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Lattice structures of the earth magnetic field
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6 - Derivation of the wavelength

6.1 - Conditions

The basic condition after chapter 5.2 is that a whole number of oscillations fits once around the earth and thus forms a standing wave, as a steady state. In a first attempt counts to the surface:

Wavelength = circumference/number of oscillations

The wavelength corresponds to a certain distance on the circumference of the earth. In the following illustration 6.1 the situation is illustrated for an oscillation.
Both points
A and C are the end points of the circular arc s, which is at the same time also a wavelength l

 AC = s and  s = Lambda
If one looks at this situation from the earth centre, a certain angle a can be assigned to every oscillation.
 alpha = 2Pi/n
Then all together as a basis appears:
 Basic equation
For reasons of the illustration a factor has remained unnoticed, in the previous representation, which must be included now in the consideration: the ball figure of the earth.
It must be still taken into consideration that the wave propagation linearly takes place, and not along the curved earth surface. Hence, in the consequence it requires a modification of the basic equation.
 Wavelength and circumference
Illustration 6.1 - Wavelength and circumference



6.2 - The derivation

After the Huygens principle the points A, B, C in Illustration 6.1, so the extreme points, serve as source points of the standing wave.

Electromagnetic waves spread out spherically from a point and if point B is the starting point, corresponds the distance BDC = h to the way of the wave.

Because there is a steady state and it is enough to look the way of the wave from a source to the next source.
 2h = Lambda dash
The triangle MCD is right-angled in the point D and it counts:
 Equation for the sine in the triangle MCD
 Equation for h
Change after lambda dash produces the corrected wavelength:
 Equation for the corrected wavelength



6.3 - The frequency equation

The connection between wavelength and frequency is:
 connection between wavelength and frequency
With thus arises the equation for the earth frequencies:
 Equation for the earth frequencies
c stands for the speed of light, R for the radius of the earth, n n for the number of oscillations

Because the earth is, however, no perfect sphere, there to the poles flattened. Here exist two radii: the pole radius and the equator radius. And from it also result two (easily divergent of each other) basic frequencies.

The geodetic reference system WGS84 is based on a spheroid, so to a rotation ellipsoid. See in addition
chapter 17.1.
The data of the geodetic reference system WGS84 are:

Pole radius : 6356752 m
Equator radius : 6378137 m
Flattening : 1:298,2572

c = 299792458 m/s as the speed of light arises with n=1 for the single radii:

Pole radius : 11,79 Hertz
Equator radius : 11,75 Hertz
The appearing difference may seem unimportant at first sight. However, in case of farther consideration a significant difference arises. In the next chapter 7 this will become still clearly recognizable in the subject sferics.

Nikolas Tesla declared at his time that the earth frequency lies near 12 hertz.



6.4 - A relation to the pole axis of the earth

Another connection to the pole axis is given by the basic oscillation. If one declares for the first basic frequency the general solutions, these are:
n f Pole radius Equator radius
1  c/4r 11,79 Hz 11,75 Hz
2  c/(2r*sqrt(2)) 16,674 Hz 16,618 Hz
3  c/2r 23,58 Hz 23,5 Hz
For n=1 the general equation for the wavelength is: lambda = 4R
And one receives the identical result if a rod of the length of the Pole axis freely swung at both ends. As well as in the following illustration 6.2 shown.
 Earth and basic oscillation
Illustration 6.2 - Earth and basic oscillation
Here the analogy is a massive rod which both ends are free. If this rod is moved now into oscillation, the basic oscillation starting longitudinal flexural vibration corresponds to the picture 6.2.
(see also „Physik“ von Gerthsen, Kneser, Vogel – Kapitel 4.1.5)



6.5 - The frequencies of the earth

If one puts successively for n the values 1,2.3 ... in the above equation for the frequency, the basic frequencies of the earth arises. The following table contains the first 30 frequency oriented on the pole radius.


n Pole radius
1 11,7903 Hz
2 16,6740 Hz
3 23,5806 Hz
4 30,8095 Hz
5 38,1542 Hz
6 45,5542 Hz
7 52,9850 Hz
8 60,4350 Hz
9 67,8975 Hz
10 75,3688 Hz
11 82,8465 Hz
12 90,3289 Hz
13 97,8149 Hz
14 105,3038 Hz
15 112,7949 Hz
16 120,2880 Hz
17 127,7825 Hz
18 135,2783 Hz
19 142,7752 Hz
20 150,2730 Hz
21 157,7715 Hz
22 165,2708 Hz
23 172,7706 Hz
24 180,2709 Hz
25 187,7717 Hz
26 195,2729 Hz
27 202,7744 Hz
28 210,2762 Hz
29 217,7784 Hz
30 225,2807 Hz


To this basic frequency suitable harmonivs still appear, i.e. one must form in addition only the integer multiples.
(One calls harmonics as general integer multiples of an elective basic frequency)
In the book the frequencies are still declared which are oriented on the equator radius.



6.6 - A further derivation

In the book the derivation is still declared for a farther frequency equation.


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