Chapter 13. Mach angle (13-37)

Review

The angle of the Mach cone depends on the airplane’s speed, which we call \(v_{\mathrm{airplane}}\). To see how to calculate this angle, inspect parts (a) and (b) of Figure 13-27. In a time \(\Delta{t}\) the airplane travels a distance \(v_{\mathrm{airplane}}\Delta{t}\); this is shown by a blue line in Figure 13-27a and Figure 13-27b. The distance \(v_{\mathrm{airplane}}\Delta{t}\) forms the hypotenuse of a right triangle, one leg of which is the distance \(v_{\mathrm{sound}}\Delta{t}\) that the sound travels in time \(\Delta{t}\). The sine of the angle \(\theta_{\mathrm{M}}\) in Figure 13-27b, known as the \(\textbf{Mach angle}\), equals the side opposite this angle (\(v_{\mathrm{sound}}\Delta{t}\)) divided by the hypotenuse (\(v_{\mathrm{airplane}}\Delta{t}\)). The factors of \(\Delta{t}\) cancel, leaving