A **wave** is a disturbance that propagates. Apart from electromagnetic radiation, which can travel through vacuum, waves have a medium through which they travel and can transfer energy from one place to another without any of the particles of the medium being displaced permanently. Instead, any particular point oscillates around a fixed position.

- Sea-waves, which are perturbations that propagate through water (see also surfing and tsunami).
- Sound - a mechanical wave that propagates through air, liquid or solids, and is of a frequency detected by the auditory system. Similar are seismic waves in earthquakes, of which there are the S, P and L kinds.
- Light, radio waves, x-rays, etc. make up electromagnetic radiation. In this case propagation is possible without a medium, through vacuum.

- Reflection - when a wave turns back from the direction it was travelling, due to hitting a reflective material.
- Refraction - the change of direction of waves due to them entering a new medium.
- Diffraction - the spreading out of waves, for example when they travel through a small slit.
- Interference - the addition of two waves that come in to contact with each other.
- Dispersion - the splitting up of a wave up depending on frequency.

Waves can be described using a number of standard variables including: frequency, wavelength, amplitude and period.
The amplitude of a wave is the measure of the magnitude of the maximum disturbance in the medium during one wave cycle, and is measured in units depending on the type of wave. For examples, waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter). The amplitude may be constant (in which case the wave is a *c.w.* or *continuous wave*) or may vary with time and/or position. The form of the variation of amplitude is called the *envelope* of the wave.

The period (*T*) is the time for one complete cycle for an oscillation of a wave. The frequency (*F*) is how many periods per unit time (for example one second) and is measured in hertz. These are related by:

- <math>f=\frac{1}{T}</math>.

When waves are expressed mathematically, the *angular frequency* (*ω*, radians/second) is often used; it is related to the frequency *f* by:

- <math>f=\frac{\omega}{2 \pi}</math>.

<math>y=A(z,t) \cos (\omega t - kz + \phi)</math>,

where *A*(*z*,*t*) is the amplitude envelope of the wave, *k* is the *wave number* and *φ* is the *phase*. The velocity *v* of this wave is given by:

<math>v=\frac{\omega}{k}= \lambda f</math>,

where *λ* is the *wavelength* of the wave.

In the most general sense, not all waves are sinusoidal. One example of a non-sinusoidal wave is a pulse that travels down a rope resting on the ground. In the most general case, any function of x, y, z, and t that is a non-trivial solution to the wave equation is a wave. The wave equation is a differential equation which describes a harmonic wave passing through a certain medium. The equation has different forms depending on how the wave is transmitted, and on what medium.

The Schrödinger equation describes the wave-like behaviour of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.

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