Physics 36 / Music 36 Duke University Spring 2008 Handout 16
In designing a wind instrument, we'd like to be able to sound notes based on its different normal modes (as with a bugle, for instance), and to fill in notes between available modes by changing the effective length (by lengthening the instrument with valves or a slide, or shortening it by opening tone holes).
So the modes we use must differ by musically appropriate intervals, even as the overall length is varied. These criteria may be stated formally as the following two rules:
I. The frequency ratios of the various normal modes must be independent of the length of the instrument.
II. The higher mode frequencies must be integral multiples of a fundamental mode frequency.
It can be shown [A. H. Benade, J. Acoust. Soc. Am. 31, 137 (1959)] that all shapes satisfying criterion I. are among the Bessel horns, i.e. their cross-sectional areas S(x) at each position x along their lengths can be expressed as S(x) = c(x-xo)e, where e is called a "flare parameter".
Each such shape has a "dual" with cross section Sd(x) such that the product SSd is constant for all x. If the narrow end of the dual is closed, the two shapes will share the same set of normal mode frequencies (though not, in general, the same standing wave patterns or coupling to the air outside). Since S (proportional to xe) and Sd (proportional to x-e) have the same frequencies, then, we need consider only positive values of e.
| flare parameter e | bore type | dual |
| 0 | open cylinder | (no dual) |
| 0 | semi-closed cylinder | self-dual (e = -e) |
| 2 | cone | S(x) proportional to 1/x2 |
Bessel horn segments with 1 < e < 2 are used in wind instruments -- duals of this range (-1 > e > -2), especially, are used as bell shapes -- but not as primary bore shapes, because of their non-harmonically related normal modes.
Among the infinite number of Bessel horn shapes, only certain values of e turn out to satisfy criterion II. The smallest are: e = 0, 2, 7, 13, and 19.5 (and, of course, the corresponding duals).
| flare parameter e | S(x) | shape | 2nd register interval | nth mode frequency |
| 0 | indep. of x | cylinder, semi-closed
cylinder, open |
twelfth
octave |
(2n-1)v/4L
nv/2L |
| 2 | proportional to x2 | cone | octave | nv/2L |
| 7 | proportional to x7 | (see drawing) | fifth | (n+1)v/2L |
Consider examples of each of these shapes with lengths chosen so they share the same fundamental frequency.
| flare parameter e | bore | example | length | maximum diameter if d = 1/8 in at x = 2 in |
| 0 | open cylinder | flute | 2 ft | |
| 0 | closed cylinder | clarinet | 1 ft | |
| 2 | cone | oboe | 2 ft | 1.5 in |
| 7 | impractical horn | (none) | 4 ft | 705 ft |
[A one-foot-long e = 7 horn, sounding two octaves above the instruments in the table above, would still reach 66 inches in diameter for any reasonable "blowing end"!] e = 13, 19.5, et c., have even more rapid flares!
