How to calculate amplification of a cone?

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First off very professional looking project. The sensitivity you talk about is why they use parabolas for distance listening. Though some here don't believe.

I've no problem with parabolas, but they are pretty large (to give the same gain as a horn) and also need accurate construction as a horn does.

There were sound reasons (no pun intended) why they used horns more than parabolas during WW2 - there have been a few TV programmes on during the last few years, which have shown some of the sites used, and what's still there now.
 

Tom Scott (a quite smart young person in my opinion) in UK, made this video.

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I've no problem with parabolas, but they are pretty large (to give the same gain as a horn) and also need accurate construction as a horn does.

The gain is not worth a thing if the voices aren't able to be understood. The electronic amplification makes up for the loss of gain. The accuracy of the dish is overcome by getting the mic point right.

There were sound reasons (no pun intended) why they used horns more than parabolas during WW2

I do agree with you there but not for the reason you're thinking. Those WW2 horns were looking to hear the drone of an airplane, not what Hitler was saying. The drone was a low frequency wave that traveled far and would travel well in the horn.
 
there's another type of highly directional microphone that's simple to build and has a very narrow beamwidth, called an "organ pipe" microphone. you use 37 sections of straight (1/4" id) aluminum tubing, arranged in a spiral/hexagonal pattern. the tubing sections are cut in one inch increments with the longest being 36" and the shortest 1" (to make 37 sections, the last two are both cut to 1". once the bundle of tubing has been glued together a funnel of the correct size to mate with the outer diameter of the tubing bundle is put on at the bottom, with an electret element in the nose of the funnel. i'll try to find the original magazine article this weekend
 

Which was exactly the same thing the parabolas were doing as well?, I don't see your point?.

For low frequencies you need a HUGE horn, just as you need a HUGE parabola - a horn of easily useable dimensions would be perfect for speech frequencies.

The war time detectors tended to be parabolas for long distance ('somethings' coming from the rough direction of France) as they were generally fixed, due to their massive size. With steerable horns to give accurate locations, and track the aircraft so you could intercept.
 
Watch video. After doing lots of reading & research online I learn the long straight section of horn adds high Hz to sound plus horn is about 50% lighter weight. A straight side horn like my cardboard horn will not have much high Hz sound. England made long straight horns they were listening for low Hz German aircraft engines 20 miles away.

 
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Today I built a 3 piece horn each section is, 12", 13', 16" total 41" long. I am having trouble finding a wire with a plug that will solid to speaker then plug into the TV or computer speaker amp. Speaker on this horn is 1.5w 25 ohms. Horn makes no sound connected to TV. Horn works good connected to computer speaker 1w amp. I have 2 of these 1.5w speakers with speaker only connected to computer sound is very hard to hear but horn makes it louder but 1w is not much. I can see horn makes sound louder. First section of the horn is a cardboard tube 1.625" diameter 12" long. Second section is square 13" long. Third section 16" long, 14" wide. If I can get some 41" cardboard I might build a 41" horn must to compare 1 piece horn to 3 piece horn. Cardboard kinks when bent I need different building material. Good I had balsa wood scraps left over from model airplane building days 20 years ago. I saved head phone wires to solid to speaker but wires are tiny and solid does not stick. This was an 8 hour project. If I had taken the time to put radius in square corners inside the 2nd section and bottom of 3rd section I wonder if horn will work better.











 
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You might notice, that your nice little animated GIF at the bottom is an actual 'horn' where as your cardboard one is made with straight sides so isn't really a 'horn' at all, and won't work anywhere near as well.
 
You might notice, that your nice little animated GIF at the bottom is an actual 'horn' where as your cardboard one is made with straight sides so isn't really a 'horn' at all, and won't work anywhere near as well.

If I had flexible material to work with I could build a real horn. Here is a drawing of my horn & drawing of animated GIF horn drawn to scale. Information online says, the long small diameter horn section adds lots of high Hz and small low Hz near the end, low Hz travels much better long distance while high Hz is lost. This was my hurricane rainy day project it was FUN something to do while being a prisoner in the house. Learning project too.

I found pictures where college students built horns by cutting paper into strips then glue 100s. together in tube shaped each slightly larger than the next. All these paper tubes telescope together glued in place to form the diameter & length they want. Very cool idea 30 students all making paper tubes could build a horn fairly quick the final wall thickness of the horn was about 1/8" very strong shape.

Look what military did, photo shows easy to make cone on the end of a jet engine. 4 metal tubes telescope together and welded.





 
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one characteristic of acoustics is that pressure and velocity are analogous to voltage and current. you can model acoustic structures using similar rules to those of antenna and RF transmission line models. for instance you can model an air column in a conduit in a similar way you would model a transmission line.
you might want to look at this: https://en.wikipedia.org/wiki/Acoustic_transmission_line
 
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