Shock wave and the Mach number

The experience of sonic booms caused by an airplane flying overhead having speed greater than that of sound is very common. The cause of this is what we are going to study here. The motion of an airplane through the air produces sound. If V be the speed of sound and Vs be that of the airplane, then the wavelength of the associated wave, as long as Vs is well less than V, is given by,

Where fs is the frequency of sound.

As the velocity of the plane approaches the speed of sound, this wavelength gets shorter and the waves crests pile up on each other as shown in the figure (a) below.

The air-plane must exert a large force to compress the air in front of it. According to Newton’s third law, the air exerts an equal force back on the airplane. Hence there is a large increase in air resistance as the airplane approaches the speed of sound V, this phenomenon is known as the “sound barrier.”

The wavelength equation in eq.(1) is derived from the Doppler effects. But as the speed of the airplane Vs goes beyond that of sound, the source of sound becomes supersonic and the wavelength is not governed by equation (1) and the Doppler effect does not describe the sound wave in front of the source any more. The figure (b) elaborates what happens in this part. As the airplane moves, it produces sound. A series of wave crest is thus emitted each spread out in a circle cantered at the position of the airplane when it emitted the crest. After a time, t the crest emitted from point S1 has spread to a circle with radius Vt, and the airplane has moved a greater distance Vst to position S2. As a result, there will be interference of the sound waves. Let the waves interfere constructively along the bold line as shown in figure (b), which makes an angle β with the direction of the airplane velocity. As the waves interfere constructively, the amplitude of the resultant wave crest increases along the bold line. This increased amplitude wave crest is known as shock wave.

According to the figure,

Here, the ratio   is known as the Mach number. This number is greater than unity for all supersonic speeds. These shock waves are 3D in nature as shown in the figure below.

It’s the arrival of this shock wave that causes the sonic boom to our ears.

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January 31, 2018

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