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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.

Characteristic properties All waves have common behaviour under a number of standard situations. All waves can experience the following:
Transverse and longitudinal waves Transverse waves are those with vibrations perpendicular to the wave's direction of travel; examples include ripples on the surface of a pond, waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations along the wave's direction of travel; examples include sound waves.
Physical description of a wave
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:
When waves are expressed mathematically, the angular frequency (ω, radians/second) is often used; it is related to the frequency f by:
<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 nonsinusoidal 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 nontrivial 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 wavelike 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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