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Subject: LF: VLF propagation (continued..)
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Hello "Dreamers" , LF Group,
A few weeks ago Paul Nicholson has brought our attention to a nice paper
dealing with a EM wave propagating in earth-ionosphere waveguide:

http://dspace.mit.edu/bitstream/handle/1721.1/16973/53712465.pdf?sequence=1

This work presents a set of equations describing E-field strength of an 
EM wave
propagating in a waveguide where its dispersive and dissipative properties
are taken into account. I have used these equations to calculate the E-field
 strength to add other(possibly better) way of calculating E-field with 
respect
 to some simple models and estimations already presented and discussed 
here,
on this group after Stefan, dk7fc 2nd VLF transmission.
 I will refer to original notation  of these equations as it can be seen
 via the pdf viewer instead of repeating it here, in a text-style form.
The leading equation is 5.2  with auxiliary 5.4; 5.5; 5.6 along
 with the attenuation and phase velocity fits presented in chapters 
5.3.1 and 5.3.2
Athough these equations may seem 'complicated' there is only one, 
computationally
 nontrivial issue i.e. calculation of Legendre function of a complex 
degree 'nu'.
However, the paper of Jones and Burke  [ J.Phys.A: Math. Gen. Vol.23 
(1990) pp.3159-3168]
entitled; "Zonal harmonic series expansions of Legendre functions and 
associated Legendre
functions"   provides formulae which can be easily programmed.
Now,  some numbers:  input data: f=8970 Hz,antenna current and height: 
0.48 Amp and 100 meters
tx-rx distance: 858.7 km, R(earth): 6370 km (Paul's reception of 2nd 
Stefan transmission)
I had to use the height of D-layer so i assumed its three values: 
60,70,80 km
and  obtained the following values of  effective E-field strength expressed
in microVolts/m
60 km   0.738
70 km   0.632
80 km   0.553

For those who are interested in more details of the present calculations 
some
auxilary  complex values:
{complex number z = a + b*i is denoted as (a,b) or (real,imaginary) }
for f=8.97kHz  "nu" = (349.34204, -0.39951862)
theta = 0.1348
Legendre function of complex degree nu:
P(nu)[-cos(theta)] = (-0.053431176, 0.17015796)
SIN(pi*nu) = (-1.6678783, 0.76733224)

In  1-2 forthcoming weekends i will try to prepare  more detailed 
description
of the calculational scheme including some graphs , fortran code etc... 
and will make it
 available via internet as pdf file

                                      73, Piotr, sq7mpj

qth: Lodz /jo91rs/