1. The University of Delaware
ELEG 648 Plane waves
Mark Mirotznik, Ph.D.
Associate Professor
The University of Delaware

2. SUMMARY
Time Domain
Frequency Domain

3. Wave Equation Time Dependent Homogenous Wave Equation (E-Field)
Vector Identity

4. Wave Equation
Source-Free Time Dependent Homogenous Wave Equation (E-Field)
Source Free

5. Wave Equation
Source-Free Time Dependent Homogenous Wave Equation (E-Field)
Source Free
Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)
Lossless

6. Wave Equation: Time Harmonic
Time Domain
Frequency Domain
Source Free
Source Free
Lossless
Lossless
“Helmholtz Equation”

7. General Solution Case: Time Harmonic Rectangular Coordinates
Wave Number

8. Separation of Variable Solutions
Assume Solution of the form:

9. Separation of Variable Solutions
Assume Solution of the form:

10. Separation of Variable Solutions
Assume Solution of the form:
function of x
function of y
function of z
constant

11. Separation of Variable Solutions
function of x
function of y
function of z
constant

12. Separation of Variable Solutions

13. Separation of Variable Solutions
b purely real
b purely imaginary
b complex
Traveling and
standing waves
Exponentially modulated
traveling wave
Evanescent waves or or

14. Wave Propagation and Polarization
TEM: Transverse Electromagnetic Waves
“A mode is a particular field configuration. For a given electromagnetic boundary value problem, many
field configurations that satisfy the wave equation, Maxwell’s equations, and boundary conditions usually
exits. A TEM mode is one whole field intensities, both E and H, at every point in space are contained in
a local plane, referred to as equiphase plane, that is independent of time” E H
Plane Waves
“If the space orientation of the planes for a TEM mode are the same (equiphase planes are parallel) then the fields
form a plane wave. E H E H E H E H
Uniform Plane Waves
“If in addition to having planar equiphases the field has equiamplitude (the amplitude of the field is the same
over each plane) planar surfaces then it is called a uniform plane wave.”

15. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
Let’s begin by assuming the solution is only a function of z and has only the x component of electric field.
Let’s also look at the term that represents a traveling wave moving in the +z direction

16. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
For uniform plane wave assume the solution is only a
function of z and has only the x component of electric
field.
Lets find H

17. Several observations:
Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
Several observations:
E and H are orthogonal to each other and to the direction of energy propagation
E and H are in phase with each other
H is smaller in amplitude than E by the term
(for a uniform plane wave)
Wave Impedance

18. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
How fast does the wave move?

19. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
How much power does the wave carry?

20. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
How fast does the power flow?

21. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation
Relationship between phase and group velocity

22. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

23. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

24. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

25. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

26. Uniform Plane Waves in Unbounded Lossless Medium Principal Axis Propagation

27. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

28. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

29. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

30. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

31. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

32. Uniform Plane Waves in Unbounded Lossy Medium Principal Axis Propagation

33. Example: Skin Depth in Sea Water
Skin Depth, cm
Frequency, GHz

34. Example: Skin Depth in Copper
Skin Depth, microns
Frequency, GHz

35. Example: Skin Depth in Teflon
Skin Depth, meters
Frequency, GHz

36. Polarization

37. Polarization

38. Polarization

39. Polarization

40. Polarization

41. Polarization

42. Uniform Plane Waves: Propagation in Any Arbitrary Directionz H f E q y x

43. Uniform Plane Waves: Propagation in Any Arbitrary DirectionH y x z q f
Since E and b are at right angles
from each other.
where
and

44. Uniform Plane Waves: Propagation in Any Arbitrary Direction
Summary and Observations:
Frequency Domain
Time Domain
Observation 1. E, H and b vectors are pointing in orthogonal directions.
Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance