What part of "mathematically" are you having trouble understanding?
Yes, and many, many times.
What I have never done is have occasion to use the inverse square law in an RF link calculation.
-- Jim Pennino Remove .spam.sux to reply.
Not even then, if the beam is required to have finite energy flux [1] and be a solution of the Helmholtz equation (or even the paraxial wave equation).
I think a better phrasing of your original point would have been "The inverse square law applies to the far field of a radiator." Don't like the inverse square law? Stay in the near field, where beams can be collimated rather than becoming spherical waves, or you can get faster than 1/r^2 fall-off from higher-order multipole sources.
But while many (including myself) are picking on the details, your original point that high frequencies can allow highly directive transmission is spot on. Also the point made by another that the Earth-ionosphere waveguide avoids the inverse square law too. Tough luck for the frequencies in the middle - all they're good for is broadcast transmission when you want more bandwidth than you're allowed at the lower frequencies.
[1] Infinite energy flux can deliver unto you infinite plane waves and Bessel beams, which are perfectly collimated, but not physically realisable.-- Timo
Your results must have been very inaccurate.
The limited aperture dimensions of metal dishes, horns, splash-plates, helices, Yagis, etc., used for terrestrial microwave links, in terms of the wavelength, means beam widths are generally measured in degrees for frequencies below EHF. Furthermore, intensive frequency re-use has led to extensive use of absorber-lined tubs shrouding dishes to constrain sidelobe/backlobe levels - this technique can increase the edge taper, increasing the beamwidth for a given aperture size.
A beamwidth of one degree, or for that matter, any angle other than zero requires use of the inverse square law for PFD, or an inverse law for field strength.
Chris
And you ignore distance?
(Microwave, not RF)
John
Most reasonable response yet.
The geometry of the inverse square law is predicated upon radiation from a point source.
The field from a real radiator approaches following the inverse square law as the distance increases such that the divergence approaches that of a point source.
Given a good enough beam former, that distance may become an astronomical distance.
The collimating characteristics of many antennas are such that you can not be on this planet and be far enough away to have enough divergence to approximate inverse square law behavior.
-- Jim Pennino Remove .spam.sux to reply.
Sorry ... you have a screw loose.
Chris
I was on the Dew line Extension project in the Aleutians and worked with the tropo radios, etc. Five Watts (the exciter) into a 60 foot parabolic antenna gave us usable communications at 100+ miles, with occasional frequent deep fades. Twenty-five kW got rid of the fade effects.
Math and nature sometimes fit nicely together, but are not the same. Any "REAL" beam does not have a 100% math description. To do that you have to know it all, and only trolls claim such. So your perfect beam does not exist. Of course you can think of one, you can almost make one, but in the end all math used in describing the real world is an approximation.
All beams have some divergence, and somewhat obey the inverse square law. Just try and aim any laserbeam on a target 3 miles away, and you will find out. Did that once, to make a "really" straight path at a 3 mile airfield. At 3 miles the laserbeam was about 3-6 yards diameter, and moving all over the place,because a mathematically rigid beam also does not exist.
Sorry you can't understand junior high school geometry.
Let's try it this way:
The inverse square law is predicated upon radiation from a point source.
Any radiator with real area doesn't become a point until you are an infinite distance from it.
Therefor, for a real area at less than an infinite distance, the inverse square law is an approximation to what really happens.
The quality of the approximation depends on the real area and the distance from it.
Got it now?
-- Jim Pennino Remove .spam.sux to reply.
only if it has infinite width.
consider the difraction that when your perfectly collimated beam passes through an aperature that exactly matches its size
contrast that with what happens when it doesn't.
Bye. Jasen
- snip -
Radiation from an aperture antenna is a matter of diffraction. At any point of inspection, the field strength (and therefore the PFD) is the result of integrating contributions from the distribution of current in the aperture. If it's a one-dimensional aperture, like a thin wire dipole, then the integral is over that one dimension; if it's a two-dimensional aperture, like a dish, the integral is over both dimensions. This is a little more complicated than 'high school geometry' but the calculus involved can often be kept quite simple - it all depends on the form of the current distribution.
Away from the antenna, simple geometry indicates that the path lengths over which the different components act are different, so the result of the integral will depend on the distance between the point of inspection and the antenna. However, it has been found for many practical implementations of dish, etc., that when the difference between path lengths from the centre and edges of the (normal) aperture is one sixteenth of the wavelength then the result is essentially independent of the distance. The distance at which this condition is achieved is the well-known yardstick 2D^2/lambda, and, for example, this equates to 360 metres for a 3 metre diameter dish working at 6 GHz; perhaps the largest size used for terrestrial microwave links. Therefore, radiation pattern measurements need to be carried out using a measurement range at least this long for the patterns to be generally applicable.
Of course you have the prerogative to apply different interpretations of the physics to your own experiments and if your interpretation brings theory and practice together for the general case then it has some validity. What I've described above is a small part of the physics that's actually used in the design of communication links, broadcasting, radar, etc. There are well-known simplifications like 'ray optics' that neglect diffraction phenomena, but these are strictly applicable only to the case of infinitesimal wavelength, so they might be useable in optics but they cause approximation when applied to systems with apertures of only 60 wavelengths - the above example. When the radiation from a dish is predicted using ray optics, the ray paths from the feed to the reflector all end up parallel in the aperture so this gives the impression of a 'collimated' parallel-sided beam ... but this is a major approximation and is certainly not what is measured in practice.
Chris
The inverse square law is predicated upon a spherical wave front.
A true spherical wave front is obtained only from a point source.
Point sources don't exist in the real world.
At sufficient distance, the difference between the real wave front and a true spherical wave front approaches the limits of measurment, and therefor the inverse square law provides a value indistiguisable by MEASUREMENT from the "real" value at sufficient distance.
The use of the inverse square law is always an approximation in the real world, though granted sometimes a very good approximation.
The inverse square law is a THEORETICAL relationship and 2D^2/lambda is a PRACTICAL rule of thumb.
Got the point yet?
-- Jim Pennino Remove .spam.sux to reply.
Which is quite possible mathematically.
Concider what happens when the perfectly collimated beam impacts a half gallon of Chunky Monkey ice cream.
However, I'm not sure what either has to do with a wave front propogating in free space.
-- Jim Pennino Remove .spam.sux to reply.
The problem seems to be, assuming photons have something to do with it: in particular, the energy of photons. If what you say is true, it would be a purely classical effect: ELF waves are larger than higher frequency waves, and it is _plausible_ that they need proportionally more power to excite a larger region of space to have a sufficient number of waves to support a signal.
I didn't say that is necessarily true, but it has a better chance of being true than your argument about photons.
Congratulations. Out of messages 2-7 in this thread, yours was the first which at least tried to contribute some constructive information, instead of a random insult: that is, unless "bullshit", "snip crap" and "doesn't know what he is talking about" now constitute constructive information when written by approved posters.
Geez... I'm beginning to sound like Tom Potter -- but you guys should be ashamed of yourselves. The Wikipedia article does mention some efficiency issues, and you could have discussed whether these were intrinsic to any use of ELF, or just to the transmitting stations used by the USN.
Of course the issue of photon energy seems like a completely misguided idea... although it _might_ have been nice to discuss whether there were any basis for the implicit idea of "constant information per carrier photon", or not -- and why.
Since when does being on the right side of most discussions (let's say for the sake of argument) justify churlish, not to say loutish, behavior. You gentlemen can contribute much more than that. There are other readers besides the OP.
?
contains some interesting looking pictures.
Come on! This was a "Radium" question! If you don't know that that means "open season", then search the archives a bit!
OK. We are all properly admonished. NOW!!! How about we start discussing the "Jewish Question" !!!! :-)
Makes sense to me. It's like saying that cutting a suit down the middle makes an instantly more narrow suit, but that there is nothing inherently narrow about a half-suit: it could be half a size 58, vs. a child's size.
ote:
world
Has everything to do with it. Mathematics is not "reality". In the real world, things are not "infinite" or "point sources" or other mathematical concepts. So for the most part this argument is stupid. But you were the one saying that the inverse square law did not apply to an anisotropic beam. We say it pretty much did at a distance. Then you say only at "infinity" and then we have to answer how close of an approximation to reality do you want your math to be? So you say how far if not infinity? Astronomical distances? And we say "could be", and we note the distance needed is that which takes you into the "far field" of the source. So how far away is that? Well how large is the source? If the source is "infinite" then a "perfectly collimated" beam is mathematically possible. But who cares since "infinite objects" are not possible. All you are doing is arguing how many angels can stand on the head of a pin!
The bottom line is all of this is that radiation sources creating propagating radiation can be replaced by an "equivalent aperture" at some point and as noted, an aperture source actually has a transform relationship between the aperture "source" and the propagating beam. At a sufficient distance the expansion of the beam is always dictated by those mathematics. Hence all arguments of "collimated beams" not following inverse square laws at some location is invalid. The questions of how closely are the laws followed or how far you have to be away are merely practical fine points. It is your "natural philosophy" that were are taking issue with here.
world
Bingo, you finally got the point, almost.
Theoretical mathematics is not reality and the inverse square law is theoretical mathematics.
-- Jim Pennino Remove .spam.sux to reply.
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