• Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium

We`ll now derive the Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium from the Maxwell`s equations.

The general Maxwell`s equations are

For linearity and isotropy, we have

We will use the following identity to obtain the wave equations

To derive the wave equation for E

replacing Jfree in the equation (7)

So, we get the wave equation for E

To obtain the wave equation for B in Harmonic Isotropic Linear Stationary (HILS) medium, we follow the same principle.

Taking the divergence of B =0 (Maxwell second equation above) and writing B in terms of H as appropriate for linearity and isotropy, we obtain

From equation (4) of the Maxwell`s equations, we obtain equation (12) below, and upon simplifying and applying curl on both sides, we obtain the following form of the equation

Then, we replace the right-hand side of equation (11) and the curl of E (from the 3rd Maxwell`s equation) in equation (12)

Hence, we have obtain the wave equation for H in a HILS medium.

So, we have finally obtained the wave equations for both E and H. That`s what we wanted.



Now, let`s try an additional simplification for the wave equation for E.

let E be in the x-direction,

We can therefore replace the right-hand side of equation (9) with the result of the above equation

Equation (15) is the solution for Ex that would be obtained from (14).

As E and H are transverse, an example of a solution would be

where Z is the characteristic impedance.