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Lattice structures of the earth magnetic field
 
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5 - The approach for the basic field

5.1 - Basic oscillations

On account of the previous consideration the conclusion lies near that the magnetic field of the earth owns a symmetrical structure with regard to a three-axle ellipsoid. In this structure appear certain angles which are integer parts of 360 degrees.

If one looks away from the real field-generating elements in the earth inside and concentrates merely upon the external field reaching round the earth, so it allows the following approach:
The whole field of the earth can be explained, after the Huygens principle (Huygens - Dutch physicists, in 1629-1695), by using a spectrum of discreet frequencies and worldwide fixed "source points". Namely as a sum of an assemblage from standing spatial waves, the so-called basic oscillations (elementary waves). These spread from the "source points", and induce by superposition of all oscillations - as a steady state - the earth field.

 

The beginning occurs on the base of oscillations on or around a ball

Examples of oscillation possibilities:

 

 sine wave  cosine wave  wave
     
Sine Cosine  

 

Sine or cosine = oscillation = wave
 
This is valid for physical oscillations:

Formula:      f·λ = c     (frequency multiplied with wavelength is equal to speed of light)

 

 

5.2 - The theorie

To determine the real source points of the field and the basic oscillations, it requires, as already mentioned, a two-dimensional fourier analysis (see chapter 8). The illustration with a qualitative attempt however at first is enough here:
 
 basic oscillation   Visualise a sphere, around a standing wave has established. By analogy to the Bohr nuclear model, if one understands the running electron around as a wave by De Broglie.
Then merely an integer number of oscillations fits round the whole circumference.
.


n · λ ∼ 360° = 2π
Illustration 5.1 - basic oscillation    
     
 basic oscillation and angle   The wavelength is proportional to the angle Alpha:

λ ∼ α
Illustration 5.1.1 - basic oscillation and angle    
 
 
Condition for n oscillations around a ball:   n · α = 2π

 

A standing wave around a ball can be interpreted physically as a steady state

A ball owns 3 degrees of freedom

Only for illustration: As a source point serves the North Pole (picture 5.1). If one lets proceed the basic oscillation from North Pole through South Pole again run to the North Pole, the second wave arises around the equator. For these first both oscillations exists a mathematical draught, that is suited for a representation, namely the spherical harmonics.

Because the ball is three-dimensional, however, still another degree of freedom exists here, and thus can form, vertically to both basic oscillations, still an other standing wave. Then this runs radial – from the centre outgoing.

 

 

5.3 - Spherical harmonics

Standing waves on a ball surface are called spherical harmonics. There exist 3 kinds of oscillation forms.
(See also Torges „Geodäsie", page 41-43.)
 
 zonal spheric harmonics   Zonal spherical harmonics depend merely on the degree of latitude

sin φ
cos φ
Illustration 5.2 - Zonal spherical harmonics    
     
 sectorial spheric harmonics   Sectoral spherical harmonics depend merely on the degree of longitude

sin λ
cos λ
Illustration 5.3 - Sektoral spherical harmonics    
     
 tesseral spheric harmonics   Tesseral spherical harmonics depend on the degree of latitude and on the degree of longitude

sinφ·sinλ
sinφ·cosλ
cosφ·sinλ
cosφ·cosλ
Illustration 5.4 - Tesseral spherical harmonics    

 

 

5.4 - Addition of sine waves

Spherical harmonics can be explained as two sine - or cosine waves which stand vertically on each other and mutually interference themselves. Two such waves can be added according to the following qualitative rules:
     
 ero-Grid from two sine waves   The zero points of both waves are transfered on the consideration level So the the zero grid occurs.
Illustration 5.51 Zero-Grid from two sine waves    
     
 Addition of two sine waves   1) + and + produces +
2) - and - produces -
3) + and - produces 0

As to be seen, fields with different algebraic signs produce respectively different states. There exist three oscillation states:
positively (+), negatively (-), neutrally (0).

Illustration 5.5 - Addition of two sine waves    
     
 Origin of the basic field   It is striking that all zero fields lie diagonal to each other. If one connects now the zero fields with each other, the accompanying picture 5.6 arises.

In the farther course this grid-like (red) oscillation structure is called basic field. Then the producing (blue) sine waves are called basic oscillations.

Illustration 5.6 - Origin of the basic field    

 

 

5.5 - The basic field

If one interferences two sine waves, which stand vertically on each other, quantitatively so the illustrations 5.8 and 5.9 arise. Clearly is to be seen the exact grid forming, as well as the alternate polarity of the single fields.
 
 The basic field   The superposition of two - vertically on each other standing - waves produces the known grid pattern with the alternate polarities of the grid fields. (Illustration 5.8 and 5.9)

Here is seen that the field maxima appear as point-shaped in the middle of the squares, while the lines consist of zero values - by analogy to the Chladni-sound figures.
Illustration 5.8 - The basic field    
     
 the basic field by 3D view   Mathematical seen basic fields or tesseral spherical harmonics can be explained by the multiplication of two sine or cosine waves.

It originate in such terms how they appear in the equation by Gauß and Weber (
chapter 2.7). See also chapter 10.3
Illustration 5.9 - The intensity of the basic field by 3D view    
 

 

basic field = grid = two-dimensional oscillation structure
 
 
Well is also to be seen in illustration 5.9 that in each case two generated grid fields again prove a (generated) oscillation. This permits two views of the grid:

1) The generated grid is described in the level of the basic oscillations
2) The generated grid is described in the grid level itself

 
Possibly still a second grid cam be marked here. Namely the maximum grid: It connects the field maxima (minima) with each other and shows the extremalen course of the field.
 
Because a complete square grid on the surface of a sphere cannot be realised, so the oscillation systems are formed like the geographic grid system. There always exist two poles. Then the accompanying meridians and circles of latitude form the grid system.
 
Pictures on radar base, which show unambiguously basic fields or tesseral spherical harmonics on the solar surface, have originated from the solar satellite Soho. There the question arise whether the magnetic field of the earth also such oscillation structures has produced.
In the beginning of the 50s last century Dr. med. Ernst Hartmann described a grid system which proceeds in the magnetic north-south-direction. This Hartmann-grid is treated in the chapter 11 more in detail. With the basic field model an attempt is given to describe the Hartmann-grid as a magnetic tesseral spherical harmonic.
     
 Chladni - Figures   The Chladni-sound figures form an Analogon here. If one irradiates a sandy coated metal plate with sound waves, standing waves on the plate ocurrs in the resonance case. Then along the zero values the sand remains lying and the typical oscillation figures appear.

(see also: „Physik" from Gerthsen, Kneser, Vogel – chapter 4.1.5 – Eigenschwingungen deformierbarer Körper)
Illustration 5.7 - Chladni - Figures    
 
 
The superposition of two sine waves can be shown three-dimensional also like in illustration 5.10:
 

 basic oscillations and basic field

 
Illustration 5.10 - Basic oscillations and basic field
 
 
While the mathematical concept of the spherical harmonic does not ask for the cause of the oscillation field, the underlying waves must be incorporated in case of the physical consideration. The concept of the basic field performs this. The basic field is defined by the basic oscillations.
 
The term basic field is as a physical equivalent
to the concept of the mathematical
tesseral spherical harmonic

 


    Tesseral sherical harmonics
    = Product of two oscillations
    = 2 vertically waves standing on each other
    =
grid = basic field
    
= two-dimensional oscillation structure

 

 

5.6 - Huygens Principle
(in addition to the book)

After chapter 5.2 a ball owns 3 degrees of freedom. 2 degrees of freedom would be covered by the use of spherical harmonics. The third degree of freedom is still absent: the radial direction. In addition the understanding of a physical representation is required, with which the expansion of physical waves can be described: the Huygens principle.

 

 Huygens principle The Huygens principle goes out from a source S, which generates uniformly wave fronts in all directions.
To receive the resultant wave front in the point P, however, it is not necassary to look the whole propagation from S.

 

 Huygens principle The Huygens principle says that every point (O) of a wave front can be looked as a starting point of a new wave, the so-called elementary wave (grey).
The situation of the resultant wave front (P) arises by overlapping (superposition) of all elementary waves

 

The wave origins (O) deliver by superposition of the elementary waves the resultant wave front (P). In three dimensions elementary waves are spherical, in two dimensions circularly.

 

 Waves around a ball wave around a ball:

extrema of the wave
= source
= wave origins


1 oscillation = 2 sources

 

 Huygens principle Oscillation state for one oscillation after the Huygenschen principle with a maximum (wave mountain) as a source (green)

Besides, the
minimum fronts (blue) and the maximum fronts (red) are the elementary waves

 

In the previous picture the oscillation situation is shown in the cross section for one oscillation. In the following picture the oscillation situation is shown for n oscillations.

 

 interference of the basic waves Going out from the source points P, after the Huygens principle, around every source point arises concentric circles (elementary waves) of minimum zones (red circles) and maximum zones (blue circles).

Because a stationary state exists, the wave fronts are stationary in the spatial situation. The superposition of the elementary waves occurs according to the same rules like already in chapter 5.4 described.

 

From the interference overlapping of positive wave fronts arises Oscillation maxima (with plus marked) and in the following as positive poles called
From the interference overlapping of negative wave fronts arises Oscillation minima (with minus marked) and in the following as negative poles called

namely where several wave fronts form an intersection or an area of concentration

By the superposition of positive and negative elementary waves also form zero Pole

The oscillation extrema, resulted by the superposition, lie again on ball surfaces which the generating ball (e.g., the earth) sketched concentrically, in the following layers = L called

Between these extreme layers exist zero Poles which also lie on concentric ball bowls - in the following zero walls called
 
 radial standing wave It turns out this simplistic view of the resultant field:
black sketched lines = extreme lines = layers
magenta lines = zero lines = zero walls

The generated layers L form a radial standing wave

 

The mathematically, physical inquiry of the layers, also the quantitative regulation, occurs in Kapitel 12

 

 

5.7 - Stratification structure, oscillation structure
(in addition to the book)

One standing wave on a ball generates a rotation-symmetrical spatial structure:

 

 rotation symmetric 1  rotation symmetric 2
The poles lie circularly on concentric balls

The zero walls form concentric cones
and concentric balls
 zero areas

 

Stratification structure = from one standing wave generated radial stratification structure

 

Two standing waves on a ball generate a radial grid-shaped structure:

 

 layers  space angle  cube

 

There arise two view possibilities:

 

The zero surfaces form the walls of a grid-shaped radial oscillation system
The poles lie in the centre of the zero cube
 cube system
 
The poles also form a grid-shaped radial oscillation system like a molecular grid (e.g., NaCl) in the following pole grid called
The pole connections behave like sticks which swing at both ends freel
 grid system

 

Space grid = from two standing waves generated radial stratification structure

 

Oscillation structure = sum of all possible space grids around a ball

 

Remark:
The stratification structure generated by one wave is
identical with the stratification structure that two waves generate.

 

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