Wave

See also: Wave (homonymy)

A wave is the propagation of a Perturbation producing on its passage a reversible variation of Propriété S Physique S local. It transports energy without transporting Matière.

As any unifying concept the wave recovers a large variety of very different physical situations.

• the oscillating wave, which can be Périodique, is well illustrated by the wrinkles caused by stone which fall into water.
• the solitary wave or Soliton finds very an good example in the Mascaret S.
• the Shock wave, perceived acoustically with the passage of the Mur of the sound by a plane, for example.
• the electromagnetic Onde does not have in certain cases not a material support. In quantum physics, an electromagnetic wave is a wave caused by a particle. For example, the photon, is, according to Einstein, a particle which corresponds to the electromagnetic wave of the light.
• the acoustic Wave which has one of them.
• the wave of probability

Examples

Let us illustrate the concept of “transport of energy without transport of matter”. In the case of a mechanical Wave, one observes small local and transitory displacements elements of the medium which support this wave, but not total transport of these elements. It is thus for a marine Vague which corresponds to a roughly elliptic movement of the water particles which, in particular, agitate a Bateau in Mer. In this context, a horizontal displacement of matter is a Courant; however, one can have a wave without current, even a vague energy with counter-current. The wave transports Horizontal ement the energy of the Vent which gave rise to him the Broad and, this independently of the total transport of water.

In the cord musical instruments the disturbance is brought various manners: bow (violin), hammer (piano), finger (guitar). Under the effect of the excitation applied transversely, all the elements of the cords of these instruments vibrate transversely around a position of balance which corresponds to the cord at rest. The energy of vibration of the cords transforms into its because the transverse movements of the cords put moving the air which bathes them. A sound corresponds to the propagation in the air of a wave of pressure of this air. In a point of space, the pressure of the air oscillates around the value of its pressure at rest, it grows and it decrease alternatively around this value. In a sound wave the local movement of the molecules of air is done in the same direction as the propagation of energy, the wave is longitudinal. It should be noted that the longitudinal and transverse directions refer to the direction of propagation of the energy which is taken as longitudinal direction.

The electromagnetic waves are transverse waves which can move in the Vide as in a medium. Optics is a particular case of propagation in mediums Diélectrique S, while the propagation in a Métal corresponds to an electric current in alternative mode.

The gradually transmitted signal can as for him be illustrated using the dominos: the latter receive a signal and transmit it while falling on dominating according to. A file of car advancing with the signal of a green light does not constitute an example of transmission gradually.

Dimensionnality

Are $\backslash \; overrightarrow\; \{U\}$ the displacement of energy and $\backslash \; overrightarrow\; \{v\}$ the speed of the wave:

• $\backslash \; overrightarrow\; \{U\}\; \backslash \; parallel\; \backslash \; overrightarrow\; \{v\}$: the wave is longitudinal.

example: Arises with roll. If one brutally moves a whorl of such a spring tended between two supports one sees being formed one compression wave of the whorls. In this case the movement of the whorls is done in the same direction as the propagation of energy, according to the line which the axis of symmetry of the spring constitutes. It is about a longitudinal wave to a dimension.

• $\backslash \; overrightarrow\; \{U\}\; \backslash \; club-footed\; \backslash \; overrightarrow\; \{v\}$: the wave is transverse.

Examples: When a drum is struck, one creates on his skin a transverse wave with two dimensions, as in the case of water surface.

When one moves electric charges, the magnetic fields and electric local vary to adapt to the variation of position of the loads producing a electromagnetic Onde. This wave is transverse and can be propagated in the three directions of space. Let us note that in this case the wave is not a displacement of matter.

• a wave can be at the same time longitudinal and transverse.

Example: On the Sea, a Vague is created by a Vent in general of a storm which causes a variation height of water. It is the same for the rounds in water caused by the fall of a stone. In this case one can easily see that the wave propagation is done in two dimensions of water surface.

Temporal periodicity and space periodicity

The simplest case of periodic travelling wave is a wave known as “monochromatic”.

If one takes a stereotype of the medium at a given time, one sees that the properties of the medium vary in a sinusoidal way according to the position. There is thus a space periodicity; the distance between two maximum is called Wavelength, and is noted λ. If successive photographs are taken, it is seen that this “profile” moves at a named speed Speed of phase.

If one places oneself at a given place and that one records the intensity of the phenomenon according to time, one sees that this intensity varies according to a law, it also sinusoidal. The time which passes between two maximum is called period and is noted T .

Modeling of a travelling wave

A wave is modelled by a function has ( X , T ), of amplitude has , X being the position in space (vector) and T being time.

Very an big family of the solutions of equations of propagation waves is that of the sinusoidal functions, sine and cosine (they are not only). It is also shown that any continuous periodic phenomenon can break up into functions sine oïdales (Fourier series), and in a general way any continuous function (Transformée of Fourier). The sinusoidal waves are thus object of a simple and useful study.

Within this framework, a sinusoidal wave can be written:

$A\; (X,\; T)\; =A\_0\; \backslash \; sin\; (\backslash \; Omega\; T\; -\; \backslash \; mathbf\; \{K\}\; \backslash \; cdot\; \backslash \; mathbf\; \{X\}\; +\; \backslash \; varphi)$

(Demonstration)

One calls

• amplitude the factor $A\_0$,
• phase the argument of the sine $\{\backslash \; Omega\; T\; -\; \backslash \; mathbf\; \{K\}\; \backslash \; cdot\; \backslash \; mathbf\; \{X\}\; +\; \backslash \; varphi\}$,
• while φ is the phase in the beginning when T and X is null.

The absolute phase of a wave is not measurable. The Greek letter ω indicates the Pulsation wave one notes that it is given by the derivative of the phase compared to time:

$\{\backslash \; partial\; \backslash \; over\; \backslash \; partial\; T\}\; (\backslash \; Omega\; T\; -\; \backslash \; mathbf\; \{K\}\; \backslash \; cdot\; \backslash \; mathbf\; \{X\}\; +\; \backslash \; varphi)\; =\; \backslash \; Omega$.

The vector K is the Vecteur of wave. When one places oneself on only one axis, this vector is a scalar and is called Nombre of wave: it is the number of oscillations which one counts on 2$\backslash \; pi$ units of length.

One has for the standard of the vector of wave:

The pulsation is written according to the frequency $\backslash \; nu$:

The speed of phase is worth finally:

Types of waves

One distinguishes several categories of waves:

• longitudinal waves, where the points of the propagation medium move locally according to the direction of the disturbance (typical example: the compression or the decompression of a spring, the its in a medium without shearing: water, air…)
• transverse waves, where the points of the propagation medium move locally perpendicularly within the meaning of the disturbance, so that it is necessary to utilize an additional size to describe them (typical example: waves, waves of the earthquakes, the electromagnetic waves). One speaks to describe this of polarization.

The propagation medium of a wave can be three-dimensional (sound wave, luminous, etc), two-dimensional (wave at water surface), or unidimensional (wave on a vibrating cord).

A wave can have several geometries: planes, spherical, etc It can also be progressive, stationary or évanescente (see Propagation of the waves). It is progressive when it moves away indefinitely from its source.

From a more formal point of view, one also distinguishes the scalar waves which can be described by a variable number in space and time (the sound in the fluids for example), and the vectorial waves which require a vector with their description (the light for example)…

If one defines the waves as associated with a material medium, the electromagnetic waves are excluded! In the latter case it is an electromagnetic disturbance which can be propagated in the vacuum (matter).

Celerity of a wave, frequency

Two speeds can be associated with a wave: the speed of phase and speed of group. The first is the speed to which the phase of the wave is propagated, while the second corresponds to the propagation velocity of the envelope (possibly deformed during time). The speed of group corresponds to what is called the celerity of the wave.

For a periodic travelling wave, there is a double periodicity: at a given moment, the size considered is spatially periodic, and at a given place, the size periodically oscillates with the court of the naked temps.
Fréquence $\backslash$ and period T are bound by the relation $T\; =1/\backslash \; nu$.
For a travelling wave being propagated with celerity C , the corresponding wavelength $\backslash \; lambda$ is then determined by the relation: $\backslash \; lambda=c/\backslash \; nu$ where $\backslash \; lambda$ is in m, $\backslash \; nu$ in hertz (Hz), and C in m.s- ¹.
$\backslash \; lambda$ is the space period of the wave.

The celerity of the waves depends on the properties of the medium. For example, the sound in the air with 15°C and 1 bar is propagated with 340 m.s- ¹.

• For a material wave, plus the medium is rigid, plus celerity is large. On a cord, the celerity of a wave is all the more large as the cord is tended. The speed of sound is larger in a solid than in the air. In addition, more inertia of the medium is large, more celerity decreases. On a cord, celerity is all the more large as the linear density (mass per unit of length) is low.

• For an electromagnetic wave, the propagation velocity will be generally all the more large as the medium is diluted (in the general case, it is however advisable to consider the electromagnetic properties of the medium, which can complicate the physique of the problem). Thus, the speed of light propagation is maximum in the vacuum. In glass, it is approximately 1,5 times weaker.

In a general way, celerity also depends on the frequency of the wave. Such mediums are qualified the dispersive ones, the others, those for which celerity is same whatever the frequency are known as not-dispersive.
Extremely fortunately, the air is a nondispersive medium for our sound waves! With regard to the light, the phenomenon of Dispersion is also at the origin of the Arc-en-ciel: the various colors are propagated differently in water, which makes it possible to break up the sunlight according to its various components. Dispersion by a prism is also classically used: by breaking up the light, one can thus make spectroscopy (the interferential methods give however now results much more precise).

Examples of waves

• mechanical Waves:

• the Vague S are disturbances which are propagated in the Eau (see also Tsunami).
• Onde on a vibrating cord
• the its is a wave of Pression which is transmitted in the fluids and the solids, and which is detected by the auditive Système
• the seismic waves is similar to the sound waves and is generated at the time of a Earthquake
• electromagnetic Ondes:
• the electromagnetic Light and, in general, waves result from the electromagnetic disturbances
• a Onde radio is a variable electromagnetic field, often periodic, produced by a antenna
• the gravitational waves

See also

Related articles

On the various undulatory phenomena

• Wave on a vibrating cord
• Mascaret, Tsunami, Vague scélérate
• Package of wave
• Beat

Physical theoretical elements

• Propagation the waves, Diffusion of the waves, Interference
• Diffraction, reflection, Refraction
• Soliton

To the measures of the waves

• Frequency, Hertz;
• Wavelength;
• Phase, Dephasing, Harmonic

Mathematical theoretical elements

• goniometrical Function, periodic Phenomenon

• Joseph Fourier, Transformed of Fourier.
• Pierre-Simon Laplace, Transformée of Laplace
• Produces convolution

External bonds

• Waves of surface, Michel Heel

Random links:

Compatible Time Sharing System | Honore Sebastien Vial de Clairbois | Buzz (rock group) | Osipaonica | Žamberk | Étude_de_la_musique_par_l'oreille

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