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Re: LF: Ionospheric VLF propagation

To: <[email protected]>
Subject: Re: LF: Ionospheric VLF propagation
From: "Antonio Tagliavini" <[email protected]>
Date: Thu, 1 Apr 2010 01:47:20 +0200
In-reply-to: <[email protected]>
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Dear ultra-lowfers,
why not Mathematica?  It is a very powerful tool containing all special functions you need (e.g..: Legendre). No recipes, no subroutines, no Fortran...
Best regards
Antonio
 
----- Original Message -----
From: "Piotr Młynarski" <[email protected]>
Sent: Thursday, April 01, 2010 1:25 AM
Subject: Re: LF: Ionospheric VLF propagation

> 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/
>
>
>
>
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