Summary

• If a physical phenomenon that occurs at one place at a given time is reproduced at other places at later times, the time delay being proportional to the space separation from the first location then the group of phenomena constitutes a wave.
• Three-dimensional wave equations (Helmholtz equations) in terms of electric and magnetic fields are given as,

• For a perfect dielectric medium, the wave equations reduce to

• For a free space, the wave equations reduce to

• For time-harmonic fields, the wave equations reduce to

where, Î³ is defined as the propagation constant.

• The real part of the propagation constant (Î±) is defined as the attenuation constant (Neper/m). It is given as,

• The imaginary part (Î²) is defined as the phase constant (radians/m). It is given as,

• A wave is said to be a plane wave, if
1. The electric field  and magnetic field  lie in a plane perpendicular to the direction of wave propagation
2. The fields  and  are perpendicular to each other
• A plane wave is said to be uniform plane wave if
1. The electric field  and magnetic field  lie in a plane perpendicular to the direction of wave propagation
2. The fields  and  are perpendicular to each other
3.  and  are uniform in the plane perpendicular to the direction of propagation (i.e.,  and  vary only in the direction of propagation)
• Standing waves can be formed by confining the electromagnetic waves within two perfectly reflecting conductors. Unlike the travelling electromagnetic wave in which the electric and the magnetic fields are always in phase, in standing waves, the two fields are 90Â° out of phase.
• The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is given as,

• The velocity with which the overall shape of a wave amplitude, known as the modulation or envelope of the wave, propagates through a medium is known as the group velocity or energy velocity of the wave. This is given as,

• The intrinsic impedance of the wave is defined as the ratio of the electric field and magnetic field phasors (complex amplitudes). It is given as,

• Velocity of electromagnetic wave propagation is given as,

• The phenomenon that the alternating fields and hence currents are confined within a small region of a conducting medium inside the surface is known as the skin effect and the small distance from the surface of the conductor is known as skin depth. It is given as,

• The surface impedance of a conductor is defined as the ratio of the tangential component of the electric field to the tangential component of the magnetic field. The surface impedance for a thick conductor is,

• The real part of the intrinsic impedance is known as surface resistance or skin resistance,

Rs (Î©/m2). It is given as,

• The ratio of the imaginary part of the complex permittivity (Îµ'' ) to the real part of the complex permittivity (Îµ â€™ ) is the ratio of the magnitude of the conduction current density to the magnitude of the displacement current density. This ratio is defined as the loss tangent or loss angle of the medium.

The loss tangent gives a measure of how lossy a medium is. For a good (lossless or perfect) dielectric medium (Ïƒ << Ï‰Îµ), loss tangent is very small. For a good conducting medium (Ïƒ >> Ï‰Îµ), loss tangent is very large. For a lossy dielectric, loss tangent is of the order of unity.
• The polarisation of a uniform plane wave refers to the time-varying behaviour of the electric field at some point in space, i.e., the orientation of the electric field vector at a given instant of time in space.
• A plane electromagnetic wave in which the electric field vector vibrates harmonically along a fixed straight line perpendicular to the direction of wave propagation without changing its orientation is known as plane polarised electromagnetic wave. When two orthogonal plane polarised electromagnetic waves are superimposed then the resultant vector rotates under certain conditions, giving rise to different polarisations like linear, circular and elliptical.
• If a plane electromagnetic wave is incident normally from the medium 1 to the medium 2, the reflection and transmission coefficients are given as,

Reflectance,

Transmittance,
• The plane of incidence is the plane containing the incident wave and the normal to the interfacing surface.
• The angle of incidence is defined as the angle between the direction of propagation and the normal to the boundary.
• For a plane electromagnetic wave incident obliquely from the medium 1 to the medium 2. When the electric field vector  is perpendicular to the plane of incidence, i.e., the electric vector is parallel to the boundary surface,it is called perpendicular or horizontal polarisation. On the other hand, when the electric field vector  is parallel to the plane of incidence, i.e., the magnetic field is parallel to the boundary surface, it is called parallel or vertical polarisation.
• The reflection and transmission coefficients for perpendicular polarisation are given as,

Reflection coefficient,

Transmittance,
• The reflection and transmission coefficients for perpendicular polarisation are given as,

Reflection coefficient,

Transmittance,
• If a plane wave is incident normally upon the surface of the conducting medium, the wave is entirely reflected. For a field that varies with time, neither  nor  will exist within the conductor. Thus, in this case, where the medium 1 is a perfect dielectric and the medium 2 is a perfect conductor,

Transmission coefficient, = 0, and

Reflection coefficient, = -1

Hence, the amplitudes of  and  of reflected wave are same as those of the incident wave, but they differ in the direction of flow.
• Poynting vector can be thought of as representing the energy flux (in W/m2) of an electromagnetic field. It is given as,

• Poynting theorem states that the vector product  at any point is a measure of the rate of energy flow per unit area at that point. The direction of power flow is in the direction of the unit vector along the product  and is perpendicular to both  and