UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
LECTURE NOTES 10.5
EM Standing Waves in Resonant Cavities






One can create a resonant cavity for EM waves by taking a waveguide (of arbitrary shape)
and closing/capping off the two open ends of the waveguide.
Standing EM waves exist in (excited) resonant cavity (= linear superposition of two counterpropagating traveling EM waves of same frequency).
Analogous to standing acoustical/sound waves in an acoustical enclosure.
Rectangular resonant cavity – use Cartesian coordinates
Cylindrical resonant cavity – use cylindrical coordinates to solve the EM wave eqn.
Spherical resonant cavity – use spherical coordinates
A.) Rectangular Resonant Cavity: ( L  W  H  a  b  d ) with perfectly conducting walls
(i.e. no dissipation/energy loss mechanisms present), with 0  x  a , 0  y  b , 0  z  d .
n.b. Again, by convention: a > b > d.
Since we have rectangular symmetry, we use Cartesian coordinates - seek monochromatic
EM plane wave type solutions of the general form:


E  x, y, z , t   Eo  x, y, z  e it


B  x, y, z , t   Bo  x, y, z  e  it
Subject to the boundary conditions
E||  0 and B  0
at all inner surfaces.
Maxwell’s Equations (inside the rectangular resonant cavity – away from the walls):
(1) Gauss’ Law:
 
 E  0

(3) Faraday’s Law:

 
B
 E  

t
(2) No Monopoles:
 
 B  0

(4) Ampere’s Law:

  1 E
 B  2

c t
 
 Eo  0
 

  Eo  i Bo
 
 Bo  0
 

  Bo  i 2 E o
c
Take the curl of (3):
= 0 {Gauss’ Law}
2
  
 
  

 
   
2 
    Eo  i  Bo     Eo   Eo  i   Bo    E {using (4) Ampere’s Law}
c




2

  
 Eox     Eox
c
2

  
2
  Eoy     Eoy
c
2

 
 2 Eoz     E oz
c


2
 E ox  E ox  x, y, z  


 E oy  E oy  x, y, z   i.e. each is a fcn  x, y, z 



 Eoz  Eoz  x, y, z  
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede

For each component  x, y, z of Eo  x, y, z  we try product solutions and then use the
separation of variables technique:
2
 
E oi  x, y, z   X i  x  Yi  y  Z i  z  for  2 E oi  x, y, z      E oi where subscript i  x, y, z .
c
2 X i  x
 2Yi  y 
 2 Zi  z 
 c 


 
Yi  y  Z i  z 
X
x
Z
z
X
x
Y
y








i
i
i
i
 X i  x  Yi  y  Z i  z 
2
2
2
x
 y
 z
 c 
2
Divide both sides by X i  x  Yi  y  Z i  z  :
2
2
2 X i  x
1  Yi  y 
1  Zi  z 
 


  
The wave equation becomes:
2
2
2
X i  x  x
Yi  y  y
Z i  z  z
c

 

 


1
 fcn  x  only
 fcn  y  only
2
 fcn z  only
0  x  a 


This equation must hold/be true for arbitrary (x, y, z) pts. interior to resonant cavity 0  y  b 
0  z  d 


This can only be true if:
1
Xi  x
2 X i  x
x 2
1
Yi  y 
 2Yi  y 
1
 2 Zi  z 
Zi  z 
y
2
z 2
2 X i  x
 k x2 X i  x   0
  k  constant 
2
x
2
x
  k y2  constant 
 2Yi  y 
 k y2Yi  y   0
2
y
  k z2  constant
 2 Zi  z 
 k z2 Z i  z   0
2
z

n.b. We seek
oscillatory
(not damped)
solutions !!!
2
 
with: k  k  k  k     characteristic equation
c
2
2
x
2
y
2
z
General solution(s) are of the form:  i  x, y, z  :
E oi  x, y, z    Ai cos  k x x   Bi sin  k x x    Ci cos  k y y   Di sin  k y y     Ei cos  k z z   Fi sin  k z z  
n.b. In general, k x , k y and k z should each have subscript i  x, y, z , but we will shortly find out
that k xi  same for all i  x, y, z , k yi  same for all i  x, y, z , and k zi  same for all i  x, y, z .
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
Boundary Conditions: E||  0 @ boundaries and B  0 @ boundaries:
 E x  0

Cx  0
 A y  0
 x  0, x  a 
Eoy  0 at 
  coefficients  
 z  0, z  d 
 E y  0
 A z  0
 x  0, x  a 
Eoz  0 at 
  coefficients  
 y  0, y  b 
Cz  0
 y  0, y  b 
Eox  0 at 
  coefficients
 z  0, z  d 
k y  n b , n  1, 2,3,....
and 

k z   d ,   1, 2,3,....
k x  m b , m  1, 2,3,....
and 

k z   d ,   1, 2,3,.... 
k x  m a , m  1, 2,3,....
and 

k y  n d , n  1, 2,3,.... 
n.b. m = 0, and/or n = 0 and/or   0 are not allowed, otherwise E oi  x, y, z   0 (trivial solution).
Thus we have (absorbing constants/coefficients, & dropping x,y,z subscripts on coefficients):
E ox  x, y, z    A cos  k x x   B sin  k x x   sin  k y y  sin  k z z 
E oy  x, y, z   sin  k x x  C cos  k y y   D sin  k y y   sin  k z z 
E ox  x, y, z   sin  k x x  sin  k y y   E cos  k z z   F sin  k z z  
 
E ox E oy E oz


0
But (1) Gauss’ Law:  E  0 
x
y
z
Thus:
k x   A sin  k x x   B cos  k x x   sin  k y y  sin  k z z 
 k y sin  k x x   C sin  k y y   D cos  k y y   sin  k z z 
 k z sin  k x x  sin  k y y    E sin  k z z   F cos  k z z    0
This equation must be satisfied for any/all points inside rectangular cavity resonator.
In particular, it has to be satisfied at  x, y, z    0, 0, 0  .
We see that for the locus of points associated with (x = 0,y,z) and (x,y = 0,z) and (x,y,z = 0),
we must have B  D  F  0 in the above equation.
Note also that for the locus of points associated with  x  m 2k x , y, z  and  x, y  n 2k y , z 
and  x, y, z   2k z  where m, n,   odd integers (1, 3, 5, 7, etc. …) we must have:
  Ck
  Ek
 0.
Ak
x
y
z
Note further that this relation is automatically satisfied for m, n,   even integers (2, 4, 6, 8, etc. …).
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede

 m 
kx  
 m  1, 2,3, 4,

 a 
E ox  x, y, z   A cos  k x x  sin  k y y  sin  k z z   


 n 
Thus: E oy  x, y, z   C sin  k x x  cos  k y y  sin  k z z   k y  
 n  1, 2,3, 4,
b



E oz  x, y, z   E sin  k x x  sin  k y y  cos  k z z   
  
k z  
   1, 2,3, 4,
 d 

With: E o  x, y, z   E ox xˆ  E oy yˆ  E oz zˆ
n.b.: m  n    0 simultaneously is not allowed!

Now use Faraday’s Law to determine B :

i  
B    E



E
i  E
Box    oz  oy
z
  y
E
i  E
Boy    ox  oz
x
  z

i 


    Ek
y sin  k x x  cos  k y y  cos  k z z   Ck z sin  k x x  cos  k y y  cos  k z z  



i 


    Ak
z cos  k x x  sin  k y y  cos  k z z   Ek x cos  k x x  sin  k y y  cos  k z z  


i  E oy E ox 
i 



Boz   
   Ck
x cos  k x x  cos  k y y  sin  k z z   Ak y cos  k x x  cos  k y y  sin  k z z  

y 
  x


Bo  x, y, z   Box xˆ  Boy yˆ  Boz zˆ
or:

 Ek
i

y


  Ck

 Ak
z
x

  cos  k x  sin  k y  cos  k z  yˆ
 Ek
  cos  k x  cos  k y  sin  k z  yˆ
 Ak
 sin  k x  cos  k y  cos  k z  xˆ
 Ck
z
x
y
z
x
x
y
x
y
y
z
z

This expression for Bo  x, y, z  (already) automatically satisfies boundary condition (2) B  0 :
Box  0 at x  0, x  a
 m 
with k x  

 a 
m  0,1, 2,
4
Boy  0 at y  0, y  b
Boz  0 at z  0, z  d
 n 
with k y  

 b 
n  0,1, 2,
  
with k z  

 d 
  0,1, 2,
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
 
Does  Bo  x, y, z   0 ???
 
B
B
B
 Bo  x, y, z   ox  oy  oz
x
y
z
i
 cos  k x  cos  k y  cos  k z 
  Ck

k x Ek
y
z
x
y
z
 



 k  Ck

 k y Ak
z
z
x

  cos  k x  cos  k y  cos  k z 
 Ek
  cos  k x  cos  k y  cos  k z 
 Ak

i

 k k E  k k C
x
y
x z
x
x
y
z
 k y k z A  k x k y E
y
x
y
z
 k x k z C  k y k z A

  cos  k x x  cos  k y y  cos  k z z  
 
  Bo  x, y, z   0 YES!!!
For TE Modes:


  Ck
  Ek
  Ck
  0 or: C   A  k x
  0 tells us that: Ak
Ez  0  coefficient E  0 . Then Ak
x
y
z
x
y
 ky

Thus:
The
lowest
TEm ,n ,
mode
a  b  d 



E ox  x, y, z , t  
A cos  k x x  sin  k y y  sin  k z z  e it
 m
kx  
 a

 , m  1, 2,3,

k
E oy  x, y, z , t    A  x
 ky


 i t
 sin  k x x  cos  k y y  sin  k z z  e

 n
ky  
 b

 , n  1, 2,3,

E oz  x, y, z , t   0
  
kz  
 ,   1, 2,3,
 d 
(n = 0 is NOT allowed for TE modes!!!)
i k 
 A  x  k z sin  k x x  cos  k y y  cos  k z z  e it
  k y 
i
 k z A cos  k x x  sin  k y y  cos  k z z  e it
Box  x, y, z , t  
is: TE111
Boy  x, y, z , t  

i  k
Boz  x, y, z , t    A  x
  k y


 it
 k x  k y  cos  k x x  cos  k y y  sin  k z z  e


The angular cutoff frequency for m, n,  th mode for TE modes in a rectangular cavity is:
2
mn
2
 m   n    
c 
 
 

 a   b   d 
2
and: vg 

k
 c  v no dispersion.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
For TM Modes:

  Ak


  k y 
Bz  0  Ck
x
y  0 or: C   A 
 kx 
Thus:


E ox  x, y, z , t  
E oy  x, y, z , t  
The
lowest
TM m, n ,
mode
a  b  d 
is: TM 111
 ky
A 
 kx
A cos  k x x  sin  k y y  sin  k z z  e it
 m
kx  
 a

 , m  1, 2,3,


 it
 sin  k x x  cos  k y y  cos  k z z  e

 n
ky  
 b

 , n  1, 2,3,

  
kz  
 ,   1, 2,3,
 d 
(m = 0 is NOT allowed for TM modes!!!)
 k   k   k  
E oz  x, y, z , t    A  x    y   y   sin  k x x  sin  k y y  cos  k z z  e it
 k z   k x   k z  
k
i  
Box  x, y, z , t     A  k x   y
  
 kx
   ky
 ky  
   kz
   ky
  A

 kx


i  
  k   k y  k   k x
Boy  x, y, z , t     Ak
A

x
y
  z
 kx    kx

B  x, y, z , t   0
 
 it
 k z  sin  k x x  cos  k y y  cos  k z z  e
 
 
 i t
  cos  k x x  sin  k y y  cos  k z z  e
 
oz
 
For either TE or TM modes: k  k  k  k   
c
2
 m
kx  
 a

 , m  1, 2,

2
x
2
y
2
z
 n
ky  
 b
2
with:

 , n  1, 2,

 
kz  
 d

 ,   1, 2,

The angular cutoff frequency for m, n,  th mode is the same for TE/TM modes in a rectangular cavity:
2
2
 m   n    
mn  c 
 
 

 a   b   d 
6
2
and: vg 

k
 c  v no dispersion.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
B.) The Spherical Resonant Cavity:
The general problem of EM modes in a spherical cavity is mathematically considerably more


involved (e.g. than for the rectangular cavity) due to the vectorial nature of the E and B -fields.
 For simplicity’s sake, it is conceptually easier to consider the scalar wave equation, with a

2
1   r ,t 


2
0
scalar field   r , t  satisfying the free-source wave equation    r , t   2
c
t



which can be Fourier-analyzed in the complex time-domain   r , t      r ,   e  it d 


with each Fourier component   r ,   satisfying the Helmholtz wave equation:
 2  k 2   r,    0 with: k 2   c 2 i.e. no dispersion.
In spherical coordinates, the Laplacian operator is:


  r ,   
 2  r ,  
1 2
1
 
1


   r ,  
 r  r ,     r 2 sin    sin     r 2 sin 2   2
r r 2


2
To solve this scalar wave equation – we again try a product solution of the form:

  r ,  
R r 
P   Q   e  it 
r
 , 
 f  r  Y
m
,m
m
spherical
harmonics
The Ym  ,   satisfy
the angular portion of
scalar wave equation…

Plug this   r ,   into the above scalar wave equation, use the separation of variables technique:
Get radial equation:
Let: f   r  
    1 
 d2 2 d
 k2 
 2
 f   r   0 where:  = 0, 1, 2, 3, . . .
r dr
r2 
 dr
1
u  r  . Then we obtain Bessel’s equation with index v    12 :
r
2
 d2 1 d
   12  
2
k 
 2
 u  r   0
r dr
r 2 
 dr
Solutions of the (radial) Bessel’s equation are of the form: f m  r  
Am
B
J   1  kr   m N   1  kr 
2
2



r 
r 
Bessel fcn of 1st
kind of order   12
Bessel fcn of 2nd
kind of order   12
It is customary to define so-called spherical Bessel functions and spherical Hankel functions:
1
  2
j  x     J   1  x 
2
 2x 
where:
x  kr
1
  2
n  x     N   1  x 
2
 2x 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
1
1,2 
and:
h
 2
 x      J  12  x   iN  12  x    j  x   i n  x 
 2x 
Prof. Steven Errede
n.b. If x = kr is real, then
h 2  x   h*1  x 
 1 d   sin  x  
j  x     x  

 
 x dx   x 


 1 d   cos  x  
n  x      x  

 
 x dx   x 


sin  x 
x
cos  x 
n0  x   
x
eix
ix
 ix
e
h0 2  x  
ix
j0  x  
h01  x  
j1  x  
sin  x  cos  x 

x2
x
h1   x   
n1  x   
cos  x  sin  x 

x2
x
h1
1
2
 x  
eix 
1 
x 
i

x
e  ix  i 
1  
x  x
For x  1,  :
j  x  


x
x2

 ...  where:  2  1 !!   2  1 2  1 2  3 ...  5  3  1
1

 2  1!!  2  2  3 
n  x   
 2  1!! 1 
x  1



x2
 ... 
2 1  2 

For x  1,  :
1
 

j  x  
sin  x 

x
2 

1
 

n  x    cos  x 

x
2 

The general solution to Helmholtz’s equation in spherical coordinates can be written as:





  r , t    
A1m h1  kr   Am2 h 2  kr   Ym  ,  

 ,m
Coefficients are determined by boundary conditions.
For the case of EM waves in a spherical resonant cavity we will (here) only consider TM modes,

which for spherical geometry means that the radial component of B, Br  0 . We further assume


(for simplicity’s sake) that the E and B -fields do not have any explicit  -dependence.
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Ym  ,   
Hence:
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
2  1    m  ! m
P  cos   eim

4    m  ! 
Will have some restrictions imposed on it
Associated Legendré Polynomial
 

If Br  0 and B  explicit function of  , then:  B  0  B  0 {necessarily}

 
B
But:   E  
requires: E  0
t
 TM modes with no explicit  -dependence involve only Er , E and B


 
  1 E
B
Combining   E  
and   B  2
with harmonic time dependence e it of solutions,
t
c t
2
     
We obtain:   B      B  0
c
The  -component of this equation is:
2
2
1   1 
 

rB
rB   2
sin  rB    0





 
2 

r   sin  
r
c

But:
  rB   rB
1  
  1 



sin
sin


rB




  sin  
  sin 2 
 sin   



~Legendré equation with m 1
 Try product solutions of the form: B  r ,   
u  r  1
P  cos  
r
Substituting this into the above equation gives a differential equation for u  r  of the form of:
2
d 2u  r         1 

Bessel’s equation:
  
 u  r   0 with  = 0, 1, 2, 3, . . . defining the
dr 2
r 2 
 c 
angular dependence of the TM modes.
Let us consider a resonant spherical cavity as two concentric, perfectly conducting spheres of
inner radius a and outer radius b.
u  r  1
P  cos   , the radial and tangential electric fields (using Ampere’s Law) are:
r
u r 

ic 2
ic 2
Er  r ,   
P  cos  
sin  B   
    1 

r
 r sin  
r
ic 2 
ic 2 u  r  1
E  r ,   
rB   
P  cos  

 r r
 r r
If B  r ,   
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
But E  E which must vanish at r = a and r = b

Lect. Notes 10.5
u  r 
r
r a

Prof. Steven Errede
u  r 
r
r b
0
The solutions of the radial Bessel equation are spherical Bessel functions (or spherical Hankel
functions).
u  r 
The above radial boundary conditions on 
r  a  0 lead to transcendental equations for
r r b
the characteristic frequencies,  {eeeEEK}!!!
However {don’t panic!}, if: (b – a) = h is such that h  a then:
    1     1

 constant!!!
r2
a2
And thus in this situation, the solutions of Bessel’s equation:
2
2
d 2u  r         1 
d 2 u  r 
       1
2
2
   
 k u  r   0 where: k    
 u  r   0 
dr 2
dr 2
a 2 
a2
c
 c 
i.e. u  r   A cos  kr   B sin  kr 
are simply sin (kr) and cos (kr) !!!
Then:
u
r
r a
 kA sin  ka   kB cos  ka   0 and
u
r
r b
 kA sin  kb   kB cos  kb   0
For  b  a   h  a an approximate solution is: u  r   A cos  kr  ka 
with: kh  k  b  a   n , n = 0, 1, 2, . . .
       1  n 

Thus: k    
 , n = 0, 1, 2, 3, . . . and  = 0, 1, 2, 3, . . .
a2
c
 h 
2
2
2
n
The corresponding angular cutoff frequency is:
    1
 n      1
n  c k 
c 
for h  a , n = 0, 1, 2, 3, . . . and  = 0, 1, 2, 3, . . .
 
2
a
a2
 h 
2
2
n
Because h  a , we see that the modes with n = 1, 2, 3, . . . turn out to have relatively high
 n 
frequencies n  c 
 for n  1 . However, the n = 0 modes have relatively low frequencies:
 h 
0   c
    1 c

    1 for h  a .
a2
a
An exact solution (correct to first order in (h/a) expansion) for n = 0 is: 0  
c
    1
 a  12 h 
These eigen-mode frequencies are known as Schumann resonance frequencies.  = 1, 2, 3, . . .
(W.O. Schumann – Z. Naturforsch. 72, 149, 250 (1952))
10
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
For n = 0, the EM fields are: E0   0 , Er0  
Lect. Notes 10.5
1
1
P cos   and B0   P1  cos  
2 
r
r
ẑ
Very Useful Table:
r̂  ˆ  ˆ
ˆ  r̂  ˆ
ˆ  ˆ  r̂
ˆ  ˆ  r̂
ˆ  r̂  ˆ
Prof. Steven Errede
ˆ , B 0 ˆ
 0 
rˆ, Er0  rˆ
S
r̂  ˆ  ˆ

a
b
ˆ
ŷ

x̂
Poynting’s vector:


1 
1
1
S0  
E0   B0    rˆ  ˆ  3 P  cos   P1  cos    3 P  cos   P1  cos   ˆ 
r
r
0

 

Circumpolar
N-S waves!
The Earth’s surface and the Earth’s ionosphere behave as a spherical resonant cavity (!!!)
with the Earth’s surface {approximately} as the inner spherical surface: a  r  r  6378 km
 6.378 106 m (= Earth’s mean equatorial radius), the height h (above the surface of the Earth)
of the ionosphere is: h  100 km  105 m (  a ) → b = a + h  6.478 x 106 m.
c
    1 for h  a .
For the n = 0 Schumann resonances: 0  
 a  12 h 
  1: 01 
c 2
 a  12 h 

f 01 
01
 10.5 Hz
2
  2 : 02 
c 6
 a  12 h 

f 02 
02
 18.3 Hz
2
  3 : 03 
c 12
 a  12 h 

f 03 
03
 25.7 Hz
2
  4 : 04 
c 20
 a  12 h 

f 04 
04
 33.2 Hz
2
  5 : 05 
c 30
 a  12 h 

f 05 
05
 46.7 Hz
2
n.b. For the n = 1
Schumann resonances:
f1  1.5 KHz
(. . . etc.)
The n = 0 Schumann resonances in the Earth-ionosphere cavity manifest themselves as peaks
in the noise power spectrum in the VLF (Very Low Frequency) portion of the EM spectrum →
VLF EM standing waves in the spherical cavity of the Earth-ionosphere system!!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede

Schumann resonances in the Earth-ionosphere cavity are excited by the radial E -field
component of lightning discharges (the frequency component of EM waves produced by
lightning at these Schumann resonance frequencies).
Lightning discharges (anywhere on Earth) contain a wide spectrum of frequencies of EM
radiation – the frequency components f01, f02, f03, f04, . . excite these resonant modes – the Earth
literally “rings like a bell” at these frequencies!!! The n = 0 Schumann resonances are the
lowest-lying normal modes of the Earth-ionosphere cavity.
Schumann resonances were first definitively observed in 1960. (M. Balser and C.A. Wagner,
Nature 188, 638 (1960)).
 Nikola Tesla may have observed them before 1900!!! (Before the ionosphere was known to
even exist!!!) He also estimated the lowest modal frequency to be f01 ~ 6 Hz!!!
n = 0 Schumann Resonances:
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
The observed Schumann resonance frequencies are systematically lower than predicted,
 1 i 

(primarily) due to damping effects:  2  02 1 
where Q = Quality factor  0 = “Q”

Q 


of resonance, and  = width at half maximum of power spectrum:
The Earth’s surface is also not perfectly conducting.
Seawater conductivity  C  0.1 Siemens!!
Neither is the ionosphere! → Ionosphere’s conductivity  C  104  107 Siemens

On July 9, 1962, a nuclear explosion (EMP) detonated at high altitude (400 km) over
Johnson Island in the Pacific {Test Shot: Starfish Prime, Operation Dominic I}.
- Measurably affected the Earth’s ionosphere and radiation belts on a world-wide scale!
- Sudden decrease of ~ 3 – 5% in Schumann frequencies – increase in height of ionosphere!
- Change in height of ionosphere: h  h  h  2   0.03  0.05  R  400  600 km !!!
- Height changes decayed away after ~ several hours.
- Artificial radiation belts lasted several years!


Note that # of lightning strikes, (e.g. in tropics) is strongly correlated to average temperature.
 Scientists have used Schumann resonances & monthly mean magnetic field strengths to
monitor lightning rates and thus monitor monthly temperatures – they all correlate very well!!!
Monitoring Schumann Resonances → Global Thermometer → useful for Global Warming
studies!!
Earth Coordinate System:
f0 
c
    1
2  a  12 h 
E0   0
(north – south)
1
P  cos   (up – down)
r2
1
B0   P1  cos   (east – west)
r

1
S0   3 P  cos   P1  cos   ˆ (north – south)
r
Er0  
 
For the n = 0 modes of Schumann Resonances:



E  rˆ
(up – down) B  ˆ (east – west) S  ˆ (north – south)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
We can observe Schumann resonances right here in town / @ UIUC!! Use e.g. Gibson P-90
single-coil electric guitar pickup  LP 90  10 Henrys, ~10K turns #42AWG copper wire  for
detector of Schumann waves and a spectrum analyzer (e.g. HP 3562A Dynamic Signal Analyzer)
– read out the HP 3562A into PC via GPIB.
 Orientation/alignment of Gibson P-90 electric guitar pickup is important – want axis of

pickup aligned  B  ˆ (i.e. oriented east – west) as shown in figure below. n.b. only this
orientation yielded Schumann-type resonance signals {also tried 2 other 90o orientations
{up-down} and {north-south} but observed no signal(s) for Schumann resonances for these.}
Electric guitar PU’s are very sensitive – e.g. they can easily detect car / bus traffic on street
below from 6105 ESB (6th Floor Lab) – can easily see car/bus signal from PU on a ‘scope!!!
n.b. PU housed in 4 closed, grounded aluminum sheet-metal box to suppress electric noise.
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 10.5
Prof. Steven Errede
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15