equations by taking their dot product with the wavevector . Equation (4.5.31a) further shows that is normal to both vectors, and equation (4.5.31b) shows that is normal to both vectors. In brief, the vectors are orthogonal to each other, and there is no field component along the wavevector , i.e. the wave is a transverse electromagnetic (TEM) type. Also, in an isotropic medium,is parallel to vectorandis parallel to vector. This statement does not hold for the anisotropic medium. Maxwell equations (4.5.31e)–(4.5.31h) apply to an anisotropic medium. In an anisotropic medium, equations (4.5.31e) and (4.5.31f) show that is normal to vectors and is normal to vectors . However, is not parallel to . Also, is not parallel to . It is discussed in subsection (4.2.3).
Likewise, the wave equation for could be written that is useful for the EM‐wave propagation in an anisotropic medium. Equation (4.5.32b) is an eigenvalue equation, and the nontrivial solution for E ≠ 0 provides the eigenvalue of k belonging to the propagating waves in the unbounded isotropic medium. The medium supports two numbers of linearly y‐polarized waves known as normal modes propagating in ±x‐directions with same phase velocity (vp) given below:
(4.5.33)
In the above equation, is the wavenumber in free space. Using the intrinsic impedance with equation (4.5.31a), the magnetic field vector and Poynting vector are obtained below:
In the case of the propagation of waves in an isotropic medium, the wavevectorand Poynting vectorboth are in the same direction. It provides the phase and group velocities in the same direction.
4.5.5 Uniform Plane Waves in Lossy Conducting Medium
The loss‐tangent (tan δ), given in equation (4.5.10), is much greater than unity, i.e. tan δ ≫ 1 for a highly conducting medium. It means a contribution of the conduction current is much more than that of the displacement current in a conducting medium, i.e.. However, the approximation for a low‐loss medium is taken differently. The propagation constants of the EM‐wave in a highly conducting and also in a low‐loss medium are obtained from equation (4.5.4) as follows:
In a lossy medium, the plane wave propagates in the x‐direction with the uniform field components in the (y‐z)‐plane as shown in Fig. (4.9a). The field components are given by equation (4.5.24), incorporating the conductivity σ of a medium. They are modified as,
(4.5.36)
Using the field solutions from equations (4.5.20), the above equations are reduced to the following forms:
In the above equations, the complex propagation constant γ is given by equation (4.5.4).
The conducting medium is highly dispersive, whereas the low‐loss medium is nondispersive. Using equations (4.5.35a,b) with equation (4.5.12a), the wave equation and the phase velocity in a conducting medium are given below:
equations by taking their dot product with the wavevector 