Brass Instruments: Why Bells Allow Efficient Energy Transfer

[michaelp7]

Hi everyone,

I'm working on a presentation for a physics class, and I've run into some difficulty. Our group is trying to explain why the bells on brass instruments allow more efficient energy transfer into the environment. I've tried a number of sources, including the book by Barry Parker and that by Rossing Moore & Wheeler, as well as the following sites:

http://www.victor-victrola.com/Fundamentals of Horns.htm
http://www.jstor.org/stable/84052

(The first is non-mathematical but on-topic. The second is somewhat dense, and I couldn't bring its results into a more digestible or pertinent framework.)

If anyone knows of an accessible mathematical explanation of this topic, I would really appreciate it. Apologies if this counts as homework-- I wasn't sure how to classify the question.

 

[Curl]

From the second paper you published, it seems like the material has the right stiffness so the natural frequency is somewhat close to what you want to hear from a bell.

 

[AlephZero]

The basic issue here is impedance matching. If you have a wave traveling through a medium with a sudden large change of impedance, most lf the energy in the wave will be reflected back and only a small part transmitted. However if you want to maike a loud sound, you want most the energy to be transmitted out of the instrument, not reflected back inside it.

The fact that sound is reflected is sort of obvious if the wave is gonig from low to high impedance (the limiting case is 100% of te sound being reflected back from a solid object) but it's not so obvious that the same thing happens when the impedence suddenly decreases, which is what happens at the open end of cylindrical pipe.

This might help, and the demonstrations are fairly easy to set up (e.g. use two ropes of different thickness). One problem with undestanding acoustics is that you can't actually see what the air is doing!
http://paws.kettering.edu/~drussell/Demos/reflect/reflect.html

This is a big site on the acoustics of musical instruments, but (as you might expect from a university research group) it tends to assume you already know the basics of acoustic theory:
http://www.phys.unsw.edu.au/jw/brassacoustics.html
http://www.phys.unsw.edu.au/jw/z.html

The basic equations for reflected and tranmitted waves have much wider application than musical instruments - try websites on acoustics, rather than sites specifically about music.

 

[maimonides]

One standard textbook on musical acoustics is "The physics of musical instruments" by Fletcher and Rossing. You might try there (and it has lots of references).
Btw: from what I remember, the bell has more to do with achieving a harmonic series of resonances than with efficient energy transfer.

 

[sophiecentaur]

If you look at the standard picture of the standing wave in a pipe with one end closed at resonance, you get a node at the closed end and an antinode at the open end. But the actual position of this open end is not well defined - the so-called End Effect. The position of this 'end' will vary with the pitch of the note and the diameter of the pipe. So, the simple notion of the pipe being λ/4 long doesn't really hold. However, if you add another half wavelength of pipe (exactly) you will get another resonance. The open pipe is a good example of a situation in which the Overtones (odd numbers of half wavelengths) don't coincide with the harmonics of a fundamental. To get a nice sounding note, you need the appropriate harmonics to be resonate at the same time as the note you are trying to blow. To achieve this, you need to fool around with the openend of the pipe so that the 'effective' open end of the pipe is different for all the wanted harmonics. A bell at the end seems to do the trick.
Otoh, matching is also important, if you want to be able to blow a loud sound. A tapered horn also achieves impedance matching as it results in a bigger area of air in the end of the tube 'pushing' against the surrounding air. A bigger piston / loudspeaker cone will have the same effect of allowing more power to be coupled into the air.
Horns are used in non-resonant sound sources as well as in brass instruments.

 

[AlephZero]

If you look at the standard picture of the standing wave in a pipe with one end closed at resonance, you get a node at the closed end and an antinode at the open end. But the actual position of this open end is not well defined - the so-called End Effect. The position of this 'end' will vary with the pitch of the note and the diameter of the pipe. So, the simple notion of the pipe being λ/4 long doesn't really hold.

I assume that by "standard picture" you mean a cylindrincal pipe.

Most wind instruments that are closed at one end are conical pipes, not particularly because of impedance matching or end effects, but because the harmonic series from a conical pipe includes all the harmonics, while the series from a cylindrical pipe closed at one end only includes the odd harmonics. The only "mainstream" western european wind instrument with a cylindrical pipe closed at one end is the clarinet.

The basic form of the pressure distribution in a conical pipe is p = (sin λx)/x, compared with p = cos λx for a cylindrical pipe closed at one end.

Incidentally, the reason brass instruments tend to have large bells and woodwind instruments do not becomes rather obvious, if you try to make a large bell out of wood

 

[sophiecentaur]

My "standard picture" was for an organ pipe (the model for many lab experiments).
I seem to remember hearing that a cornet has a cylindrical bore but a trumpet has a conical bore. They certainly sound different; could that be why?

Is it really true that the bell-end of a brass instrument is 'only' there for the timbre and not to get a good loud sound out? A bugle that couldn't be heard in battle would not be much use, whatever the sound of its note.

 

[AlephZero]

My "standard picture" was for an organ pipe (the model for many lab experiments).

That's what I thought - though real organ pipes are not all cylindrical. There are scale drawings between about page 20-30 of this http://www.laukhuff.de/images/stories/downloads/katalog/11.pdf if you are interested.

I seem to remember hearing that a cornet has a cylindrical bore but a trumpet has a conical bore. They certainly sound different; could that be why?

The last part of the cornet bore is wider diameter and has a bigger cone angle than a trumpet. But the a bigger percentage of the complete length of a cornet is cylindrical compared with a trumpet. The shape of the bells are also different. So there isn't really a simple answer to that question.

Comparing separate pictures can be misleading, because the lengths of the instruments are different (though the length of the tube is the same) so pictures are often not to the same scale. From left to right, flugelhorn, trumpet, cornet:

http://www.inglesm.com/wp-content/uploads/2008/04/trumpet_cornet_bugle_p.jpg

Is it really true that the bell-end of a brass instrument is 'only' there for the timbre and not to get a good loud sound out? A bugle that couldn't be heard in battle would not be much use, whatever the sound of its note.

You get both at the same time. The sound radiation becomes very efficient for the higher harmonics when the wavelength is similar to the diameter of the bell, so the high harmonics are effectively amplified compared with the low ones. This also coincides with the most sensitive part of the human hearing range, around 3kHz.

 

[michaelp7]

Wow, thanks for all the replies. That definitely helps.