1  Introduction

1.1 What are waves?

It would be wonderful if we could start this part of the course with a reliable and universally agreed upon definition of the concept of a wave. Alas, waves are rather difficult to define in a way that is neither too vague nor rules out examples of wave phenomena. A wave might, for example, be defined as

a disturbance that transfers energy through a medium or vacuum without the transfer of matter1

1 Google AI.

but this already seems a rather awkward definition2. Moreover, standing waves, which are considered by most to be waves, would not fit this definition. Wikipedia perhaps does better with

2 If a wave is neither in a medium nor in a vacuum, what is it in?

a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities

though again, the adjective ‘propagating’ does not seem to apply to a standing wave. Perhaps, like other difficult to define concepts, it may be pragmatic to say of waves that “I know one when I see one”.

Fortunately, in this course we will only be examining a subset of wave phenomena and at each stage we will ensure that the properties of the waves being discussed are clear. Hence, it should be unnecessary to have a single comprehensive definition of a wave for this course.

1.2 Types of wave

To give an indication of just how pervasive wave phenomena are, we will list a few types of waves and their properties. Looking first at types of physical wave, we have

  • Electromagnetic waves, e.g. light, radio waves, gamma rays, infra-red,
  • Sound waves, including shock waves,
  • Seismic waves,
  • Surface waves, including water waves3,
  • Waves on a string or wire,
  • Gravitational waves,
  • Waves in traffic flow,
  • etc.

3 In turn, water waves may be wind drive, tidal waves, tsunamis, standing waves, capillary waves, gravity waves, breaking waves, etc.

Furthermore, fundamental particles have wave like behaviour and are described using wave functions. Looking at their mathematics and properties, waves may be

  • Linear or non-linear,
  • Dispersive or non-dispersive,
  • Transverse or longitudinal (or both),
  • Propagating or standing,
  • Solitary waves (solitons),
  • Wave packets,
  • etc.

It should be clear one can have a healthy research career studying just waves.

1.3 What sort of waves will we focus on?

Naturally, we will need to focus our attention on just a subset of the phenomena listed above. We will start be considering linear4, non-dispersive waves in one dimension, discovering the wave equation and extending it to three-dimensions. However, from then on we will focus on electromagnetic waves, i.e. light. Being an electromagnetic phenomenon, light waves satisfy Maxwell’s equations5 which are linear, ensuring that the study of light waves will not be too strenuous. However, being transverse waves propagating in three dimensions, there will be plenty of phenomena, e.g. polarisation, reflection, refraction, diffraction an interference, to study.6

4 Non-linear waves are considerably more challenging to study

5 You will discover Maxwell’s equations later in this unit.

6 Note that studying electromagnetic waves before studying electromagnetism will inevitably lead to some challenges. I will attempt to minimise use of knowledge of electromagnetism that you have not yet seen, but there may be the occasional formulae or equations that you will need to take on trust. I will also use appendices to provide material worth revisiting after you have studied electromagnetism.