Uniform Plane Waves - Derivation

Section 9.2 showed how Maxwell’s Equations could be reduced to a pair of phasor-domain “wave equations,” namely:

Let us identify the Cartesian components of each of these fields as follows:

Now Equation 9.4.3 may be interpreted in detail for

as follows: and for

as follows:

and for

as follows:

The wave equation for

(Equation 9.4.1 written explicitly in Cartesian coordinates is

Decomposing this equation into separate equations for each of the three coordinates, we obtain the following:

Applying the constraints of Equations 9.4.6 and 9.4.7, we note that many of the terms in Equations 9.4.11-9.4.13 are zero. We are left with:

Now we will show that Equation 9.4.3 also implies that

must be zero. To show this, we use Ampere’s Law for a source-free region (Section 9.2):

Again applying the constraints of Equation 9.4.3, the left side of Equation 9.4.18 must be zero; therefore,

. The exact same procedure applied to

(using the Maxwell-Faraday Equation; also given in Section 9.2) reveals that

is also zero.² Here is what we have found:

This conclusion is a direct consequence of the fact that Maxwell’s Equations require the electric field to be proportional to the curl of the magnetic field and vice-versa.

The general solution to Equation 9.4.14 is:

where

and

are complex-valued constants. The values of these constants are determined by boundary conditions – possibly sources – outside the region of interest. Since in this section we are limiting our scope to source-free and homogeneous regions, we may for the moment consider the values of

and

to be arbitrary, since any values will satisfy the associated wave equation.³

Similarly we have for

:

where

and

are again arbitrary constants. Summarizing, we have found

where

and

are given by Equations 9.4.19 and 9.4.20, respectively.

Note that Equations 9.4.19 and 9.4.20 are essentially the same equations encountered in the study of waves in lossless transmission lines; for a reminder, see Section 3.6. Specifically, factors containing

describe propagation in the

direction, whereas factors containing

describe propagation in the

direction. We conclude:

The same procedure yields the uniform plane wave solution to the wave equation for

, which is

where

and where

,

,

and

are arbitrary constants. Note that the solution is essentially the same as that for

, with the sole difference being that the arbitrary constants may apparently have different values.

To this point, we have seen no particular relationship between the electric and magnetic fields, and it may appear that the electric and magnetic fields are independent of each other. However, Maxwell’s Equations – specifically, the two curl equations – make it clear that there must be a relationship between these fields. Subsequently the arbitrary constants in the solutions for

and

must also be related. In fact, there are two considerations here:

• The magnitude and phase of must be related to the magnitude and phase of . Since both fields are solutions to the same differential (wave) equation, they may differ by no more than a multiplicative constant. Since the units of and are V/m and A/m respectively, this constant must be expressible in units of V/m divided by A/m; i.e., in units of , an impedance.
• The direction of must be related to direction of .

Let us now address these considerations. Consider an electric field that points in one of the cardinal directions – let’s say

– and make the definition

for notational convenience. Then the electric field intensity may be written as follows:

We may now determine

from

using the Maxwell-Faraday Equation (Section 9.2):

Solving this equation for

, we find:

Now let us apply the curl operator. The complete expression for the curl operator in Cartesian coordinates is given in Section B.2. Here let us consider one component at a time, starting with the

component:

Since

, the above expression is zero and subsequently

. Next, the

component:

Here

, so we have simply

Note that

points in the

direction. So, as expected, both

and

are perpendicular to the direction of propagation. However, we have now found a more specific relationship:

and

are perpendicular to each other. Just as

, we see that

points in the direction of propagation. This is illustrated in Figure 9.4.1.

Now let us resolve the question of the factor relating

and

. The factor is now seen to be β/ωμβ/ωμ in Equation 9.4.31, which can be simplified as follows:

The factor

appearing above has units of

is known variously as the wave impedance or the intrinsic impedance of the medium. Assigning this quantity the symbol “

,” we have:

The ratio of the electric field intensity to the magnetic field intensity is the wave impedance

(Equation 9.4.33; units of

). In lossless media,

is determined by the ratio of permeability of the medium to the permittivity of the medium.

The wave impedance in free space, assigned the symbol

, is

Wrapping up our solution, we find that if

is as given by Equation 9.4.35, then

• 1

  By the way, this is a highly-recommended exercise for the student.

• 2

  Showing this is a highly-recommended exercise for the reader

• 3

  The reader is encouraged to confirm that these are solutions by substitution into the associated wave equation.