# Coherent wave

Coherent waves - vibrations is a constant phase difference. Of course, the condition is not in every point in space, only in some areas. Obviously, to meet the definition of the oscillation frequency also foreseen equal. Other waves are coherent only on a certain region of space, and then the phase difference is changed, and this definition is used anymore.

## Rationale for

Coherent waves are considered simplistic, does not occur in practice. Mathematical abstraction helps in many branches of science: space, fusion and astrophysical research, acoustics, music, electronics and of course optics.

For real-world applications are used simplified methods, including last trohvolnovaya system, applicability of the basics are outlined below. To analyze the interaction may specify, for example, a hydrodynamic or kinetic model.

Interference of waves

Solution of the equations for coherent waves allows to predict the stability of systems that operate using plasma. Theoretical calculations show that sometimes the amplitude of the result in a short time grows infinitely. That means creating a potentially explosive situation. Solving the equation for coherent waves, the selection of conditions possible to avoid unpleasant consequences.

## define

At first, we introduce a number of definitions:

• Monochromatic wave is called a single frequency. The width of its spectrum is zero. In the graph it is the only harmonic.
• signal spectrum - graphical representation of the amplitude of harmonics compose, wherein the abscissa (X-axis, horizontal) delayed frequency. The spectrum of a sine wave (a monochromatic wave) becomes the only spektrinka (vertical dashes).
• Fourier transformations (direct and inverse) are called decomposition of complex harmonic oscillations into monochromatic and inverse addition integer from disparate spektrinok.
• Wave analysis circuits is not conducted for complex signals. Instead, there is a decomposition into individual sinusoidal (monochromatic) harmonics for each relatively easy to make a formula describing the behavior. When calculating on a computer this is enough to analyze any situation.
• Any non-periodic infinite spectrum signal. Boundaries are cut to its reasonable limits before analysis.
• Beam deflection called diffraction (wave) from a straight path due to interaction with the propagation medium. For example, it is shown at overcoming the obstacle in front of the slit.
• Interference phenomenon known as superposition of the waves. Because of which there is quite a bizarre pattern of alternating bands of light and shadow.
• Refraction is called refraction of waves travel at the interface of two media with different parameters.

wave oscillation

## The concept of coherence

Soviet Encyclopedia says that the waves of the same frequency always coherent. This is true only for the individual fixed points in space. Phase determines the result of adding vibrations. For example, the anti-phase waves of the same amplitude gives a straight line. Such fluctuations cancel each other. The largest wave amplitude at in-phase (phase difference is zero). On this fact is based the principle of lasers, mirror and focusing system of light beams, especially receiving radiation makes it possible to transmit data at a whopping distance.

According to the theory of interaction of the oscillation coherent waves form an interference pattern. In the beginner there is a question: the light bulb does not appear striped. For the simple reason that the radiation is not the same frequency, but lies within the spectrum segment. And the land, and, decent width. Due to the heterogeneity wave frequency random, do not exhibit their theoretically and experimentally in the laboratory based and proven properties.

It has good coherence laser beam. It is used for communication over long distances with the line of sight, and other purposes. Coherent waves then propagate in space and on the receiver are mutually reinforcing. The beam of light scattered frequency effects can be deducted. Possible to select conditions that the radiation coming from the source, but the receiver does not register.

Ordinary light bulbs, too, does not work at full capacity. Achieve 100% efficiency at the present stage of the art is not possible. For example, discharge lamps suffer strong frequency dispersion. As for the light-emitting diodes, the founders of the concept of nanotechnology promise to create basic elements for the production of semiconductor lasers, but in vain. A significant part of the development is classified and not available to the ordinary man in the street.

Only a coherent waves exhibit wave quality. Acting in concert, as a splinter broom: one easy to break, combined - sweep garbage. Wave properties - diffraction, interference and refraction - are characteristic of all vibrations. Just register the effect is difficult because of the randomness of the process.

Coherent waves do not exhibit dispersion. They show the same frequency and equally deflected prism. All examples in the physics of wave propagation are usually for coherent oscillations. In practice, we have to take into account the present low spectral width. Which imposes especially on calculation process. How does the real result of the relative coherence of the waves - try to answer numerous books and publications scattered with intricate names! Single answer does not exist, it is highly dependent on the specific situation.

## wave packets

To facilitate the solution of practical tasks, you can enter, for example, the definition of a wave packet. Each one of them is broken down further into smaller parts. These subsections interact coherently between similar frequencies of another package. Such an analytical method is widespread in radio engineering and electronics. In particular, the spectrum concept was originally introduced to give into the hands of engineers reliable tool to evaluate the complex signal behavior in certain cases. Estimated a little bit of the impact of each harmonic oscillation on the system, then the net effect is their full addition.

Therefore, when assessing the real processes that are not even close to be coherent, it is permissible to break the object of analysis into the simplest components to assess the outcome of the process. The calculation is simplified by using computer technology. Computer experiments show the accuracy of the formulas for the current situation.

At the initial stage of the analysis assume that the packets with small spectral width perhaps conditionally replace harmonic oscillations and subsequently to use the direct and inverse Fourier transform for the evaluation result. Experiments have shown that the variation in phase between selected packets gradually increases (varies with a gradual increase scatter). But for the three waves of the difference is gradually smoothed out, consistent with the stated theory. There are some limitations:

1. The space should be infinite and homogeneous (k-space).
2. The amplitude of the wave is not attenuated with increasing distance, but changes over time.

It is proved that in such an environment, each wave manage to pick up a finite spectrum, which automatically makes it possible to machine analysis and the interaction spectrum of the resulting wave packet broadens. Fluctuations in fact not considered coherent, but are described by superposition, shown below. Where the wave vector ω (k) is determined by the dispersion equation; The EC recognized the amplitude of harmonics considered packet; k - wave number; r - spatial coordinate for the exponent represented by the equation is solved; t - time.

superposition equation

## coherence time

In reality, the diverse packages are coherent only on a single interval. A further discrepancy between the phases becomes too large to apply the above-described equation. To display the conditions of the possibility of calculations, the concept of coherence time is introduced.

It is believed that at the initial time phase, all the packages are identical. Selected elementary fraction of coherent waves. Then the required time is as the ratio of the number Pi to the spectral width of the package. If time exceeded coherent, in this portion can not be used for superposition formula addition oscillations - phase differ too much from each other. Wave is no longer coherent.

The package may be considered, if it is characterized by a random phase. In this case the wave interaction goes on different scheme. Then are Fourier components of said formula for further calculations. And taken to calculate the two other components are taken from the three packages. This is a case of coincidence with the theory mentioned above. Consequently, the equation shows the dependence of all the packages. More precisely - the addition result.

For best results, you need to package the spectrum width does not exceed the number of pi divided by the time of solving the problem of coherent superposition of waves. When detuning frequency harmonic amplitudes begin to oscillate, to obtain accurate results is difficult. Conversely, for the two coherent oscillations addition formula is simplified to the maximum. The amplitude is the square root of the sum of the original harmonics in the square erected and folded with its own double product multiplied by the cosine of the phase difference. For coherent quantities angle is zero, the result is, as already indicated above, a maximum is obtained.

Along with the time and the coherence length is used, the term "length of a train", which is analogous to the second term. To sunlight, this distance is one micron. The spectrum of our star is very wide, so that explains the meager distance where the radiation is considered to be coherent to itself. For comparison, the gas discharge length of a train reaches 10 cm (more than 100,000 times), while the laser radiation maintains properties and kilometer distances.

Since radio waves are much easier. Quartz resonators can achieve high wave coherence, which explains the spots on the reliable reception area bordering zones of silence. A similar change is manifested in the existing picture of the course of the day, cloud movement, and other factors. Changing propagation conditions coherent wave, and the superposition of the interference affects the full. In the radio frequencies at low coherence length may exceed the diameter of the solar system.

adding conditions strongly depend on the shape of the front. The simplest problem is solved for a plane wave. In fact, the front is usually spherical. The points are in phase at the surface of the sphere. The infinitely distant from the source area condition the plane might take for granted, and to conduct further calculations are willing to take the postulate. The lower the frequency, the easier it is to create the conditions to carry out the calculation. Conversely, the light sources with a spherical front (remember the Sun) is difficult to fit a coherent theory, written in textbooks.

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