

(a) Method and calculation

Differentiate the arrival map:$$\frac{dt_r}{dt_e}=1+\frac{v^2t_e}{c\sqrt{v^2t_e^2+h^2}}.$$The minimum requires $t_e<0$ and gives$$t_e^*=-\frac{h}{v\sqrt{M^2-1}}=-0.84904\,\mathrm s.$$Substitution gives the steady-cone arrival $t_r^*=h\sqrt{M^2-1}/v=2.54713\,\mathrm s$.

(b) Interpretation and further calculation

If emission begins at $t_e=0$, the minimizer above is unavailable. For all allowed $t_e\ge0$, the derivative is positive; the earliest arrival comes from the initial pulse:$$t_{r,\mathrm{first}}=h/c=2.94118\,\mathrm s.$$The nominal cone arrival requires a disturbance emitted about $0.849\,\mathrm s$ before overhead passage. Geometry cannot supply a wavefront that was never emitted.

Physical check

This is a constraint on source history, not a modification of local sound speed. The unconstrained minimum is valid only if emission began early enough.