Fourier Transforms

CHAPTER 4 THE FOURIER SERIES

Oscillatory phenomena are ubiquitous in physical systems. Here are some examples. When a guitar string is plucked, it vibrates. The vibration of the string sets up an acoustic wave in the air which, when it reaches your eardrum, causes it to vibrate. This vibration is mechanically transmitted to a fluid in the cochlea. The oscillating fluid stimulates sensory cells, which transmit to the brain electrical signals that are, in turn, interpreted as the sound of the guitar. The light emitted by a laser comes from an electromagnetic standing wave oscillating in the laser's resonant cavity. The wave itself is fed by atomic processes described, in the language of quantum mechanics, by waves. Numerous optical devices rely on the interactions of optical waves with periodic structures. The electromagnetic fields radiated by highly directive antenna arrays and the acoustic fields used in medical ultrasound imaging are combinations of waves. The shaking of the ground during an earthquake and the resulting motions of buildings are also periodic vibrations. Moreover, even systems that display no oscillatory behavior, like the evolution of the temperature distribution in a bar of metal as it is heated, can be described mathematically as portions of periodic functions.

The mathematical models for these diverse physical systems are based on a simple idea, proposed by Fourier in 1807, that any function defined on a finite interval [0, L], including certain discontinuous ...