Why do megaphones still sound terrible?

[PointyOintment]

Isn't the idea to get the message as far as you can in the direction it is pointed?

If so, then reduce the bandwidth to what you really need so you can get more power out.  Second, make it as loud as you can, so compress, smash the &^&$ out of the signal so the average is much closer to the peaks.

How about discretizing the frequencies in the audio, so that you run a spectrum analysis on the input signal, with (say) 200 Hz bins, and then reconstruct the signal for output with only one central frequency peak per bin? That would, I think, reduce the total bandwidth of the output sound by a lot, allowing more power at each of the output frequencies. Plus, free autotune.

Basically, transducer efficiency is inversely proportional to bandwidth. That is true for speakers as well as aerials.  Thus, fidelity and efficiency seldom come in the same package unless it's a big package with multiple radiating elements for different frequencies. To create a small handheld unit with a loud output from a limited power source, well, go figure what has to be left out.

The transducer used is coupled to a horn, same shape  as a trumpet, which gives excellent coupling to the surrounding air but has a natural resonance, such that it will only handle a narrow range of frequencies.

Would it be possible to engineer some grooves/ridges or something into the horn to make it an acoustic metamaterial that presents a different effective horn to each of the above-described different discrete frequencies? I'm imagining it would work somewhat like the coupled tubes of different sizes in this thesis [PDF], where the each size of grooves/ridges would interact with one of the discrete frequencies, so each frequency would see its own optimal horn shape.