loud trumpet...where does the energy come from?

I was thinking the other day about how loud trumpets and other horns can be, but it really seems like the sound energy going into them is considerably less that what comes out.
Sure, we say that the horn “magnifies” or maybe focuses the sound, but the energy has to come from somewhere. The only energy input is from the vibrating lips and breath, which without the horn seems considerably smaller. The horn is much louder even when it isn’t pointed toward the ear so its louder in all directions…much louder than the lips without the horn.

So am I wrong? What’s up with the increased energy? where does it come from? or more accurately, how does the lower energy imput get adjusted to higher energy output.

Here’s an extremely technical paper describing the physics of a trumpet:

http://mafija.fmf.uni-lj.si/seminar/files/2013_2014/The_physics_of_the_trumpet_(Bostjan_Berkopec).pdf

Here’s a more accessible paper with lots of visual aids on the same subject:

http://newt.phys.unsw.edu.au/jw/brassacoustics.html

Horns are very good at matching impedance. This is the bottom line of why they are so loud.
To expand. Most sound producing devices couple their energy to the air very poorly. The problem is that a vibrating mass - like the wooden top to a violin or guitar, a vibrating speaker cone etc, is made of a dense (relative to the air) material, and almost all of the energy you put into the system is spend making the radiator move, and very little of the energy couples into the air. The difference in material properties that matter are their impedance - and the thing that matters is that when you have a difference in impedance between the materials, you don’t get good energy transfer. The bigger the difference the worse things are.) A horn provides a gentle constant change of impedance along its length, matching the open air at one end, and a much lower impedance at the small end, which allows the actual sound source - the lips blowing out air - to couple better to the air in the horn.
The horn has resonant effects that make the whole production of a musical tone work well, and that helps the initial sound from the lips to be loud in the first place, so you get a double benefit. The sound from the lips isn’t just the feeble sound of someone blowing a raspberry, it is energising a resonant system, and that is also going to become much louder. The lips see rising and falling pressure in the horn throat that acts to make the lips vibrate at the frequency of the resonance, and thus put energy at the required frequency into the system. Controlling these resonances is the art of playing the horn, and (especially for a natural horn) part of how you play different notes.
For things like speaker systems, the important bit is the impedance matching function of the horn. Horns tend to be pretty directional at higher frequencies as well, so that helps further, although typically you try to design the horn to have a good spread - the wider the end of the horn the better here.

I can think of two answers:

  1. The sound that an instrument makes is magnified because the instrument is designed with resonant frequencies around the sound in question. My best non-sound example of this is sloshing in a bathtub. As any child realizes at some point, very small motions in a bathtub can result in a cascade of water because each small energy input reinforces the existing oscillating energy, causing the wave to grow each time.

When you go bbbpbpbpbp with your lips without a horn, each bpbp dissipates into the surrounding chaos of the air. When you do it into the horn, the chambers of the horn are the right size that the first bp echos back to your mouth in time to gain some energy from the next bp, and so on.

  1. There’s very little energy in the sound we hear from instruments, but it is energy that we are specifically adapted to sense.

Excellent YouTube video on exactly this topic from Mark Rober (he of recent glitter bomb fame): https://m.youtube.com/watch?v=pFEB0chiuJA

Sound doesn’t carry nearly as much energy as people think. When you use a lot of energy to make sound (for instance, using high-wattage speakers), the vast majority of that energy just gets turned into heat and wasted. So since sound-makers are so inefficient, there’s lots of room for improvement, and so a sound-maker that’s just a little more efficient can produce a lot more sound.

A trumpet is more efficient than a human voicebox, and so it can produce more sound from the same energy input (and correspondingly less heat, but the difference in heat is much less noticeable).

The numbers I’m familiar with say that a single instrument can’t create more than a few watts of sound energy, and that most instruments only turn about 1% of their energy input into sound energy.

A few more things to note:

[ul]
[li]Perceived loudness is a psychological quantity[/li][li]The trumpets range from F♯3 ~ (184.997 Hz) to F♯6 (1479.978 Hz) is in the band of the most loud perceived tones for humans. [/li][li]Trumpets waveform are pretty close to a offset square wave, which will have all odd harmonics, which will sound “louder” in some cases and cut through in other cases.[/li][/ul]