# From ripping tape to the magic Mach cone

To seal the carton, it is often necessary to fix it with tape. When the tape is torn off, a “squeak” sound will be heard. As everyone knows, this sound is the same principle as the sonic boom produced by supersonic aircraft flying. What kind of scientific principle is behind this phenomenon of life?

To clarify this problem, we need to analyze the stress on the tape when the tape is torn off.
First of all, the tape at the initial position will be affected by the adhesive force of the separation point and the pulling force of the left and right hands of the person. Since the angle α between the viscous force and the right-hand pulling force is relatively large, the radial component of the pulling force is smaller than the viscous force, and the tape will not continue to separate immediately, but will form a rotational moment, which is comparable to the moment generated when the left hand fixes the tape. balance.
As we tear the tape, the angle between the direction of the pulling force and the viscous force decreases, and when the radial component of the pulling force exceeds the viscous force, the tape will be torn off. When this small piece of tape is torn off, it will lose its viscous force in an instant, and it will move away with a large acceleration under the action of pulling force.
Continue to tear the tape, and the tape will repeat the above separation process. It can be seen that tearing the tape seems to be a breeze, but its stress is different at every moment.

The researchers observed under the high-speed camera that the time it takes for the tape to “jump” 1 mm is only a few microseconds (1 microsecond is equal to one millionth of a second), so the moving speed of the tape along the tearing direction can reach 600~ 900 m/s (the speed of sound in air is 340 m/s at 1 standard atmospheric pressure and 15 degrees Celsius), which has reached the range of supersonic speed.
The supersonic movement of the tape will strongly squeeze the air in the forward direction, thereby creating a shock wave (shock wave), with the tape as the wave surface, which will continue to spread upwards, and then form a shock wave overpressure and emit a sonic boom. After several tiny sonic booms are superimposed, it will form the harsh sound heard when tearing tape every day.
Mach cone behind supersonic

In order to reflect the compression of the air by the tape more clearly, here we introduce three concepts: Mach number, Mach cone and Mach wave.
Mach number M=the ratio of air velocity V to sound velocity a, that is, M=V/a. When tearing the tape, V is 600-900 m/s, a is 340 m/s, and the calculated Mach number is 1.7-2.6. The Mach number of a supersonic aircraft is between 1 and 5, that is to say, the compression speed of the air when the tape moves along the tearing direction exceeds the compression speed of the air when some supersonic aircraft fly at high speed.
When an object moves at supersonic speed, the sound waves always lag behind the object, forming a cone behind it, and all sound waves are confined in this cone, which is called Mach cone. In simple terms, Mach cones form when the velocity of an object (the source of the wave) is greater than the velocity of the sound wave.

high speed fighter

The Mach cone is the interface between the disturbed and undisturbed medium. This wave is called bow (shǒu) wave, also known as Mach wave.
When a supersonic aircraft flies, the sound waves are confined in the Mach cone. Only when the Mach cone sweeps by, can we hear the sonic boom on the ground (the shock wave generated by the aircraft flying at supersonic speed is transmitted to the ground to form an explosion).
Mach cones around us

We often see Mach cones in real life. For example, when we stand by the sea and watch a speedboat passing by, we will find a “V” shaped wave with the boat as the top on the water surface behind the boat. , the speed of the ship can easily exceed the propagation speed of the water wave, and the bow wave can be observed on the water surface.
When the speed of the light source is greater than the speed of light traveling around, the photon Mach cone will also be generated theoretically. In recent years, researchers have used ultrafast scanning cameras to record photonic Mach cones. The discovery of the photonic Mach cone is expected to develop a new imaging mode, which brings great hope for the development of the next generation of biomedical imaging instruments.

Schematic diagram of speedboat wake

Photonic Mach Cone

Inconspicuous items in daily life may contain profound scientific principles, and the application of these principles can create high-tech products that change human life. We must pay attention to observing life with a scientific perspective and discover science from life.