I've been reading up about the Austin-Cohen formula, originally
an empirical determination, valid only for long waves (> 200m):
signal strength proportional to exp( -alpha * r/sqrt(lamda))
where alpha is about 0.0015. This is an attenuation factor for
daylight propagation over water, and if we plug in the numbers
for 9kHz, we obtain an answer very close to 3db per 1000km,
which is the figure we've been using. There is reference to a
1919 paper by Cambridge mathematician G.N. Watson in which the
Austin-Cohen formula is derived theoretically for propagation
in the cavity formed between two spheres. Will have to look
for a copy of that paper (Proc Royal Society 95 p546, 1919,
I believe).
I found in another paper a formula for field strength at VLF.
It uses Legendre functions to model propagation in terms of
cavity modes, combined with an empirical model of attenuation.
Formula is too long to post here but the paper is at
http://dspace.mit.edu/bitstream/handle/1721.1/16973/53712465.pdf?sequence=1
Maybe I will program this if I get time, but a formula you
can run on a calculator is much nicer.
Very frustrating searching for useful info on the web. There
are many references but they all lead to journal or archive
sites where you have to pay a subscription to download, or a
large fee for single articles. Very annoying since we pay
for this research with our taxes and then have to pay again
to see the results.
Here is another paper which would be nice to have,
James R Wait
'A study of VLF field strength data both old and new'
Pure and applied physics, vol 41 sept 1958 pp73-85
First page is visible at
http://www.springerlink.com/content/p31412563642105w/
I think the introductory equation there is the one used by
Piotr. Both equations have the basic dependence of 1/sqrt(r)
for field strength appropriate for power confined to 2D expansion.
I plot here a comparison between the 'flat earth' formula
B = sqrt( 9.5e-21 * ERP/r) * exp( -r/a)
and Wait's spherical formula, both for 70km daytime D layer
and to concentrate on the geometry I've used the same attenuation
factor, 3db/1000km for both,
http://abelian.org/vlf/wait9-day.png
At 1mW ERP we expect 2.5fT (flat earth) or 2.1fT (sphere)
at 850km. I think either of these formulas is a good
match for our observations, considering combined error of ERP and
rxed flux density measurements. The spherical one is obviously
the more correct, geometrically.
At short range, we may have a mixing of multiple paths, so the
signal strength will differ from above. In mode theory terms
this is what is meant by the phrase 'uncontaminated by higher
modes' on Wait's page 1 - simply at long range, we just get a
single incident ray with variable attenuation but no multi-path
fading.
Well I think we've squeezed everything possible out of page one!
Here is a nice little essay to read on a tea break,
http://web.mit.edu/sts/pubs/pdfs/MIT_STS_WorkingPaper_37_Yeang.pdf
It describes a bygone era in which radio amateurs played a
significant role in scientific research into radio propagation.
--
Paul Nicholson
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