Alex Cros said:
... My reasoning is as follows: the EM wave induces an alternating current in the wire which in its turn is connected to the speaker which produces the sound signal. ...
The substantial answer to your query has (I think?) been given. The frequency of the EM wave is much higher than audio frequency. So the AC in the wire would drive a suitable loudspeaker at supersonic frequencies. The audio frequencies, which are "carried" by the radio wave, are "modulated" onto it and need to be "demodulated" in some way to produce an AF current which could produce sound in a speaker.
As Marcusl nearly said, you don't need to add much to your design: a diode is enough, if your radio wave is simple AM. Selectivity, as Marcusl did say, is an issue, if you want to choose which station to listen to. If you have a strong local signal (as I do), it is like listening to someone near to you with others talking in the background, but you don't have much choice. If you try to listen to someone else, loudmouth just keeps talking on top!
What I actually wanted to comment on, was the "very long antenna", picked up by Berkeman with his 30m "long wire antenna". Presumably your VLA was intended to capture as much of the EM wave as possible, to give enough power/current to drive the speaker. And indeed as boys (or girls) we were urged to use as long a piece of wire as we could for the antenna (or in my case, aerial) of our crystal sets.
But your conclusion from this may not be accurate. The current from an antenna does not simply increase with length. A "short" antenna has a very high impedance and can supply little current. As it gets longer, the impedance falls to a minimum when it is 1/4 wavelength long. This is usually taken as the optimum length. As you further increase the length, the impedance rises again up to 1/2 λ, then falls to another minimum at 3/4 λ and so on.
The principle, "go longer, young man" works for our crystal sets, because the medium wave AM band starts at about 180m wavelength. So any antenna less than about 45m is actually short of the optimum for any station. For my favourite station, then the BBC Home Service on 200kHz long wave band, the optimum length single wire antenna is 375m . Very few of us had anything near long enough to be an optimum antenna, so increasing length would improve it.
If you do go on increasing the length well beyond 1/4 λ, then the effect on your received current will depend on the impedance of your detector circuit. If you don't adjust this to account for the variation of antenna impedance, you could decrease the signal by increasing the length.
As length increases, the directional sensitivity also changes. So unless you change the orientation, you could increase the antenna length and make the antenna less sensitive to waves from the particular direction you need.
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BTWs
Nice to join in v.happy reminiscences about crystal sets (mine had a Ge diode rather than a crystal, but I think it was a point contact diode - OA90). They depended on AM and would not resolve the more complex modulation systems around today. UK is threatening to switch off AM now that the evil DAB is taking over. Then crystal sets will be no more.
Sophiecentaur celebrates the limitation on the range of frequencies we can hear, but is this right? Am I to be grateful that I can't hear half the range of frequencies that a young person can? It does not seem to be an advantage to me. Perhaps if I could hear up to 100kHz, I'd be annoyed by all the bat clicks? If dogs can hear ultrasonic sounds (is that a self-contradictory term?) presumably there should be something to hear that makes these sounds. WikiP suggests small rodents make such sounds, but don't say how. So perhaps I'd be totally confused by hearing small rodents? But perhaps I'd be good at finding the punctures in my bike tyre?
Now if my hearing were even better (or for SC, worse) and I could hear up to 200kHz or more, I don't know what else I'd hear. But I don't think it would be radio stations. They broadcast radio waves, not sound waves. I don't know what VLF and ULF transmissions are around, but I believe they happen and I've never heard one AFAIK. Since they are very penetrative and long range, I'd have thought we would hear them, if it were ONLY a question of frequency range.
If, as SC mentions, potentially we could use the long "short" wire to drive an acoustic transducer and convert the radio wave into an acoustic wave (very distorted I suspect.) , I don't think we'd hear the AM sound. As has been discussed in other threads, AM is not like the beats produced by adding signals of different frequencies. The only frequencies present in a broadcast 198 kHz AM radio signal are in the range 189 kHz to 207 kHz, inaudible even to dogs and children. No audio frequency radio waves are present.
