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

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.

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