WavePropagation

Version 2.0
2/12/92
by Mark Steeds
and John Schneider

Introduction

Warning:  The text in this NoteBook is not yet presentation quality.  However, the graphics may help you better understand wave propagation.  The graphics are generally presented without the commands that generated them.  Those commands will show up in later versions.  Enjoy.

   In this Notebook, the propagation of a uniform plane wave in free space, a perfect dielectric, a lossy dielectric, and a good conductor will be examined.

What is a Plane Wave?

A plane wave is a wave that, if you were to examine any plane perpendicular to the unit vector in the direction of propagation, the wave has the same phase throughout the plane.  Figure 1 shows a plane wave that is propagating in the x direction.  Notice that at x = 0, the wave has the same phase for any value of y.

Figure 1

Uniform plane wave

The Wave Function

A uniform plane wave varies with time and position.  Variation with time is dependent only on the frequency at which the wave is propagating.  Variation with distance is dependent on the medium in which the wave is propagating.
		
   The uniform plane waves we are concerned with are those that constitute the time-varying electric and magnetic field intensities.  For reasons that will be shown later, these waves are also known as transverse electromagnetic waves, or TEM waves.  One possible situation for the propagation of the electric and magnetic field intensities is the one shown in Figure 2.  In this case, we are assuming  that the waves are propagating in the z direction, that the electric field intensity varies only with time and in the x direction, and that the magnetic field intensity varies only with time and in the y direction.

Figure 2

Orientation of propagating electric and magnetic field intensities.

The following functions describe the propagation of the electric and magnetic waves discussed above and whose orientation is shown in Figure 2.

			Ex = E0 Exp[±αz] Cos[ωt ± βz]			(1)
and
			Hy = ( E0 / Z ) Exp[±αz] Cos[ωt ± βz]		(2)
			
Where α, β, and Z are functions of the permittivity, permeability, and conductivity of the medium in which the wave is propagating and the frequency of the propagating wave.

Table 1 gives some information on the constants used in equations 1 and 2.

	Name of Constant	Definition			Units           
	Eo			magnitude of electric		V / m
				field intensity

	α			attenuation constant		Np / m

	β			phase constant		rad / m
			 
	ω			angular frequency		rad / s

	Z			intrinsic impedance                  Ω

Table 1

Constants found in the wave functions

Figure 3 shows the electric and magnetic field intensities at a fixed moment in time.  The electric field intensity is represented by the solid line and the magnetic field intensity is represented by the dashed line.  Animating figure 3 will show both fields varying with time and distance.  The purpose of figure 3 is to show the orientation of the fields, that the fields are in phase, and that the fields transverse the axis of propagation.  For the sake of visual clarity, the amplitude of the magnetic field intensity is shown the same as the amplitude of the electric field intensity, a rare occurrence.  Also, no attenuation of either field is shown.

Figure 3

Not Animated:  Electric and magnetic field intensities at time t = 0.
		Animated:  Time-varying electric and magnetic field intensities
		                    propagating in the z direction

The following is a summary concerning the orientation and propagation of the electric and magnetic field intensities.
		
1.  The electric field intensity is always perpendicular to the magnetic field
intensity.
2.  The electic field intensity is always in phase with the magnetic field
intensity.
3.  The direction of propagation is defined by taking the cross product of the
unit vectors that define the orientation of the electric field intensity and
the magnetic field intensity.

Due to the similarities between the electric and magnetic field intensities, only the electric field intensity will be discussed in the remainder of this Notebook.

Free Space

The propagation constant γ is defined by:
		
	γ = jω Sqrt[μϵ] Sqrt[ 1 ± j (σ/ (ωϵ))] 				(3)

where j is Sqrt[ ±1 ], μ and ϵ are the permeability and permittivity of the medium in which the wave is propagating, respectively, and σ is the conductivity.  The real part of this constant is the attenuation constant α and imaginary part  is the phase constant β.

     For free space, the conductivity is zero.  In this case, Gamma will not have a real part and the attenuation constant, Alpha, will be zero.  The result of having Alpha equal to zero is that there will be no loss of energy as the wave propagates in the z direction and the magnitude of the electric field intensity will not be attenuated as the value of z increases.  The wave function for the time-varying electric field intensity bcomes:
		
	Ex = Eo Cos[ωt ± βz]						(4)

Figure 4 shows this function for free space, a magnitude of 100 V/m, and a frequency of 1 MHz at time t = 0.  Animating Figure 4 shows the time-varying electric field intensity propagating in the z direction.

Figure 4

Not Animated:  Electric field intensity at time t = 0.
			Animated:  Time-varying electric field intensity propagating 
				        in the z direction.

Note that an unattenuated wave propagating toward infinity represents the propagation of an infinite amount of energy, Figure 4 represents an ideal case.

Perfect Dielectrics

For the most part, the propagation of plane waves in a perfect dielectric is the same as in free space.  Again, the conductivity is zero, which leads to an attenuation constant of zero.  The only change is that the relative permability and permittivity are not necessarily both equal to one.  For waves of the same frequency, different material constants will cause different wavelengths for the waves. In general, a larger value of the relative permittivity will result in a smaller wavelength.  This is shown in figures 5a and 5b, where figure 5a shows a wave propagating in free space and figure 5b shows a wave propagating in teflon.

Figure 5a

Figure 5b

a)  Electric field intensity at time t = 0 in free space
			      (μr = 1 and ϵr = 1).
			b)  Electric field intensity at time t = 0 in teflon
			      (μr = 1 and ϵr = 2.1).

Lossy Dielectrics

For lossy dielectrics, the conductivity is no longer zero.  Thus, the attenuation constant has a nonzero value.  In general, the larger the conductivity a medium has, the faster a wave will dissipate as it propagates in the medium.  This is shown in Figures 6a and 6b.  Where Figure 6a is the electric field intensity propagating in distilled water and Figure 6b is the electric field intensity propagating in a steak cooking in a microwave.  In the case of the steak, the energy of the wave that is attenuated becomes heat and cooks the steak.

Figure 6a

Figure 6b

a)  Electric field intensity propagating in distilled water with μr = 1,
                             ϵr = 80, and σ = 10±4.
                        b)  Electric field intensity propagating in a steak with μr = 1,
                              ϵr = 47, and σ = 2.17.
                        Both a) and b) can be animated

Good Conductors

Media with high conductivity and large conduction currents are classified as good conductors.  For this classification of materials, a new constant, called the skin depth, becomes important.  The skin depth is the distance inside a conductor at which a propagating field has attenuated to Exp[±1 ] times its original value.   The skin depth is inversely proportional to the square root of the frequency of the wave and the conductivity and permeability of the medium in which the wave is propagating.  For a wave that has a frequency of 1 kHz and is propagating in a medium with a relative permeability of 1 and a conductivity of 6.17  107, the skin depth is approximately 2 mm.  In this case,  the electric field intensity would not exist only a few millimeters away.  This is shown in Figure 7.

Figure 7

Not animated:  Electric field intensity in silver at time t = 0.
                              Animated:  Time-varying electric field intensity propagating
                                                          in silver along the z axis.

User Controllable Graphics and Calculations

Do-it Yourself Graphics

In this section, the user can create three-dimensional animations of time-varying electric fields propagating in a medium.  The function that plots three dimensional grahpics is:
	
	PropGrap3D1[ E0, ur, er, cond, freq, zstart, zend ].
		
where, E0 = magnitude of electric field,
	ur  = relative permeability,
	er  = relative permittivity,
	cond  = conductivity,
	freq   = frequency,
	zstart  = starting value of z,
	zend = ending value of z.

An example for silver is PropGraph3D1[100, 1, 1, 6.17 10^7, 10^3, 0, .01].

Example 1

PropGraph3D1[100, 1, 1, 6.17 10^7, 10^3, 0,.01]

Do-it Yourself Constants

There are several constants invloved in wave propagation.  Many have been set up as functions in this Notebook of Mathematica.  The following is a list of those available.
		
				WaveVelocity
				WaveLength
				IntrinsicImpedance
				PhaseConstant
				PropagationConstant
				AttenuationConstant
				SkinDepth

To find out more about each function, type "?NameofConstant" and then hit the enter key.

Example 2

?WaveVelocity

WaveVelocity[ 1, 1 ]

Initialization Cells

<</Users/schneidj/NoteBooks/msteeds/Mathematica/Packages/Waveprop.m

Off[General::spell1];
Clear[PropGraph3D1]
PropGraph3D1[ E0_ ,ur_, er_, cond_, freq_, xmin_, xmax_]:=
   Block[{alpha,beta,tmax,tstep,wavelength},	
      alpha = AttenuationConstant[ ur, er, cond, freq];
	  beta = PhaseConstant[ur, er, cond, freq];
	  wavelength = WaveLength[ur, er, freq];
	  tmax = (1 / freq);
	  tstep = tmax / ( 2 N[Pi,10] );
	  Do[
	     Plot3D[E0 Exp[-alpha x] Cos[freq 2 N[Pi,10] t - beta x],
	      {x, xmin, xmax}, {y, -10, 10},
		  PlotPoints -> 30, 
		  PlotRange -> {-E0,E0}, 
		  AxesLabel -> {z,y,x}],
		 {t,0,tmax,tstep}]
	];
On[General::spell1];