Wave

From Free net encyclopedia

This article is about waves in the most general scientific sense; a separate article focuses on ocean waves.

For other meanings see wave (disambiguation).

A wave is a disturbance that propagates through space, often transferring energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic radiation, and probably gravitational radiation can travel through vacuum, that is, without a medium. Waves travel and transfer energy from one point to another, with little or no permanent displacement of the particles of the medium (there is little or no associated mass transport); instead there are oscillations around fixed positions.

Periodic waves are characterized by crests (highs) and troughs (lows), and may usually be categorized as either longitudinal or transverse. Mechanical waves are longitudinal if the particle displacements are parallel to the direction of propagation of the wave crests (example: acoustic waves, or sound), and transverse if the displacements are perpendicular to that direction (example: waves on a string).

1 The medium which carries a wave
2 Examples of waves
3 Characteristic properties
4 Transverse and longitudinal waves
- 4.1 Polarization
5 Physical description of a wave
6 See also
7 External links

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The medium which carries a wave

A medium that can carry a wave is classified by one or more of the following properties:

A linear medium if the amplitudes of different waves at any particular point in the medium can be added.
A bounded medium if it is finite in extent, otherwise unbounded.
A uniform medium if its physical properties are unchanged at different locations in space.
An isotropic medium if its physical properties are the same in different directions.

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Examples of waves

Image:Waves crashing.jpg

Ocean surface waves, which are perturbations that propagate through water (see also surfing and tsunami).

Radio waves, microwaves, infrared rays, visible light, ultraviolet rays, x-rays, and gamma rays make up electromagnetic radiation. In this case, propagation is possible without a medium, through vacuum. These electromagnetic waves travel at 299 792 458 m/s in a vacuum.

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.

Gravitational waves, which are fluctuations in the gravitational field predicted by general Relativity. These waves are nonlinear, and have yet to be observed empirically.

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Characteristic properties

All waves have common behaviour under a number of standard situations. All waves can experience the following:

Reflection – the change of direction of waves, due to hitting a reflective surface.
Refraction – the change of direction of waves due to them entering a new medium.
Diffraction – the circular spreading of waves that happens when the distance between waves move through an opening of equal distance.
Interference – the superposition of two waves that come into contact with each other.
Dispersion – the splitting up of waves by frequency.
Rectilinear propagation – the movement of waves in straight lines.

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Transverse and longitudinal waves

Image:Elliptical trajectory on ripples.png

Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.

Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow elliptical paths.

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Polarization

Only transverse waves can undergo polarization. A wave can be polarized by using a polarizing filter; this wave will only oscillate in one direction.

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Physical description of a wave

Image:Wave.png

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 crest is the highest point of a wave, while the trough is the lowest point. The wavelength (λ) is the distance between two crests or two troughs that are beside each other. For electromagnetic radiation, it is usually measured in nanometres.

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:

In other words, the frequency and period of a wave are reciprocals of each other.

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>.

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Travelling waves

Waves that remain in one place are called standing waves - e.g. vibrations on a violin string. Waves that are moving are called travelling waves, and have a disturbance that varies both with time t and distance z. This can be expressed mathematically as:

<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.

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Propagation through strings

The speed of a wave travelling along a string (v) is directly proportional to the square root of the tension (T) over the linear density (ρ):

This equation can be found using dimensional analysis

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The wave equation

The wave equation is a differential equation which describes a harmonic wave passing through a medium, discussed above. The equation has different forms depending on how the wave is transmitted, and through what medium.

Not all waves are sinusoidal. One example of a non-sinusoidal wave is a pulse that travels down a rope resting on the ground, extending in direction x, travelling at velocity c. The height of the pulse above the ground is φ. The distance the pulse travels between some time t and time 0 is ct.

In one dimension the wave equation has the form

<math>\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=\frac{\partial^2\phi}{\partial x^2}. \ </math>

A general solution, given by d'Alembert is

considered to be the shapes of two pulses travelling down the rope, F in the +x direction, and G in the -x direction. If we substitute for x above, replacing it with directions x, y, z, we then can describe a wave propagating in three dimensions.

A non-linear wave equation can cause mass transport.

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. Quantum mechanics also describes particle properties that other waves, such as light and sound, have on the atomic scale and below.

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