Background on sound wave propagation

1. Definitions

A slight change of pressure added to a compressible flow is transmitted as a pressure wave travelling with the speed of sound. This speed of sound "a" is defined as:
       a = (K / )1/2 = (dp / d)1/2          (1)    

where K is the bulk modulus of elasticity [kgf/m2], is the density and p the pressure of the medium.

In this adiabatic changing process, friction is neglible. Then, the relationship between density and pressure is given by:

            p / = constant          (2)    

With the ratio of specific heats. ()
By substituting equation (2) into (1), the latter yields

          a = ( p /  )1/2               (3)    

Using the equation of state, (3) becomes

         a = ( R T )1/2                  (4)     

According to (4), the speed of sound is a function of the temperature.
The non-dimensional parameter "M", called Mach number, is defined as:

               M = U / a                      (5)    

In this function, U is the speed of the source of sound, e.g. an aircraft.


2. Air pressure waves


M = 0
Stationary : Mach Number = 0

When the source of sound is stationary (its speed is zero), waves of air pressure spread uniformly, and form a circle (2-dimensional case) or a sphere (3- dimensional).


M = 0.4


M = 0.7
Subsonic speed : Mach Number < 0.3

If the source of sound or an aircraft moves slower than the speed of sound, waves of air pressure develop around it. A surface that has different pressures on each side, is called a wave front. These pressure waves, wave fronts, move ahead of the aircraft because the waves travel faster than the sound source.


M = 1

Transonic speed : Mach Number = 0.3-1

As the aircraft approaches the speed of sound, it catches up with the pressure waves. These dense air pressure waves form a shock wave, shown as a red line. 


M = 1.4


M = 2.0
Supersonic speed : Mach Number > 1

As the aircraft accelerates beyond the speed of sound, it breaks through the shock wave. In this case, the shape of the shock wave forms either a wedge (2 dimensional) or a cone (3 dimensional). The shock wave angles back behind the aircraft, reaching the ground as a sonic boom. (You may be caught off-guard because you do not hear any noise until the aircraft has passed overhead, often at high altitude.)

3. Mach angle

As the aircraft flies faster than the speed of sound (M > 1), the shock wave forms either a wedge or a cone. Mach waves are very weak shock waves when the disturbance is very small. If this airplane travels at the speed of  U [m/s], the aircraft will move U*t [m] (shown in the figure below as the distance O-A), while the pressure wave which is formed at the initial position, will travel the distance of a*t [m] (shown as O-H). Hence the half angle of the Mach wave,"", should satisfy the following equation.

          = sin-1( (a*t)/(U*t) )                         (6)    

 By using (5), equation (6) gives,

               = sin-1(1/M)                                (7)    

M = 2.0 ( = 30 [deg] )

4. Example

If the speed of an aircraft is Mach 2, the half angle of the Mach wave is calculated as follows:
= sin-1(1/M)
     = sin-1(1/2)
      = 30.0 [deg]

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