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868 ENGINEERING THERMODYNAMICS

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the projectile at point ‘B’ is outside the spheres formed due to formation and growth of disturbance
at t = 0 and at the intermediate points (Fig. 16.2 (c)). If the tangents are drawn (from the point B)
to the circles, the spherical pressure waves form a cone with its vertex at B. It is known as Mach
cone. The semi-vertex angle α of the cone is known as Mach angle which is given by,

sin α =

Ct

Vt

C

VM

==^1 ...(16.16)

In such a case (M > 1), the effect of the disturbance is felt only in region inside the Mach

cone, this region is called zone of action. The region outside the Mach cone is called zone of silence.

It has been observed that when an aeroplane is moving with supersonic speed, its noise is

heard only after the plane has already passed over us.

Example 16.5. Find the velocity of a bullet fired in standard air if its Mach angle is 40°.

Sol. Mach angle, α = 40°

γ = 1.4

For standard air : R = 287 J/kg K, t = 15°C or T = 15 + 273 = 288 K

Velocity of the bullet, V :

Sonic velocity, C = γRT = 1 4 287 228.×× = 340.2 m/s

Now, sin α =

C

V

or, sin 40° =

340 2.

V or V =

340 2

40

.
sin ° = 529.26 m/s (Ans.)
Example 16.6. A projectile is travelling in air having pressure and temperature as
88.3 kN/m^2 and – 2 °C. If the Mach angle is 40°, find the velocity of the projectile. Take γ = 1.4 and
R = 287 J/kg K. [M.U.]
Sol. Pressure, p = 88.3 kN/m^2
Temperature, T = – 2 + 273 = 271 K
Mach angle, M = 40°
γ = 1.4, R = 287 J/kg K
Velocity of the projectile, V :
Sonic velocity, C = γRT = 1 4 287 271.×× ~− 330 m/s

Now, sin α =

C

V or sin 40° =

340

V

or, V =

330
sin 40 ° = 513.4 m/s (Ans.)
Example 16.7. A supersonic aircraft flies at an altitude of 1.8 km where temperature is
4 °C. Determine the speed of the aircraft if its sound is heard 4 seconds after its passage over the
head of an observer. Take R = 287 J/kg K and γ = 1.4.
Sol. Altitude of the aircraft = 1.8 km = 1800 m
Temperature, T = 4 + 273 = 277 K
Time, t = 4 s