The plane wave equation:
An diagram of a plain wave is shown below:
The solution to Maxwell’s equations for a plane wave are:
E = E0 cos (φ)x and
H = H0 cos (φ)y
where x and y are the unit vectors in their respective directions – not quite the correct notation but this is HTML.
We are not going to do the maths for this in any detail but if we substitute the above functions of E and H into Maxwell’s equations we can show that they work. E and H are orthogonal, as shown by the unit directional vectors, and also have a sinusoidal variation in amplitude. By convention the polarisation is defined by the direction of the E field. The wavelength is the distance traveled in one cycle of E and H.
We can derive some relationships from the plane wave solution that will be useful later. The field strength can be expressed as:
E = E0 cos (ωt + kz)x
H = H0 cos (ωt + kz)y
Where ω = 2πf (the angular frequency)
k = the wave number, the rate of change of phase with distance:
The wave number k is frequently found in EM theory, it describes the variation with distance along the propagation axis whereas ωt is the variation with time. The ratio of E to H is the impedance of free space (by Ohms law):
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