3.18 Sspd_chapter 1_part8_appendixxxxiv_travelling waves

Appendix XXXIV

Solving the wave equation for plane wave

Maxwell’s four differential equations are:

In free space propogation: 𝝆( charge density) =0 and

Therefore the four differential equations become:

From these four equations we obtain:

But in free space both divergences are zero therefore the above 2 equations become:

In free space €=€ ₀ and μ= μ ₀

That is free space permittivity = absolute permittivity .

And free space permeability = absolute permeability .

Also c ( velocity of light in free space ) =

Therefore the above two equations become:

We apply separation of variables to solve these two partial differential equations:

Assume E(r,t) = E1(r)E2(t)

H(r,t) = H1(r)H2(t)

In general by assuming the time dependent portion to be of harmonic nature:

That is E2(t) = E20Exp(jωt) and H2(t) = H20Exp(jωt)

The partial derivative

Hence the two equations become:

Now the partial differential equations have reduced to ordinary linear differential equations known as wave equations of 2 ^nd order and they can be solved applying operator theory.

We also assume that it is a plane wave travelling along z axis hence

Under such circumstances as we will see in Electro-Magnetic Field Theory, only x-component of E1 and y-component of H1 will remain. Hence

Applying Theory of Operator we get two roots in each case namely:

Two roots are =

Hence

So the complete solution for Electric Field is:

1 ^st Term is backward travelling wave and 2 ^nd term is forward travelling wave.