Why they were made??
In 1978 Amar Bose used one of the headphones on a airplane and noticed that he couldn’t hear the audio because of the jet engine. He started on the air flight, making the calculations for noise cancelling headphones. He introduced the first noise cancelling headphones ten years later.
How they work!!
Transverse waves are created by mechanical vibrations in a medium that produces compressions and rarefactions. A compression refers to a crest and a rarefaction refers to a trough. The medium is the part that the vibration pushes against and then pulls away. The waves create crests and troughs. The distance between two crests or troughs is the wave length, the height is the amplitude. The frequency is how many crests or troughs pass a certain point per second.
The amplitude or longitude refers to the intensity of the sound. The frequency determines the pitch or the noise. The brain can sense these characteristics but it must go through the ear first.
The way this system works is, there is a static electric charge on a very thin piece of film, that is between two metal plates. When sound comes through the two plates the electrons repel which makes the film move all by itself. Which causes the waves, which transmits the sound through the headphones.