Supersonic Speeds, Shock Waves — Question 4

Question 4

An observer off the flight line sees a different cone delay

A source flies along the $x$ direction at constant altitude $h=1200\,\mathrm m$, with $M=2.5$ and $c=340\,\mathrm{m/s}$. A ground observer is laterally displaced by $b=900\,\mathrm m$ from the flight line. Time zero is the source crossing the observer's transverse plane.

1. Find the perpendicular distance from the observer to the flight axis and the boom delay.

2. Derive the ground-intersection relation between distance $x$ behind the source and lateral coordinate $y$ at a fixed instant.

Question 4 - Solution

An observer off the flight line sees a different cone delay

(a) Method and calculation

The cone is rotationally symmetric about the flight axis. The relevant perpendicular distance is$$R_\perp=\sqrt{h^2+b^2}=1500\,\mathrm m.$$$$x=R_\perp\cot\mu=R_\perp\sqrt{M^2-1},\quad t_b=x/(Mc)=4.04345\,\mathrm s.$$

(b) Interpretation and further calculation

At ground coordinate $y$, the radial distance from the elevated axis is $\sqrt{h^2+y^2}$. Thus the cone intersects the ground where$$x^2=(M^2-1)(h^2+y^2),\qquad x>0\ \text{behind the source}.$$Equivalently $x^2/[(M^2-1)h^2]-y^2/h^2=1$, a trailing hyperbola branch. Off-axis observers receive the envelope later than an observer directly under the axis in the same transverse plane.

Physical check

Setting $b=0$ recovers the directly-under-track delay. Altitude and lateral displacement enter through their perpendicular geometric combination.