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From my limited experience with ham radio when I was a kid, I expect transmitting and receiving antennas to have lengths that are on the same order of magnitude as the wavelength, and in fact I recall having to mess around to compensate for the fact that a given antenna wouldn't be properly resonant over an entire frequency band. This also seems to match up with what we see with musical instruments, where, e.g., a saxophone's tube is half a wavelength and a clarinet's is a quarter.

For commercial FM radio with a frequency of 100 MHz, the wavelength is about 3 m, so I can believe that some of the receiving antennas I've seen are a half-wave or quarter wave. But for AM radio at 1000 kHz, the wavelength is 300 m, which is obviously not a practical length for a receiving antenna.

Can anyone explain this in physics terms, hopefully without making me break out my copy of Jackson and wade through pages of spherical harmonics? Does AM reception suffer from the length mismatch, e.g., by being less efficient? Does it benefit from it because it's so far off resonance that the frequency response is even across the whole band? Is there a dipole approximation that's valid for AM only? For both AM and FM? If the sensitivity is suppressed for the too-short antenna, is there some simple way to estimate the suppression factor, e.g., by assuming a Breit-Wigner shape for a resonance?

This question touched on this issue, but only tangentially, and the answers actually seem inconsistent with the observed facts about AM. Also related but not identical: Radio communication and antennas

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IIRC AM antennas need to be a loop whereas FM can be open. Perhaps this has something to do with it? –  Brandon Enright May 18 '13 at 21:53
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FWIW: en.wikipedia.org/wiki/Fractal_antenna –  Mike Dunlavey May 19 '13 at 1:50
    
Some possibly relevant material here: en.wikipedia.org/wiki/Antenna_%28radio%29#Resonant_antennas . In general, dipole radiation goes like $d^2\omega^4$, when the dipole is small compared to a wavelength. Out of the factor of $\omega^4$, an $\omega^2$ part is due to the mismatch between the size of the dipole and the wavelength, which makes the phase almost the same at both ends of the dipole. By reciprocity, I think something similar would hold for a receiving antenna. If $\omega$ is 100 times smaller than it "should" be for AM, it seems like you might pay a price of $100^2=10^4$. –  Ben Crowell May 19 '13 at 15:27
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Also check out en.wikipedia.org/wiki/Loop_antenna#Small_loops . It seems they're very inefficient but the loss is acceptable because the noise sources are quite high so the signal-to-noise ratio isn't affected much. –  Brandon Enright May 19 '13 at 17:00
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@BenCrowell no I meant high. The Wikipedia article suggests that for AM the noise floor can be 55db above thermal noise so even if the antenna has a 50db loss it doesn't have a significant effect on the signal-to-noise ratio. –  Brandon Enright May 19 '13 at 19:58
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http://www.antenna-theory.com/antennas/shortdipole.php is a website with useful info., including formulas.
To oversimplify, it seems to say that once the antenna is a tenth or less of the wavelength, the exact ratios don't matter so much. The antenna is inefficient, but it works for both sending and receiving. If you can detect the signal, of course you can amplify it as much as you want.

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I think putting this together with @BrandonEnright's comment, it's starting to make sense to me. When you receive, the antenna's lack of sensitivity cuts the signal strength, but it also cuts the noise, so S/N is OK. An inefficient low-power (receiving) system is not as bad as an equally inefficient high-power (transmitting) system. –  Ben Crowell May 19 '13 at 19:43
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The main reason is that in Ham radio you care about transmitting, in that case you need to make sure the antenna is in the right length so you get a standing wave inside the antenna. You can read about standing wave ratio here.

If you are just receiving then you could use any wire, loop antennas are practical for low frequency transmissions in AM, where you care about the change of amplitude.

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Hmmm...but don't exactly the same concerns apply to reception as to transmission, due to reciprocity? en.wikipedia.org/wiki/Antenna_%28radio%29#Reciprocity –  Ben Crowell May 19 '13 at 15:16
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Many ham antennas include coils that help the antenna appear to be the right length for the frequency in use, there are also trapped antennas there the coil will block frequencies above a specific point and the frequency drops the coil will allow the energy to pass to the antenna element on the other side so at high freq you have a shorter antenna and as you go down the antenna gets longer and longer.

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