# 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
*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- The electric field and magnetic field lie in a plane perpendicular to the direction of wave propagation
- The fields and are perpendicular to each other

- A plane wave is said to be
*uniform plane wave*if- The electric field and magnetic field lie in a plane perpendicular to the direction of wave propagation
- The fields and are perpendicular to each other
- 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,*R*(Î©/m_{s}^{2}). 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.*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,
- 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,
- The reflection and transmission coefficients for perpendicular polarisation are given as,
- 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,
*Poynting vector*can be thought of as representing the energy flux (in W/m^{2}) 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