Taryn Wilton, of Westland High School, asks :-
What is the function of the bell shaped end of a musical instrument such as a trombone?
Neville Fletcher, a physicist and author of the book "The Physics of Musical Instruments", responded.
A uniform straight tube closed at one end has natural frequencies of resonance very nearly in the ratio 1: 3 : 5 : 7 : 9 etc -- the odd harmonics.
The flared bell on brass instruments has several purposes, one of which is to raise the natural frequencies of the horn to something closer to 1.4 : 4 : 6 : 8 : 10 etc. Except for the lowest frequency, these form a complete harmonic series (2 : 3 : 4 : 5 etc) starting an octave higher than those of the original closed cylinder. They are thus nearly the same as the frequencies of an open cylindrical pipe (or of a conical pipe) of the same length as the trombone.
The lowest mode is not used for normal playing, but the others give good tone and a conveniently spaced set of musical intervals which can be filled in by using the trombone slide (or the valves on other instruments).
The other purpose of the bell is to couple the air column more efficiently to the outside world and thus make the sound louder.