Plane Waves and Properties Study Notes for Electronics and Communication Engineering

Assume that at any point P, the magnitudes E and B of the fields depend upon x and t only, and not upon the y or z coordinate. A collection of such waves from individual sources is called a plane electromagnetic wave.

Surface connecting points of equal phase on all waves, called as a wave front, would be a geometric plane. In comparison, a point source of radiation sends waves out in all directions.

A surface connecting points of equal phase is a sphere for this situation, so we call this a spherical wave.

In empty space, where Q = 0 and I = 0.

The following differential equations relating E and B. For simplicity, we drop the subscripts on the components E_y and B_z:

Note: that the derivatives here are partial derivatives.

For example, when we evaluate ∂E / ∂x, we assume that t is constant. Likewise, when we evaluate ∂B / ∂t, x is held constant.

The wave speed v replaced by c, where

1. Plane waves in vacuum.

Assuming that a time-harmonic propagating wave is polarized in the y-direction.

In a vacuum, the phase velocity of the wave equals to the velocity of light c.

Therefore, the 1D wave equation is:

or

Wave number (k):

Therefore, the 1D wave equation (the Helmholtz equation) is:

A solution of the second-order ODE is in a form:

Where a and b are the integration constant. Incorporating, we obtain:

The real part of the solution will be:

Since the waves are propagating in vacuum, the phase velocities for these traveling waves are:

In general, the phase velocity is a vector since it has both a magnitude and a direction. It can have a value greater than the light speed! However, there is no energy (or particles) transferred at that speed.

The wave number may also be a vector and, therefore, indicate the direction, in which the wave is traveling. In this case, it is frequently called a wave vector and the quantity kz can be replaced by

2. Magnetic field intensity and characteristic impedance.

The magnetic field intensity can be found via the Faraday's law:

Since:

 

Therefore, 

Characteristic impedance of the medium:

For a free space:

 

Therefore,

 

The Poynting vector: 

Plane wave propagation in a dielectric medium:

1. Plane wave in a lossless homogeneous dielectric.

The wave number for the wave propagating in a vacuum is a function of permittivity and permeability of free space:

Naturally, for a dielectric medium that may have different constants, the wave number will be

If a plane wave generated by a signal generator propagates through two different dielectrics

Say, with the same magnetic constants but different permittivities, the wave number will be for these two media:

Both signals travel the same distance Δz but will have different phase velocities:

This difference in velocities delays the arrival of one signal with respect to the other and causes a phase difference that can be detected:

If the total phase change in the signal passing through one of the paths is known

The relative phase difference is

Therefore, if the properties of one of the regions are known and the phase difference is measured, we can identify the other material.

The ratio of the phase velocity in a vacuum to the phase velocity in a dielectric is called the index of refraction for the material:

2. Plane wave in a lossy homogeneous dielectric.

A dielectric material can be lossy, i.e. exhibit a nonzero conductivity σ. In this situation, a conduction current must be added to the displacement current when considering the Ampere's law.

Assuming, as before, no free charges (ρ_v = 0) and following the same procedure:

Assuming, as previously, that the electric field is linearly polarized in the y direction and the wave propagates in the z direction, we arrive to:

Where the propagation constant:

The propagation constant is complex:

In a vacuum, α = 0 and β = k. In a general case, the real and imaginary parts are nonlinear functions of the frequency.

As a result, the phase velocity may depend on the waves frequency. This phenomena is called dispersion and the medium in which wave is propagating, called a dispersive medium (every lossy medium).

Another quantity we recall here is the group velocity:

The component of the electric field propagating in the +z direction is:

Wave propagates with a phase constant β but the amplitude decreases with an attenuation constant α.

Units of β are radians/m.

Units of α are nepers/m [Np/m]. If α = 1 Np/m, the amplitude of the wave will decrease e times at a distance 1m. 1 Np/m ≅ 8.686 dB/m.

The characteristic impedance is:

Two approximations are frequently used:

(A) A dielectric with small losses (σ << ωε) with a high-frequency approximation:

The approximate values for attenuation and propagation constants are:

In this situation, the phase and the group velocities are the same. Also, some attenuation is introduced.

"A pizza in a microwave oven" : water in the pizza acts as a conductor turning pizza into a complex impedance. The wave passing through it decays, therefore, the energy is absorbed and must be converted into heat.

(B) A dielectric with large losses (σ >> ωε) with a low-frequency approximation :

The conduction current is much greater than the displacement current, therefore:

In this situation:

Therefore:

We introduce a skin depth of the material:

The skin depth decreases with the increasing frequency – skin effect.

Reflection:

Reflection is the abrupt change in the direction of propagation of a wave that strikes the boundary between different mediums. At least part of the oncoming wave disturbance remains in the same medium. Regular reflection, which follows a simple law, occurs at plane boundaries. The angle between the direction of motion of the oncoming wave and a perpendicular to the reflecting surface (angle of incidence) is equal to the angle between the direction of motion of the reflected wave and a perpendicular (angle of reflection). Reflection at rough, or irregular, boundaries is diffuse. The reflectivity of a surface material is the fraction of energy of the oncoming wave that is reflected by it.

Refraction:

Refraction is the change in direction of a wave passing from one medium to another caused by its change in speed. For example, waves in deep water travel faster than in shallow; if an ocean wave approaches a beach obliquely, the part of the wave farther from the beach will move faster than that closer in, and so the wave will swing around until it moves in a direction perpendicular to the shoreline. The speed of sound waves is greater in warm air than in cold; at night, air is cooled at the surface of a lake, and any sound that travels upward is refracted down by the higher layers of air that still remain warm. Thus, sounds, such as voices and music, can be heard much farther across water at night than in the daytime.

The electromagnetic waves constituting light are refracted when crossing the boundary from one transparent medium to another because of their change in speed. A straight stick appears bent when partly immersed in water and viewed at an angle to the surface other than 90. A ray of light of one wavelength, or colour (different wavelengths appear as different colours to the human eye), in passing from air to glass is refracted, or bent, by an amount that depends on its speed in air and glass, the two speeds depending on the wavelength. A ray of sunlight is composed of many wavelengths that in combination appear to be colourless; upon entering a glass prism, the different refractions of the various wavelengths spread them apart as in a rainbow.

Polarization:

Polarization is the property of electromagnetic waves, such as light, that describes the direction of the transverse electric field. More generally, the polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization, because for these waves the direction of oscillation is along the direction of travel.

Types of polarization: Linear, Circular, and Elliptical.

Phase and group velocities

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any single frequency component of the wave will propagate. We can pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity.

Here, ω is a radial frequency and k is the wave number.

The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (modulation or envelope of the wave) propagate through space.

 

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