Wave

A wave¹ is a disturbance that travels in a medium, transferring energy and momentum from one place to another. The direction of the propagation of the wave is the direction of energy transfer. There is no large-scale motion of the medium itself.

An electromagnetic wave is a specific wave case for electric and magnetic fields.

Basics

We express a wave mathematically with a wavefunction. For a constant speed $v$, a 1D wave is propagated in the $\pm x$ direction with:

$$f\left(t\pm \frac{x}{v}\right)$$

In 3D space, where $\vec{r}=x\hat{x}+y\hat{y}+z\hat{z}$ (the position), and $\hat{n}$ is the direction of propagation:

$$f\left(t\pm \frac{\hat{n}\cdot \vec{r}}{v}\right)$$

The core idea is that we have a mathematical representation of wave motion. A wavefunction depends both on a position $\vec{r}$ and a time $t$.

The 1D wave equation is defined below. All solutions to wave equations are wavefunctions (in the same form as the above!).

$$\frac{{\partial }^{2}u}{\partial z^2}=\frac{1}{v^2}\frac{{\partial }^{2}u}{\partial t^2}$$

The 3D wave equation is defined below, where the ${\nabla }^2$ denotes the Laplacian operator.

$${\nabla }^2\vec{u}=\frac{1}{v^2}\frac{{\partial }^2\vec{u}}{\partial t^2}$$

There are two main types of planes:

• Plane waves produce planar wavefronts.
• Spherical waves emanate outwards from a single point.

Parameters

We define several key parameters:

• Temporal parameters
  • The temporal period (or just period) is the time for one complete oscillation $T=\frac{2\pi }{\omega }$.
  • The temporal frequency $\nu$ is the reciprocal of the period, measured in hertz (Hz). It indicates how many periods we have within one second.
  • The angular frequency is given by $2\pi /T=2\pi \nu$.
• Spatial parameters
  • The wavelength (spatial period) is the length of one complete oscillation $\lambda =\frac{2\pi }{k}$.
  • The wave number (spatial frequency) $\frac{1}{\lambda }$.
  • The angular spatial frequency is $k$.
• The speed of a wave is given by $v=f\lambda$.
  • This only depends on the properties of the medium it passes through and not how it’s produced.

Sub-pages

• Waveform, for practical engineering applications of waves
• Electromagnetic wave
  • Wave optics
  • Harmonic plane wave
• Poynting’s theorem
• Polarisation

Footnotes

1. “The concept is intrinsically somewhat vague” - David Griffiths, in Introduction to Electrodynamics ↩