6/6/2016
Instructor
Dr. Raymond Rumpf
(915) 747‐6958
rcrumpf@utep.edu
EE 5320
Computational Electromagnetics
Lecture #2
Maxwell’s Equations
 These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited 
Lecture 2
Slide 1
Outline
• Maxwell’s equations
• Physical Boundary conditions
• Parameter relations
• Preparing Maxwell’s equations for CEM
• The wave equation and its solutions
• Scaling properties of Maxwell’s equations
Lecture 2
Slide 2
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Maxwell’s Equations
Born
June 13,1831
Edinburgh, Scotland
Died
November 5, 1879
Cambridge, England
James Clerk Maxwell
Lecture 2
Slide 3
Sign Convention for Waves
To describe a wave propagating the positive z direction, we have two choices:
E  z , t   A cos t  kz 
Most common in engineering
E  z , t   A cos  t  kz 
Most common science and physics
Both are correct, but we must choose a convention and be consistent with it. For time‐harmonic signals, this becomes
E  z   A exp   jkz 
Negative sign convention
E  z   Aexk   jkz 
Positive sign convention
Lecture 2
Slide 4
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Maxwell’s Equations (e-jkz Convention)
Lecture 2
Slide 5
Lorentz Force Law
One additional equation is needed to completely describe classical electromagnetism...


 
F  qE  qv  B
Electric Force
Lecture 2
Magnetic Force
Slide 6
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Alternate Forms of Maxwell’s Equations
Maxwell’s Equations with Gaussian Units

  D  4 v

  B  4 v
Relativistic Maxwell’s Equations


1 B
 E  
c0 t


 F   0 J 


 12 e F  0 J free
 D 

 1   D 
  H   4 J 

c0 
t 


Maxwell’s Equations in Moving Media

  D  4 v

  B  4 v



 
1  B
 E   
   B  v 
c0  t


 1   D
 
  H   4 J 
   D  v 
c0 
t

  1

Lecture 2
Slide 7
Time‐Harmonic Maxwell’s Equations
Time‐Domain

  D  v

B  0
Frequency‐Domain
(e+jkz convention)

  D  v

B  0
Lecture 2


  E  j B
 

  H  J  j D


B
 E  
t

  D
 H  J 
t
Frequency‐Domain
(e-jkz convention)

  D  v

B  0


  E   j B
 

  H  J  j D
Slide 8
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Gauss’s Law

  D  v
 D Dy Dz
D  x 

x
y
z
Electric fields diverge from positive charges and converge on negative charges.
‐
+
If there are no charges, electric fields must form loops.
Lecture 2
Slide 9
Gauss’s Law for Magnetism

B  0
 B By Bz
B  x 

x
y
z
Magnetic fields always form loops.
Lecture 2
Slide 10
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Consequence of Zero Divergence
The divergence theorems force the D and B fields to be perpendicular to the propagation direction of a plane wave.

D  0
 
  de  jk r  0
 


jk
d  0


d

no charges
 
k d  0



k

k

B  0
 
  be  jk r  0
 

  b  jk  b  0

no charges
 
k b  0

 
k D

 
k B

k

k
Lecture 2
Slide 11
Ampere’s Law with Maxwell’s Correction

  D
 H  J 
t
  H z H y
 H  

z
 y

 H x H z

 aˆ x  
z
x


 H y H x

 aˆ y   x  y



 aˆ z

Circulating magnetic fields induce currents and/or time varying electric fields.
Currents and/or time varying electric fields induce circulating magnetic fields.
Lecture 2
Slide 12
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Faraday’s Law of Induction


B
 E  
t
  E E y
 E   z 
z
 y

 Ex Ez

 aˆ x  
x
 z

 E y Ex

 aˆ y   x  y



 aˆ z

Circulating electric fields induce time varying magnetic fields.
Time varying magnetic fields induce circulating electric fields.
Lecture 2
Slide 13
Consequence of Curl Equations
The curl equations predict electromagnetic waves!!
Magnetic Field
Electric Field
Lecture 2
Slide 14
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The Constitutive Relations
Electric Response


D E
•
•
•
•
Magnetic Response


B  H
•
•
•
•
Electric field intensity (V/m)
Initial electric “push.”
Induced electric field.
Electric energy in vacuum.
Magnetic field intensity (A/m)
Initial magnetic “push.”
Induced magnetic field.
Magnetic energy in vacuum.
• Permeability (H/m)
• Measure of how well a material stores magnetic energy.
• Permittivity (F/m)
• Measure of how well a material stores electric energy.
• Magnetic flux density (Wb/m2)
• Pretends as if all magnetic energy is tilted magnetic dipoles.
• Includes magnetic energy in vacuum and matter.
• Electric flux density (C/m2)
• Pretends as if all electric energy is displaced charge.
• Includes electric energy in vacuum and matter.
Lecture 2
15
Material Classifications
Linear, isotropic and non‐dispersive materials:


D t    E t 
We will use this almost exclusively
Dispersive materials:


D t    t   E t 
Anisotropic materials:


D  t     E  t 
A key point is that you can wrap all of the complexities associated with modeling strange materials into this single equation. This will make your code more modular and easier to modify. It may not be as efficient as it could be though.
Nonlinear materials:
D  t    0  e1 E  t    0  e 2 E 2  t    0  e 3 E 3  t  
Lecture 2
16
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Types of Anisotropy
Isotropic
Anisotropic


D t    E t 


B t    H t 


D  t     E  t 


B t      H t 
 aa
    ba
 ca
 ab  ac 
 bb  bc 
 cb  cc 
electrically anisotropic
Gryrotropic


D  t     E  t 


B t      H t 
 1
    j 2
 0
 j 2
1
0
 aa
     ba
 ca
ab
bb
cb
ac 
bc 
cc 
magnetically anisotropic
isotropic
0
0 
 3 
 1
     j 2
 0
gyroelectric
Bi‐Isotropic



D t    E t    H



B t    H t    E
 j 2
1
0
 iso
    0
 0
0
0 
3 
0
 iso
0
0 
0    iso
 iso 
uniaxial
 o
    0
 0
gyromagnetic
Bi‐Anisotropic



D  t     E  t     H



B  t      H  t     E
0
o
0
0
0 
 e 
biaxial
 a
    0
 0
0
b
0
0
0 
 c 
Lecture 2
17
All Together Now…
Divergence Equations

B  0

  D  v

  D
 H  J 
t


B
 E  
t
Curl Equations
What produces fields
Constitutive Relations


D  t     t    E  t  



B  t      t    H  t 
Lecture 2
means convolution
means tensor
How fields interact with materials
Slide 18
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Physical Boundary Conditions
Lecture 2
Slide 19
Physical Boundary Conditions
1 and 1
Tangential components of E and H
are continuous across aninterface.
2 and  2
E1,T
E2,T
H1,T
H 2,T
1 E1,N
E and H fields normal to the interface are discontinuous across 1 H1,N
an interface.
Note: Normal components of D
D1,N
and B are continuous across the interface.
B1,N
Tangential component of the wave vector is continuous across an interface.
Lecture 2
k1,T
 2 E2,N
2 H 2,N
D2,N
These are more complicated boundary conditions, physically and analytically.
B2,N
k2,T
Slide 20
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Parameter Relations
Lecture 2
Slide 21
Map of Parameter Relations

M

E

f
0
0
r




P

v
0
Lecture 2
r

D

B
0
n

c0

H
Slide 22
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The Relative Permittivity
The permittivity is a measure of how well a material stores electric energy. A circulating magnetic field induces an electric field at the center of the circulation in proportion to the permittivity.


E
 H  
     j 
t
The dielectric constant of a material is its permittivity relative to the permittivity of free space.
   0 r
 0  8.854187817 1012 F m
1  r  
r is the relative permittivity or dielectric constant
Lecture 2
Slide 23
The Relative Permeability
The permeability is a measure of how well a material stores magnetic energy. A circulating electric field induces a magnetic field at the center of the circulation in proportion to the permeability.


H
     j  
  E  
t
The relative permeability of a material is its permeability relative to the permeability of free space.
  0  r
Lecture 2
0  1.256637061106 H m
1  r  
r is the relative permeability
Slide 24
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Conductivity 
Conductivity is the measure of a material’s ability to support electric current. This term is responsible for ohmic loss in materials.
It appears in Ampere’s Circuit Law.
 

  H  J  j D

The current density is related to conductivity and the electric field J
intensity through Ohm’s Law.


J E
Lecture 2
Slide 25
’-j’’ Vs.  and 
It is redundant to have a complex dielectric constant along with a conductivity term, although it happens. We should use either a complex dielectric constant or a real dielectric constant and a conductivity.


  H  j     j   E



  H   E  j E
j     j      j
   j  
  
Lecture 2


j
  


Slide 26
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Material Impedance
The material impedance is the parameter which describes the balance between the electric and magnetic field amplitudes.

E
  
H

E

H
E
H

k
It is calculated from the permeability and permittivity of the material.

r

 0

r
0  free space impedance
 376.73031346177 
   
Phase between E and H
Amplitude between E and H
     j 
Impedance tells us that E and H are three orders of magnitude different.
Reactive component
Resistive component.
Lecture 2
Slide 27
The Complex Refractive Index
The permittivity and permeability appear in Maxwell’s equations so they are the most fundamental material properties. However, it is difficult to determine physical meaning from them in terms of how waves propagate (i.e. speed, loss, etc.). In this case, the refractive index is a more meaningful quantity.
n  r  r
In the frequency‐domain, the refractive index is a complex quantity.
n  n  j
n is the real part of refractive index. It quantifies how quickly a wave propagates.
E  z   E0 e  jk0 nz
 is the extinction coefficient. It quantifies loss and how quickly a wave decays.
* Note: when only the refractive index n is specified for a material, assume r = 1.0.
Lecture 2
Slide 28
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The Complex Propagation Constant, 
The propagation constant is very close to the complex refractive index. It describes the speed and decay of a wave.
E  z   E0 e  z
The propagation constant has a real and imaginary part.
    j
 is the attenuation coefficient. It quantifies how quickly the amplitude of a wave decays.
E  z   E0 e  z e  j  z
 is the propagation constant. It quantifies how quickly a wave accumulates phase.
It is related to the complex refractive index through
  jk0 n
Lecture 2
Slide 29
The Absorption Coefficient, 
The absorption coefficient describes how quickly the power in a wave decays.
P  z   P0 e  z
WARNING: Notice the unfortunate reuse of the symbol  for two different things. This is easily confused!!
The attenuation coefficient and absorption coefficient are related through
 abs  2 att
Lecture 2
The absorption coefficient and extinction coefficient are related through
 abs  2k0
Slide 30
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6/6/2016
Loss Tangent
Sometimes material loss is given in terms of a “loss tangent.”
tan  
  


 
P  z   P0 e  k0 nz
Recall that interpreting wave properties (velocity and loss) is not intuitive using just the complex dielectric function. To do this, we preferred the complex refractive index.
It turns out that the loss tangent and the extinction coefficient are essentially the same.
2  abs


n
k0 n
It is called a loss tangent because it is the angle in the complex plane formed between the resistive component and the reactive component of the electromagnetic field.


or

 or  
Lecture 2
Slide 31
 versus f
 is the angular frequency measured in radians per second. It relates more directly to phase and k. Think cos(t).
f is the ordinary frequency measured in cycles per second. It relates more directly to time. Think cos(2ft) and =1/f.
  2 f
Lecture 2
Slide 32
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Wavelength and Frequency
The frequency f and free space wavelength 0 are related through
c0  f 0
c0  299792458
m
s
 speed of light in vacuum
Inside a material, the wave slows down according to the refractive index as follows.
v
c0
n
Lecture 2
Slide 33
Sign Convention
How do you define forward wave propagation?
‐ z
+ z
 
E  E0 e j t  kz 
 
E  E0 e  j t  kz 
Dielectric Function
     j 
     j 
Refractive Index
N  n  j
N  n  j
Quantity
Wave Solution
Lecture 2
+z
Slide 34
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6/6/2016
Summary of Parameter Relations
Permittivity
   0 r
Permeability
  0 r
 0  8.854187817  1012 F m
0  1.256637061 106 H m
Refractive Index
Impedance
  0  r  r
n  r  r
0  0  0  376.73031346177 
Wave Velocity
Frequency and Wavelength
c0
n
c0  299792458 m s
  2 f
c0  f 0
v
Wave Number
k0 
2
0
Exact
Lecture 2
Slide 35
Table of Dielectric Constants and Loss Tangents
Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
Lecture 2
Slide 36
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Table of Permeabilities
Lecture 2
Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
Slide 37
Duality Between E‐D and H‐B
Lecture 2
Electric Field
Magnetic Field
E
H
D
B
P
M
ε
μ
38
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6/6/2016
Preparing Maxwell’s Equations
for CEM
Lecture 2
Slide 39
Simplifying Maxwell’s Equations

1. Assume no charges or current sources: v  0, J  0





D  t     t    E  t 
  H  D t
B  0





  E   B t
B  t      t    H  t 
D  0
2. Transform Maxwell’s equations to frequency‐domain:





D    E
  H  j D
B  0





  E   j B
B    H
D  0
Convolution becomes simple multiplication
Note: We have chose to proceed with the negative sign convention.
3. Substitute constitutive relations into Maxwell’s equations:



    H  0
Note: It is useful to retain μ and ε and not replace   H  j   E


them with refractive index n.

  E   j    H
    E  0
 
 
Lecture 2
Slide 40
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Isotropic Materials
For anisotropic materials, the permittivity and permeability terms are tensor quantities.  xx  xy  xz 
  xx  xy  xz 


    yx  yy  yz 
      yx  yy  yz 
 zx  zy  zz 
  zx  zy  zz 




For isotropic materials, the tensors reduce to a single scalar quantity.
 0 0 
    0  0   
 0 0  

     0
 0
0
0   
 
0

0
Maxwell’s equations can then be written as

  r H  0

  r E  0
 
 
0 and 0 dropped from these equations because thy are constants and do not vary spatially.


  H  j 0 r E


  E   j0 r H
Lecture 2
Slide 41
Expand Maxwell’s Equations
Divergence Equations
Curl Equations


  H  j 0 r E

  r H  0




  r H x 
x

  r H y 
y

  r H z 
z
H z H y

 j 0 r Ex
y
z
H x H z

 j 0 r E y
z
x
H y H x

 j 0 r Ez
x
y
0

  r E  0
 

  r Ex 
x

  r Ey 
y

  r Ez 
z
0


  E   j0  r H

Ez E y

  j0  r H x
y
z
Ex Ez

  j0  r H y
z
x
E y Ex

  j0  r H z
x
y
Lecture 2
Slide 42
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Normalize the Magnetic Field
Standard Maxwell’s Curl Equations


  E   j0  r H


  H  j 0 r E
Normalized Magnetic Field

E 377
 
n
H

 
H   j 0 H
0
• Eliminates j
• No sign inconsistency
• Just have k0
Note:
k 0    0 0
Equalizes E and H amplitudes
Normalized Maxwell’s Equations


  E  k0  r H


  H  k0 r E
Lecture 2
Slide 43
Starting Point for Most CEM
We arrive at the following set of equations that are the same regardless of the sign convention used.
Ez E y

 k0  xx H x
y
z
Ex Ez

 k0  yy H y
z
x
E y Ex

 k0  zz H z
x
y
H z H y

 k0 xx Ex
y
z
H x H z

 k0 yy E y
z
x
H y H x

 k0 zz Ez
x
y
The manner in which the magnetic field is normalized does depend on the sign convention chosen.
  j H negative sign convnetion
0

H  
  j0 H positive sign convnetion
Lecture 2
Slide 44
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The Wave Equation and Its Solutions
Lecture 2
Slide 45
Derivation of the Wave Equation
We start with Maxwell’s curl equations.


  E   j0  r H
Eq. (1)


  H  j 0 r E
Eq. (2)
Equation (1) is solved for the magnetic field.

H
j
0  r


 E

Eq. (3)
Equation (3) is substituted into Eq. (2).
 

 j
 
  E   j 0 r E
 0 r

 

1
 
  E   k02 r E
 r



Lecture 2


Slide 46
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6/6/2016
Two Different Wave Equations
We can derive a wave equation for both E and H.


  r1  E  k02 r E


   r1  H  k02 r H
It is not actually possible to simplify these equations further without making an approximation. Assuming a linear homogeneous isotropic (LHI) material, the wave equations reduce to


    E  k02 r  r E



   E   2 E  k02 r  r E


 2 E  k02 r  r E  0




    H  k02 r  r H



   H   2 H  k02 r  r H


 2 H  k02 r  r H  0


We see that these equations will have the same solution since it is the same differential equation! So, we only have to solve one of them.
Lecture 2
Slide 47
Plane Wave Solution in Homogeneous Media
Given the wave equation in an LHI material,


 2 E  k02 r  r E  0
The solution is a plane wave.
 
 

E  r   E0 exp  jk  r
 
 

H  r   H 0 exp  jk  r


Lecture 2


Slide 48
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6/6/2016
Sign Convention
SIGN CONVENTION #1
eikz  eik0 Nz  eik0  n i  z
Forward Propagation Along +z
e  ikz
ik0 nz
 k0 z
 e
 e
Refractive Index and Dielectric Constant
N  n  j
 r   r  j r
SIGN CONVENTION #2
oscillatory Decaying
term in  z exponential
e  ikz  e ik0 Nz  e  ik0  n i  z
Forward Propagation Along +z
e  ikz
 ik0 nz
 k0 z
 e
 e
Refractive Index and Dielectric Constant
N  n  j
 r   r  j r
oscillatory Decaying
term  z
exponential
Lecture 2
Slide 49
Amplitude Relation
Given plane wave functions of the form
 
 

E  r   E0 exp  jk  r
 
 

H  r   H 0 exp  jk  r




The amplitudes are related through Maxwell’s equations.


  E   j0  r H
 
 


  E0 e  jk r   j0  r H 0 e  jk r
 
 
 

 j k  E0 e  jk r   j0  r H 0 e  jk r
 

k  E0  0  r H 0

Lecture 2





 
 

k  E0 k  E0
H0 

0 r k00 r
Slide 50
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IMPORTANT: Plane Waves are of Infinite Extent
Many time we just draw rays or sometime rays with perpendicular lines to represent the wave fronts.
ray
ray + perpendicular lines
Unfortunately, this suggests the wave is confined spatially. In reality, plane waves are of infinite extent. This more this way…
Lecture 2
Slide 51
Solving the Wave Equation as a Scattering Problem
Scattering problems cast the wave equation into the following matrix form.


     E  k02 r E  g
1
r
Ax  b


A    r1  k02 r

xE
bg
• A source b is needed • Only one solution exists
Lecture 2
Slide 52
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Solving the Wave Equation as an Eigen‐Value Problem
The wave equation can also be solved as an eigen‐value problem. This approach is used when “modes” are being calculated.


  r1  E  k02 r E
Ax   Bx
A    r1 

xE
B  r
  k02
• No source is needed • Multiple solutions exists
Lecture 2
Slide 53
Wave Equation Vs. Maxwell’s Equations
Wave Equation
The most generalized wave equations are


     E  k02 r E


   r1  H  k02 r H
1
r
In LHI materials, these reduce to


 2 E  k02 r  r E  0


 2 H  k02 r  r H  0
Today, it is rare to see the wave equations solved in this form because it leads to spurious solutions.
The “fixes” to the spurious solutions problem are incorporated into Maxwell’s equations before a wave equation is derived.
Lecture 2
Maxwell’s Equations
Maxwell’s equations expanded into Cartesian coordinates are
H z H y

 k0 xx Ex
y
z
H x H z

 k0 yy E y
z
x
H y H x

 k0 zz Ez
x
y
Ez E y

 k0  xx H x
y
z
Ex Ez

 k0  yy H y
z
x
E y Ex

 k0  zz H z
x
y
These are often written in matrix form as
 0
 
 z
 
 y
 z
  xx
y   Ex 

 x   E y   k0  0
 0
0   E z 

0

x
0
 yy
0
0   H x 
 
0   H y 
 z
 0
 
 z
 
 y
 zz   H z 
0

x

 xx
y   H x 
 
 x   H y   k0  0
 0
0   H z 

0
 yy
0
0   Ex 
0   E y 
 zz   Ez 
Typically, “fixes” are incorporated here and then a wave equation is derived.
 0
 
 z
 
 y
 z
0

x
y    xx

 x   0
0   0

0
 yy
0
1
0   0

0   z
 zz    y
 z
0

x
 xx
y   E x 

 x   E y   k02  0
 0
0   Ez 

0
 yy
0
0   Ex 
0   E y 
 zz   Ez 
Slide 54
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Scaling Properties in
Maxwell’s Equations
Lecture 2
Slide 55
Scaling Properties of Maxwell’s Equations
There is no fundamental length scale in Maxwell’s equations.
Devices may be scaled to operate at different frequencies just by scaling the mechanical dimensions or material properties in proportion to the change in frequency.
This assumes it is physically possible to scale systems in this manner. In practice, building larger or smaller features may not be practical. Further, the properties of the materials may be different at the new operating frequency.
Lecture 2
Slide 56
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6/6/2016
Scaling Dimensions
We start with the wave equation and write the parameters dependence on position explicitly.
 
  
2
   E  r     0 0   r  r   E  r 
r  r 
1

Next, we scale the dimensions by a factor a.
 a  
 
 

1
2
  a   E  r a    0 0   r  r a   E  r a 
r  r a 
a  1 stretch dimensions
a  1 compress dimensions
The scale factors multiplying the operators are moved to multiply the frequency term.

2
 
  
1
 
   E  r       0 0   r  r    E  r  
r  r  
a

 r
r 
a
The effect of scaling the dimensions is just a shift in frequency.
Lecture 2
Slide 57
Visualization of Size Scaling
a = 0.5
fc = 1000 MHz
fc = 500 MHz
Lecture 2
Slide 58
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6/6/2016
Scaling  and 
We apply separate scaling factors to  and .

1
a  



  E   2 0 0   a  r   E
r
The scale factors are moved to multiply the frequency term.

Lecture 2
1
r

  E   a a


2

 0 0   r  E
The effect of scaling the material properties is just a shift in frequency.
Slide 59
30