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/
|
