Impressum  Copyright © Klaus Piontzik  
German Version 
15  Magnetic layers and frequencies 1
15.1  Distance consideration
If one looks closer to the distances (from the distance table  chapter 12.2), one recognises they all form multiple of the radius R=6355758,426 m = L_{o}. This is the basic hull radius. So the distance values can be also shown so: 
k  1  2  3  4  5  6  7  8  
n  m  
1  1  L_{o}  3L_{o}  5L_{o}  7L_{o}  9L_{o}  11L_{o}  13L_{o}  15L_{o} 
15.2  Frequency consideration
If one understands the distances l' from the distance table as a wavelength of the accompanying hull respectivly of the oscillation layer, then frequencies can be also assigned to all distances. 
15.2.1  Case 1
A possibility exists to
orientate by the illustration 6.2. The double diameter is
equivalent to the wavelength. Then it is worth λ = 4R = 4l' and: 
The distance values for n=1 are used from the distance table for the following considerations. 
k  1  2  3  4  5  6  7  8  
n  m  
1  1  11,79  3,93  2,358  1,68  1,31  1,07  0,907  0,786  Hz 
6355,76  19067,28  31778,79  44490,31  57201,83  69913,34  82624,86  95336,38  Km 
It is to be noticed here that for k=2 the half Schumann frequency originates. 
15.2.2  Case 2
A farther possibility
exists to take the whole diameter of the accompanying
hull respectively of the oscillation layer as a
wavelength. So λ = 2R = 2l' and: 
From the distance table the distance values are used again for n=1. 
k  1  2  3  4  5  6  7  8  
n  m  
1  1  23,584  7,861  4,716  3,369  2,62  2,143  1,814  1,572  Hz 
6355,76  19067,28  31778,79  44490,31  57201,83  69913,34  82624,86  95336,38  Km 
It is to be noticed here that for k=2 the Schumann frequency originates. However, this signifies also that the Schumann frequency is already included in the spectrum of the earthmagnetic frequencies. All appearing frequencies are virtually doublings of the already found basic frequencies so explaining harmonics. 
15.2.3  Case 3
For the end here exist still the possibility to interpret the radius of the accompanying hull respectively of the oscillation layer as a wavelength. So λ = R = l' and: 
From the distance table the distance values are used again for n=1. 
k  1  2  3  4  5  6  7  8  
n  m  
1  1  47,17  15,72  9,43  6,74  5,24  4,29  3,63  3,14  Hz 
6355,76  19067,28  31778,79  44490,31  57201,83  69913,34  82624,86  95336,38  Km 
All appearing frequencies are virtually quads of the already found basic frequencies, shows merely harmonics. 
15.2.4  Complete reflection
All together the distance and frequency values can be shown for n=1 then as follows: 
k  1  2  3  4  5  6  7  8  
n  m  
1  1  3/2f_{S}  1/2f_{S}  3/10 f_{S}  3/14 f_{S}  1/6 f_{S}  3/22 f_{S}  3/26 f_{S}  1/10 f_{S} 
1/f_{0}  1/3f_{0}  1/5f_{0}  1/7f_{0}  1/9f_{0}  1/11f_{0}  1/13f_{0}  1/15f_{0}  
L_{o}  3L_{o}  5L_{o}  7L_{o}  9L_{o}  11L_{o}  13L_{o}  15L_{o} 
The consequence is that all appearing frequencies in case 1 are representable as rational fraction multiples of certain basic frequencies. 
In the book the table of the extreme layers from chapter 12.2 is transformed by the described allocation (case 1) to a frequency table. 
15.3  Earth frequency and Schumann frequency
If one looks closer to
the frequencies, a connection can be derived between
earth frequency and Schumann frequency. Easier this succeeds with the frequencies f_{H2}. There exists since a common divisor frequency. 
For k=1 and k=2 exists a
common divisor frequency with: 23,584 = 3 · 7,861 23,584 = 2 · 11,792 The frequency 7,861 Hz = f_{S} corresponds to the Schumann freuqency Then the frequency can be disassembled for k=1, on the other hand, so: 
23,584 Hz =3f_{S} = 2f_{0} 
The general relation is (see chapter 15.2  case 2): 
And from it directly deducible: 
The Schumann frequency is already included in the spectrum of the earth frequencies 
From the equation for f_{H2} is also directly deducible: 
Furthermore directly deducible: 
The basic hull radius is only a little smaller then the pole or equator radius. Hence, the question arises here which dimensions f_{S} owns if the hull radius is substituted with R_{A} respectively. R_{P} 
The data of
the geodetic reference system WGS84 are: pole radius: R_{P}= 6356752 m equator radius: R_{A}= 6378137 m With c = 299792458 m/s as the speed of light arises for the single frequencies: 

For the earth frequencies:  
For the pole radius  = 11,7899 Hz  
For the equator radius  = 11,7503 Hz  
For the Schumann frequencies:  
For the pole radius  = 7,8602 Hz  
For the equator radius  = 7,8339 Hz 
Noteworthy here is still
the fact that the Schumann frequency, for historical
reasons, is won from a cavity resonator consideration.
(see in addition chapter 7.1) The concurrent derivation from the earth frequency, so, in the end, from the radii of the earth, shows that the measurements of this planet and the generated frequencies on it stands in a closed connection. 
15.4  An approximation for the equator frequencies
If one takes, in the geophysical sciences usual, middle radius of 6,371 km, one receives for f_{S} a value from 47,0558 hertz. This lies near with 47 hertz. 
If one calculates on 47 hertz back, the radius arises to 6,378,563 km. This is about 400 metres bigger than the equator radius. Approximately one can say: 
47 Hz =6f_{S} = 4f_{0} 
From it arises: 
f_{S} = 47:6 Hz = 7,8333... Hz 
f_{0} = 47:4 Hz = 11,75 Hz 
This can be used as a good approximation for the equator radius frequencies. 
15.5  Common frequencies
A general connection can
be derived by the equation from chapter 15.3 still
between earth frequency and Schumann frequency. The relation f_{0}/3 = f_{S}/2 is preserved also if one multiplies the equation by a rational number, so to a fraction. With it arises in general: 
n and k are elements of the natural numbers (1, 2, 3, 4...) 
A complete spectrum of
common frequency f_{os} originates from the equation, so
that the earth frequency and the Schumann frequency stand
with each other in a functional connection. Systematic using of the parametres n and k in the equation leads to a table in which then all common frequencies are included. The calculation of the table values occurs about the corrected basic frequency from chapter 11.2. A rounded value from f_{o} = 11,792 Hz is used. 
common frequencies for n<17, k<9
k  1  2  3  4  5  6  7  8  
n  
1  3,9307  1,9653  1,3102  0,9827  0,7861  0,6551  0,5615  0,4913  Hz 
2  7,8613  3,9307  2,6204  1,9653  1,5723  1,3102  1,123  0,9827  Hz 
3  11,792  5,896  3,9307  2,948  2,3584  1,9653  1,6846  1,474  Hz 
4  15,7227  7,861  5,2409  3,9307  3,1445  2,6204  2,2461  1,9653  Hz 
5  19,6533  9,8267  6,5511  4,9133  3,9307  3,2756  2,8076  2,4567  Hz 
6  23,584  11,792  7,8613  5,896  4,7168  3,9307  3,3691  2,948  Hz 
7  27,5147  13,7573  9,1716  6,8787  5,503  4,5858  3,9307  3,4393  Hz 
8  31,4453  15,7227  10,4818  7,8613  6,2891  5,2409  4,4922  3,9307  Hz 
9  35,376  17,688  11,792  8,844  7,0752  5,896  5,0537  4,422  Hz 
10  39,3067  19,6533  13,1022  9,8267  7,8613  6,5511  5,6152  4,9133  Hz 
11  43,2373  21,6187  14,4124  10,8093  8,6475  7,2062  6,1768  5,4047  Hz 
12  47,168  23,584  15,7227  11,792  9,4336  7,8613  6,7383  5,896  Hz 
13  51,0987  25,5493  17,0329  12,7747  10,2197  8,5164  7,2998  6,3873  Hz 
14  55,0293  27,5147  18,3431  13,7573  11,0059  9,1716  7,8613  6,8787  Hz 
15  58,96  29,48  19,6533  14,74  11,792  9,8267  8,4229  7,37  Hz 
16  62,8907  31,4453  20,9636  15,7227  12,5781  10,4818  8,9844  7,8613  Hz 
Besides, the first
smallest natural frequency is fo/3 = fs/2 =
3,9307 Hz that earth frequency and Schumann's
frequency own together. The table with the common frequencies also contains all harmonics of the Schumann frequency. With it the Schumann spectrum (from chapter 7) belongs to the frequency spectrum of the table 46. I.e. the whole Schumann spectrum can be derived from the earth frequency. 
15.6  The sferic frequencies
Another connection from the equation of chapter 15.5 arises if bigger n, so higher frequency, are looked: 
common frequencies for 1055<n<12673, k<9
k  1  2  3  4  5  6  7  8  
n  
1056  4150,784  2075,392  1383,595  1037,696  830,157  691,797  592,969  518,848  Hz 
1584  6226,176  3113,088  2075,392  1556,544  1245,235  1037,696  889,454  778,272  Hz 
2112  8301,568  4150,784  2767,189  2075,392  1660,314  1383,595  1185,938  1037,696  Hz 
2640  10376,96  5188,48  3458,987  2594,24  2075,392  1729,493  1482,423  1297,12  Hz 
3168  12452,35  6226,176  4150,784  3113,088  2490,47  2075,392  1778,907  1556,544  Hz 
7128  28017,79  14008,89  9339,264  7004,448  5603,558  4669,632  4002,542  3502,224  Hz 
12672  49809,41  24904,7  16603,14  12452,35  9961,882  8301,568  7115,63  6226,176  Hz 
The spectral maxima of
the sferic frequencies are given in general, as follows
in very much narrowbanded areas (see "Sferics" by Hans Baumer page 285): 
4150,84 Hz  6226,26 Hz  8301,26 Hz  10377,10 Hz  12452,52 Hz  28018,17 Hz  49810,08 Hz 
The comparison of the
Sferic spectrum with the table values delivers a good
correspondence of the values. About the whole spectrum
seen the maximal mistake lies less than 0.7 hertz. In his book " the cosmic octave " (page 39) produces Cousto a connection between Sferic frequency and sidereal day. The comparison with the earth rotation delivers a minimum difference of about 3 hertz and grows with increasing frequency up to a maximum value of about 36 hertz. The increasing difference clearly shows that the frequency calculation about the sidereal day only is suited as an approximate value. As by the equation from chapter 15.5 and the generated table for 1055<n<12673 have been shown that the Sferic frequencies, with enough exactness, are included in the spectrum of the common earth frequencies and Schumann frequencies. 
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