PiMath.de Planetary Systems of the Earth 1
Classic Systems
 
  Copyright ©: Klaus Piontzik  
     
 German Version    
German Version    

2.1 - Spherical harmonics

A standing wave around a sphere can be interpreted as a stationary state. Thus, each state of a wave is spatially fixed. The question now is: how many waves fit around a globe?

In classical mechanics, degrees of freedom is the number of freely selectable, independent movements of a system
A rigid body in space has a degree of freedom f = 6, because you can move the body in three independent directions and rotate in three independent planes.

Because a sphere is rotationally symmetric, so rotations are irrelevant. A ball has therefore 3 freedoms regarding a wave propagation. Therefore, 3 inde-pendent waves around the globe are possible.
Due to the spherical shape, the three freedoms can be represented as spherical coordinates.

Example earth:

1) A wave starts from North Pole via the South Pole to the North Pole

2) The second wave runs around the equator

3) The third standing wave runs radially - based from the center

There exists a mathematical concept namely for the first two examples, that is suitable for a representation - the spherical harmonics.

Standing waves on the surface of a sphere can be treated as spherical harmon-ics.

There are 3 types:

zonal spherical harmonic   Zonal spherical harmonics only depend on the latitude

sin φ
cos φ
Illustration 2.1.1 - zonal spherical harmonic    
     
sectorial spherical harmonic   Sectorial spherical harmonics depend only by the degree of longitude

sin λ
cos λ
Illustration 2.1.2 - sectorial spherical harmonic    
     
tesseral spherical harmonic   Tesseral spherical harmon-ics depend of the latitude and the longitude

sinφĚsinλ
sinφĚcosλ
cosφĚsinλ
cosφĚcosλ
Illustration 2.1.3 - tesseral spherical harmonic    

 

2.1.1 - Definition: tesseral spherical harmonic
= product of two oscillations of 2 perpendicular standing waves
= grid

 

Comment:

A complete route in terms of square grid on a sphere can not materialize.

Only grid systems are developed, which are designed as the geographic grid system. There are always two poles. The corresponding "meridians" and "circle of latitude" then make the grid.



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 Planetare Systeme 1

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Der Autor - Klaus Piontzik