Problem 3.16

An aircraft is flying at an altitude of km at Mach . Find the distance behind the aircraft at which the disturbances created by the aircraft reach sea level.

Given:
altitude = 6 km,
\( M = \) 3.

To calculate: \( d=? \)

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

Using the Mach angle equation,

\[ \sin\mu=\frac{1}{M}\ \implies\ \mu=\sin^{-1}\left(\frac{1}{M}\right) \]

Using trigonometry we can write,

\[ \tan\mu=\frac{6}{d}\ \implies\ d=\frac{6}{\tan\mu} \]

\[ d=\frac{6}{\tan\left(\sin^{-1}\left(1/M\right)\right)}=\frac{6}{\tan\left(\sin^{-1}\left(1/3\right)\right)} \]

\[ \boxed{d=16.97\,\text{km}}\ . \]

Problem 3.16:

Solution: