Fourier Transform and Wave Packets

Abstract

The use of integral transforms is a cornerstone in the analysis of partial differential equations when it comes to their linear behaviour. In this chapter, the focus is on the Fourier transform and its basic properties. The physical counterparts of this mathematical object are wave packets and their dynamics as an expression of the main linear features of a partial differential equation governing the physical phenomenon. Examples are provided relative to the d’Alembert wave equation, to the diffusion equation governing, for instance, the heat conduction in a solid material, to the advection–diffusion equation, and to the Schrödinger equation of quantum mechanics for a free particle. These sample models are here discussed with reference to the one-dimensional case. Definitions of fundamental quantities such as the wave number, the angular frequency, the phase velocity and the group velocity are given. Finally, the three-dimensional case will be surveyed.