Wave propagation

The propagation the waves is a field of the Physique being interested in displacements of the electromagnetic waves in the mediums. One generally distinguishes two categories of propagation:

• propagation in free space (vacuum, air, massive medium like glass, etc)
• guided propagation (Fiberoptic, guide of wave, etc)

Equation of wave

See also: Equation of wave

The general equation which describe the propagation of a wave $\backslash \; vec\; E$ in free space, in a medium Homogène, linear and Isotrope are:

$\backslash \; nabla^2\; \backslash \; vec\; E=\; \backslash \; Delta\; \backslash \; vec\; partial\; E=\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; ^2\; \backslash \; vec\; E\}\; \{\backslash \; partial\; t^2\}$ (establishment of the equation of propagation starting from the Maxwell's equations)

$\backslash \; vec\; E$ describes at the same time the amplitude of the wave, and its polarization (by its vectorial character). It is comparable to the propagation velocity of the wave, as we will see it low.

If one is interested in what occurs for each component of $\backslash \; vec\; E$ (by projecting the relation in each direction of space), we obtain an equation carrying on a scalar, called equation of Alembert :

$\backslash \; Delta\; partial\; U=\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; ^2\; U\}\; \{\backslash \; partial\; t^2\}$

Let us interest in the propagation according to the only direction $z$:

$\backslash \; frac\; \{\backslash \; partial\; ^2\; U\}\; \{\backslash \; partial\; z^2\}\; =\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\; U\}\; \{\backslash \; partial\; t^2\}$

For a Onde planes, the general solution of this equation is the sum of two functions:

$U\; (Z,\; T)\; =f\; (z-ct)\; +g\; (z+ct)$

Indeed, one can write:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; ^2\}\; \{\backslash \; partial\; z^2\}\; -\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; partial\; ^2\}\; \{\backslash \; partial\; t^2\}\; \backslash \; right)\; U\; (Z,\; T)\; =\; 0$

that is to say:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; Z\}\; -\; \backslash \; frac\; \{1\}\; \{C\}\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; Z\}\; +\; \backslash \; frac\; \{1\}\; \{C\}\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; U\; (Z,\; T)\; =\; 0$

And if one poses a=z-ct and b=z+ct , one obtains:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; has\}\; \backslash \; right)\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; B\}\; \backslash \; right)\; U\; (has,\; b)\; =\; 0$

Who is solved in: $U\; (has,\; b)\; =\; F\; (a)\; +\; G\; (b)$ is $U\; (Z,\; T)\; =\; F\; (z-ct)\; +\; G\; (z+ct)$

The first term is a wave being propagated in the increasing direction of Z (called travelling wave), and the second term in the decreasing direction of Z (called regressive wave).

Propagation velocity

It is interesting to see that actually, the wave $U\; (Z,\; T)$ does not depend simply on $z$ and $t$, but of the $z-ct$ quantities and $z+ct$. To include/understand what that means, let us consider the case of a progressive plane wave towards Z crescents:

$U\; (Z,\; T)\; =f\; (z-ct)$

Let us look at the structure of the wave at the point $z+\; \backslash \; Delta\; z$:

$U\; (z+\; \backslash \; Delta\; Z,\; T)\; =f\; (z+\; \backslash \; Delta\; z-ct)\; =f\; \backslash \; left\; (z-c\; \backslash \; left\; (T\; \backslash \; frac\; \{\backslash \; Delta\; Z\}\; \{C\}\; \backslash \; right)\; \backslash \; right)\; =f\; (z-c\; (T\; \backslash \; Delta\; T))$

The expression above shows us that the structure of the wave at the point $z+\; \backslash \; Delta\; z$ is the even that at the point $z$ at the moment $t-\; \backslash \; Delta\; t$, with $\backslash \; Delta\; t=\; \backslash \; Delta\; z/c$. This reasoning enables us to include/understand why a dependence in $z\; \backslash \; pm\; ct$ of the wave means that this one moves without deformation, i.e which it is about a travelling wave.

We can then define the propagation velocity of the wave by:

$\backslash \; frac\; \{\backslash \; Delta\; Z\}\; \{\backslash \; Delta\; T\}\; =c$

Harmonic wave - period and frequency

A harmonic wave is a wave Monochromatique whose expression is given in our case by:

$U\; (Z,\; T)\; =\; U\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; \backslash \; left\; (T\; \backslash \; frac\; \{Z\}\; \{C\}\; \backslash \; right)\; \backslash \; right)$

The essential property of this wave is its double periodicity, space and temporal:

$U\; (Z,\; t+2\; \backslash \; tfrac\; \backslash \; pi\; \backslash \; Omega)\; =\; U\; (z+2\; \backslash \; pi\; \backslash \; tfrac\; C\; \backslash \; Omega,\; T)\; =U\; (Z,\; T)$

The following quantities then are defined:

• the Frequency of the wave $\backslash \; nu=\; \backslash \; omega/2\; \backslash \; pi$   ($\backslash \; omega$ is called pulsation )

• the Wavelength $\backslash \; lambda=c/\backslash \; nu$
• the Nombre of wave $k=2\; \backslash \; pi\; \backslash \; lambda$

With three dimensions, the Nombre of wave is replaced by the Vecteur of wave, whose direction is that of the wave propagation.

See also: Wave, electromagnetic Radiation

Travelling waves and standing waves

It is of use in the scientific community to distinguish the travelling waves from the standing waves. The travelling waves, described previously, advance in space.

The standing waves, on the contrary, oscillate without moving. Thus, they do not depend any more of the only parameter $z-ct$, but of the parameters of space $z$ and time $t$ independently. A simple expression of a harmonic standing wave to a dimension is the following one:

$U\; (Z,\; T)\; =U\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{T\}\; \{T\}\; \backslash \; right)\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{Z\}\; \{\backslash \; lambda\}\; \backslash \; right)$

A fixed time, a standing wave resembles a travelling wave. On the other hand, its temporal evolution is completely different. A standing wave has minima (nodes) and maximum (bellies) of amplitude the fixed ones in space. Thus, if one places oneself at the nodes of this wave, the amplitude is null whatever the time. With a travelling wave, we would have seen the amplitude evolving/moving, in a sinusoidal way with time in the case of a harmonic wave.

A way simple to build a standing wave is to superimpose two travelling waves being propagated in opposite direction. It is besides what occurs when a wave is reflected on a perfect mirror.

The standing waves are very current physical objects and meet in particular in the cavities Laser or the lines ultra high frequency.

See also: Standing wave, Wave radio, Light, Propagation waves radio

Phenomena affecting the wave propagation

• Reflection

• Refraction
• Diffusion
• Interference S
• Diffraction

Several equations of wave

Traditional equation of wave

The traditional equation of wave corresponds to a nondisturbed propagation. For example, vibrating cord without friction, electromagnetic wave in the vacuum. She is written $\backslash \; Delta\; S\; =\; 0$

The relation of associated dispersion is $k^2\; =\; \backslash \; frac\; \{\backslash \; omega^2\}\; \{c^2\}$

Equation of Klein-Gordon

See also: Equation of Klein-Gordon

It is an equation of the type $\backslash \; Delta\; S\; =\; \backslash \; frac\; \{S\}\; \{a^2\}$.

One meets it in particular for the propagation of an electromagnetic wave in a plasma.

That is to say $n$ particulate density of plasma and $m$ mass of the electron. One introduces the pulsation plasma

$\backslash \; omega\_p\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{ne^2\}\; \{m\; \backslash \; varepsilon\_0\}\}$,

size characteristic of plasma.

The equation of propagation of the field $\backslash \; vec\; E$ in this plasma is

$\backslash \; Delta\; \backslash \; vec\; E\; =\; \backslash \; frac\; \{\backslash \; omega\_p^2\}\; \{c^2\}\; \backslash \; vec\; E$.

The relation of associated dispersion is

$k^2\; =\; \backslash \; frac\; \{\backslash \; omega^2\; -\; \backslash \; omega\_p^2\}\; \{c^2\}$

Remark : one can also meet this equation of dispersion within the framework of the guided propagation. For example, a wave guided between two infinite plans and perfectly distant drivers of $a$ check the traditional equation of wave between the two drivers, but the relation of dispersion is written $k^2\; =\; \backslash \; frac\; \{\backslash \; omega^2\}\; \{c^2\}\; -\; \backslash \; frac\; \{\backslash \; pi^2\}\; \{a^2\}$

One also meets this equation for the propagation of a mechanical wave: for example for a series of pendulums regularly spaced and connected to each other by springs.

Equation of the telegraphists

Equation of the type

$\backslash \; Delta\; S\; =\; \backslash \; alpha\; +\; \backslash \; beta\; \backslash \; frac\; \{\backslash \; partial\; S\}\; \{\backslash \; partial\; T\}$

It in the case of is met a wave in an electric line. For a linear electrical electric line of resistance $r$, linear inductance $l$, linear conductance of escape $g$ and linear capacity $c$, one a:

$\backslash \; frac\; \{\backslash \; partial^2\; U\}\; \{\backslash \; partial\; x^2\}\; -\; LLC\; \backslash \; frac\; \{\backslash \; partial^2\; U\}\; \{\backslash \; partial\; t^2\}\; =\; (rc+lg)\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; T\}\; +\; rg\; \backslash \; cdot\; u$

That represented a propagation with attenuation.

Relation of associated dispersion: $k^2\; =\; LLC\; \backslash \; omega^2\; +\; I\; \backslash \; Omega\; (rc+lg)\; -\; rg$

Equation of diffusion

Equation of the type

$\backslash \; frac\; \{\backslash \; partial\; S\}\; \{\backslash \; partial\; T\}\; =\; D\; \backslash \; vec\; \backslash \; nabla^2\; s$

The coefficient $D$ is called coefficient of diffusivity .

One typically meets it in two cases:

1. the equation of heat $\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; T\}\; =\; D\; \backslash \; vec\; \backslash \; T$ Delta, where the thermal coefficient of diffusivity $D$ is defined by $D\; =\; \backslash \; frac\; \{\backslash \; lambda\}\; \{\backslash \; rho\; C\}$

2. Propagation of an electromagnetic field in a metal of conductivity $\backslash \; gamma$. Equation of wave:
   $\backslash \; vec\; \backslash \; nabla^2\; \backslash \; vec\; B\; =\; \backslash \; mu\_0\; \backslash \; partial\; gamma\; \backslash \; frac\; \{\backslash \; \backslash \; vec\; B\}\; \{\backslash \; partial\; T\}$.
   Relation of dispersion: $k^2\; =\; -\; I\; \backslash \; Omega\; \backslash \; mu\_0\; \backslash \; gamma$. It is about a propagation with attenuation.

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