$1.1.11^*.$ A supersonic airplane is flying horizontally. Two microphones on the same vertical at a distance $l$ from each other register the arrival of sound from an airplane flying over the microphones with a time lag $\Delta t$. The speed of sound in air is $c$. What is the speed of the plane?
1. The more the aircraft exceeds the speed of sound, the smaller the value of $AD$, T.e. $v \sim 1/AD$.
2. If the aeroplane is travelling at sonic speed $v = s$, then $AD = AB = L$, in which case triangle $ADB$ is equilateral, or
$$v = c \frac{AB}{AD} = c \frac{L}{AD} \; (1)$$
3. Let's define the elements of the triangle $ADB$
$$DB = c \cdot \Delta t\text{; }AD = \sqrt{L^2 - DB^2} \; (2)$$
4. Substituting $(2)$ into equation $(1)$, we obtain
$${v}=\frac{{cL}}{\sqrt{{L}^{2}-{c}^{2}\Delta{t}^{2}}}.$$
NO: This problem is related to the Mach wave phenomenon. This phenomenon was written about in the journal ‘Quantum’. 2010-03.pdf (42 стр.)
$$v = cl/\sqrt{l^{2}-{c}^{2}{∆t}^{2}}$$