# Chapter 2

# Superposition of Harmonic Oscillations, Fourier Analysis

The oscillations of a system can never be exactly periodic since they always have a beginning and an end. However, the oscillations may be treated as approximately periodic if they last a very long time, compared to the period of a single oscillation. On the other hand, even if the oscillations of a system are approximately periodic, they are never exactly simple harmonic (i.e. represented by a sinusoidal function with a single frequency). For instance, even if light is a single line of the discrete atomic spectrum or a laser beam, it is always a superposition of monochromatic waves in a more or less wide band. The superposition of oscillations and waves is so real and important that it is often raised to the rank of a principle (called the *superposition principle*). It plays a very important part in the study of interference, diffraction and quantum mechanics. The validity of this principle relies on the linearity of the mechanics and electromagnetism equations.

Our purpose in this chapter is to study the superposition of simple harmonic oscillations. The *Fourier analysis* considers any function as a superposition of simple harmonic functions. We study the case of periodic functions and of nonperiodic functions, namely signals of short duration.

## 2.1. Superposition of two scalar and isochronous simple harmonic oscillations

Consider the oscillations *u*_{1} = *a*_{1} cos(ω*t* + _{1}) and *u*_{2} = *a*_{2} cos(ω*t* + _{2}) that are scalar (or vectorial ...