⇒ In this chapter we will be exploring the mathematical and physical ideas at the basis of our description of oscillations and waves. We have already explored the dynamics of the simple harmonic oscillator and we shall be building on this example to explore oscillating systems in general. We will first consider the effect of damping, and then analyse the response of the oscillating system to an external driving force (in both time and frequency domain) leading to the idea of resonance1 . We shall also be making liberal use of complex numbers, that have an important role in greatly simplifying the mathematics of oscillations, and really add a new dimension to the solution by helping us to focus our attention of the concept of phase. The strength of the simple harmonic oscillator model lies in its simplicity and widespread applicability. In fact the same mathematics that we will be developing in this chapter is used to describe motion of masses on springs, pendula (2.1), atomic resonances, perturbation to orbits of satellites and electrical circuits with reactive elements (capacitors and inductors). Indeed any physical system if displaced from equilibrium by a small enough amount will perform simple harmonic motion (as we proved in Mechanics, Part I). The logical development of the idea of a simple harmonic oscillator is to introduce a space dependants to the solution: imagine putting a number of harmonic oscillators one after the other with a mutual phase relationship depending on position: you have built yourself a wave! Waves, like harmonic oscillators, are everywhere in Physics and describe some of the most fascinating phenomena known to the inquiring mind: waves in materials (sound waves) and the electromagnetic spectrum (including visible light!).