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Date: Thu, 01 Apr 2010 01:25:50 +0200
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Subject: Re: LF: Ionospheric VLF propagation
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Paul Nicholson wrote:

>
> Piotr wrote:
> > I will also try to program these formulae
>
> It's a challenge - those Legendre functions with complex
> parameters...  there's an expression involving hypergeometric
> functions and some code for that in 'Numerical Recipes in C',
> either that or (very!) carefully transcribe/port the code in
> the paper's appendix.
>
Dear Paul, LF group

Yes, you are right - it is some kind of challenge to  calculate Legendre 
functions with a complez
degree - well, at least because you rather do not do it on , say, 
everyday basis :)
Indeed,  it  is well known that you can  compute Legendre functions 
using  2F1 function of a complex argument
and further, usingasymptotic expansion but  one must be careful with  
some values of theta and  modulus of  "nu "
( i use  notation from Lowenfels paper )   The paper has a Matlab code  
which i  am not even going to 'digest'
I do fortran  programming and i prefer  to do it  'from the scratch'  
except that  , of course, i have  subroutines for
Legendre polynomials  with  integer degree and order  which  (as it has 
appeared) were  needed  - I have found  Jones and Burke paper
[Journal of Physics: A : Mathematical and General; Vol.23, 
3159-3168(1990) ] where you can find a compact formulae
obtained by integration  etc... those  summations /formulae are really 
compact and have very elegant (mathematically)  form
so  one deals with rather straightforward programming. Paul, i have a 
pdf  copy of that article and if you wish i can send it to you
directly as attachment (540kb)

On other issues  related to Stefan  VLF experiment...  to make a long 
story a short one... i was reading Lowenfels paper
and particulary did not like the numerical fits of attenuation constants 
alpha and of phase velocity ( chapter 5.3)
those fits involve polynomials up to 12th order! it is not good from 
numerical point of view to use such a high degree etc..
anyway, i was looking for papers where  i could find those values  based 
on some experimental data ( i did find it  for Stefan freq
8.97 kHz ) with MUCH simpler  fits.as i was interested only in one freq 
and not going into ELF  regions.
as a 'side product' of my search i have found an excellent review paper 
"ELF and VLF radio waves" by  Barr, Jones
and Rodger.  published in " Journal of Atmospheric and Solar-Terrestrial 
Physics" Vol. 62, 1689-1718(2000)
some  info upon VLF antennae... for Stefan , for his consideration..
in 1989 a balloon ( not kite ;)  )  lofted antenna 3.8 kilometers  long  
was used at f =25.3khz ,
 later it was also used at 104  Hz (ELF)  with radiated pwr of 40mW 
which in terms of efficiency , was
corresponding  to 1.6 Watt radiated vs 1 MW input TX pwr
now something which should make Stefan  being even more satisfied with 
his experiments
In 1993 , the loop wire was placed  through a tunnel  1.2 km long  and  
going up along the mountain
 through which the tunnel was made  the apex  was 600 m above the tunnel 
center. this system was radiating  75 mW for every kilowatt of
input pwr at 10 khz ;  signal was detected at 200 km from TX place  i.e. 
tunnel :)
now , comes really interesting  experiment.. there was an unsuccessful 
attempt to deploy  a long wire ( a very very long wire)
from the Space Shuttle to generate electric power from its motion in the 
earth's magnetic field and to  radiate ELF signal. after
19.7 km !!! wire in space it has been fractured but  the pwr generated 
was greater than expected, however no signal has been radiated.
  it was 1996 mission.

                                                                                 
73, piotr , sq7mpj
qth; Lodz /jo91rs/