Let (x,y,z), t, E=(E_x,E_y,E_z) and H=(H_x,H_y,H_z) be the coordinates, time, electric field and magnetic field. Maxwell equation for the electro-magnetic field is given as

where µ, e and J are the magnetic permeability, dielectric constant and current density.

When the current density is equal to zero, we obtain the vector wave equation by operating rot and ?/?t and on both side of equations (18) and (19) respectively and by eliminating H:




where a formula:

is used. We also assumed the charge is distributed homogeneously and used

As one of the solutions of equation (20), we have a plane wave:
where c, w and k are the phase velocity, circular frequency and wave number. Now, H is obtained from equations (18) and (19) as

We now consider a situation when the plane wave given by equations (24) and (26) passes through an ionospheric layer with the electron density N. The equation of motion due to the electric field is given as

where m, e and v are the mass, charge and velocity of an electron. So, we have

From this, the current J_c due to the motion of the electron:


is generated. Hence, the displacement current J_i is given as
where J_d is the displacement current in vacuum. So, an equivalent dielectric constant e_p can be defined as

Hence, the refractive index n_p can be defined as:

We rewrite equation (33). Let f be the frequency, that is, f = w/(2π). Now, we have

If we define the plasma frequency f_p as (? Appendix A)

n_p can be approximated as

The phase velocity c_p of the electro magnetic wave in the ionospheric layer is obtained as

where c is the phase velocity in vacuum.The wave number k_p is now given as

Hence, the group velocity c_gp is derived from equation (11) as

From equations (37) and (39), we know that the phase velocity c_p in the ionosperic layer is faster by than the phase velocity c in vacuum. On the other hand, the group velocity c_gp is slower by than the phase velocity c_p in vacuum.

Appendix A. Physical constants
Mass of an electron: m = 9.1093897x10^-31kg Charge of an electron: e = 1.60217733x10^-19C Dielectric constant in vacuum: e = 8.854187817 x 10^-12 F/m