The Function of the Pipe and Bell -- Inside the Air Column

The tapering air column of the trumpet is the other partner in the collaboration that generates a musical tone. The history of our understanding of waves in tapered ducts (or "horns" as they are customarily called by acousticians) is a long one, and rather peculiar in that many times a basic understanding was gained and then lost, until a later researcher was led to rediscover the ideas all over again. On the other hand, a physicist who looks back over the history of his subject is struck by the prominent place that was originally occupied by musical acoustics. In fact it was one of the important sources of information about the nature of the physical world and a prime source of intellectual stimulation.

During the lifetime of Bach, the founding masters of theoretical physics took fourfold inspiration from the studies of the motions of the planets, the flow of heat, the flow of fluids, and, last but by no means least, the vibrations of musical strings and air columns. Already in the 1760s Bernoulli, Euler, and Lagrange succeeded in formulating the basic equation that enables us to make predictions about the behavior of sound waves in ducts of varying cross section. These early theoreticians discussed the behavior of sound not only in cylinders and cones but even in the family of so-called Bessel horns, to which we now know the trumpets are closely related. It is a curious quirk of history that this family of horns came to take its name from a nineteenth-century German astronomer simply because certain parts of the mathematical description of Bessel horn acoustics is based on mathematical results obtained in the course of purely astronomical calculations! Bernoulli and his contemporaries apparently did not consciously recognize that musical instruments of their day approximated the Bessel shape; this was simply the next shape following the cylinder and cone in the hierarchy of mathematical complexity.

This pioneering work, by men whose names are revered today by mathematicians, physicists, and engineers alike, lay buried for nearly a century. In 1838, the distinguished English mathematician, George Green, rediscovered the earlier results in connection with his studies of water waves in canals of gradually varying width and depth. This work by Green arose in response to a pressing practical problem, the erosion of the banks of England's transportation canals by waves set up by the canal boats as well as by tidal effects. It is in his work that we find justification for drawing an analogy between real trumpets and the water trumpet that was described in earlier pages.

In 1876 the German, Pochhammer, independently derived the equation and learned the properties of its most important solutions. In 1873 Lord Rayleigh published a brief paper on certain electrical phenomena in which he used a startlingly modern "operator method" of analysis that later he put to use in 1916 when he published a sophisticated and ingenious article on the acoustics of ducts of varying cross section. This paper included the derivation of the basic "horn equation" as an especially simple case. The implications of Rayleigh's 1916 paper have proved to be most helpful for some of us who have followed him. Finally, prehistory ends in 1919 when A. G. Webster published his derivation of the equation, and seemingly the world of science was ready to pay attention. Ever since, acousticians have referred to the basic horn equation as "Webster's horn equation," in defiance of its true history.

In the period that followed Webster, considerable practical use was made of horn acoustics in the design of phonographs and loudspeakers and for many other purposes. The subject of horn acoustics reached its contemporary maturity in the classic papers of 1946 by Vincent Salmon, whence has spring a spate of papers by many authors which has continued ever since. Readers wishing to become acquainted with the whole subject would do well to peruse the detailed and scholarly review article published in 1967 by Edward Eisner.⁶ It is this paper with its extensive bibliographical commentary that I have used as a formal basis for my remarks in the preceding two paragraphs.

Let us digress here briefly to learn what is the nature of the Bessel horn shape and its relation to actual musical instruments. The mathematical formula that gives the diameter D of a bell in terms of the distance y from its large open end is

D = B/(y + y₀)^a

where y₀ and B are chosen to give proper diameters at the large and small ends, and a is the "flare parameter" that dominates the acoustical behavior of the air column. This parameter differs from one instrument to another, depending on its mouthpiece and leader-pipe design. Trumpet bells as far back as those made by William Bull⁷ in the seventeenth century correspond closely to the shapes of Bessel horns having values of a that lie between the limits of 0.5 and 0.65. It is interesting to realize that the bell shapes that have evolved by the traditional combining of eye-pleasing artistry with practical experience are notably similar to one another in their acoustical description.

It is worthwhile to extend our digression enough to look briefly at the difference between loudspeaker horns and musical horns. The design requirements of a loudspeaker horn are of a sort that demand the best efficiency in radiating sound from a small source out into the air, whereas in musical instruments we will find quite the contrary requirements are laid upon the design - the bell flare of a brass instrument must be designed to save energy inside of the horn, giving strongly marked standing waves (sloshings of the air) at very well-defined natural frequencies.

Returning now to the musical side of "horn" acoustics, we find that Bouasse made essentially no use of the Webster equation. He gives an elegant and original derivation for it and solves it for Bessel horns and for the mathematically simpler exponential horns that find a certain application in loudspeaker design. Bouasse then drops the equation and makes no further reference to it.⁸ In dealing with brass instruments, Bouasse restricted himself to an admirably clear exposition of the acoustics of what he called "cylindro-conical" composite air columns, which have been intensively studied by other as well, before and since. These air columns have however only a rough and qualitative acoustical resemblance to the musical brasses. Bouasse seemed to be quite unaware of the extremely important role played by the mouthpiece of a brass instrument in the overall fixing of the natural frequencies of the air column. This kept him from resolving many of the serious questions that he was however perceptive enough to raise. We will take up the subject of mouthpieces and their relation to the rest of the instrument at several points later on in this chapter.

Bouasse's contemporary, the British physicist E. G. Richardson, needs mention in our account chiefly because his widely read book, The Acoustics of Orchestral Instruments,⁹ was the origin of a commonly held impression that trumpet bells are of what is known as exponential form. He also promulgated some peculiar notions about the flow of air in the mouthpieces of brass instruments. It is regrettable that such errors crept into the work of a distinguished scientist who made numerous contributions to other parts of musical acoustics.

An interesting document relating to the acoustics of brass instruments is an extremely detailed patent obtained in 1958 by Earle Kent of C. G. Conn, Ltd.¹⁰ He achieved correct tuning by joining a sequence of "catenoidal" segments instead of using a single flowing Bessel-like shape. Segmentally proportioned bores are mathematically bound to give irregular intonation patterns unless they are counterbalanced by additional irregularities of taper or cross section. Practical examples of all these matters are thoroughly discussed in the patent, which also describes the way an electronic computer may be used to aid the design process.

One other worker who has been an active contributor to the science of brass instrument air columns is Frederick Young of Carnegie-Mellon University. He has published a series of significant papers beginning in 1960.¹¹ He represents the shapes of real brass instrument air columns by a cascade of very short segments, each with an assigned taper and flare. The smallness of the segments permits him to represent the properties of the smoothly varying horn with reasonably good accuracy because it avoids mathematically introduced irregularities of the sort that are deliberately accepted in a design (such as Kent's) based on the choice of a limited number of segments.

In 1970 William Cardwell obtained a patent for a particularly simple type of brass instrument design involving the ingenious use of a single segment of catenoidal bell, attached on the one side to a cylindrical main bore, and on the other to a short, rapidly flaring bell-end.¹² This design, which is somewhat related to that of Kent, was worked out independently, and proves very practical for the construction of higher-keyed instruments in Eb and F. The two patents by Kent and Cardwell make interesting reading because of the insight they give into the practical problems o designing a brass instrument.

During the year 1967-68 Erik Jansson of the Speech Transmission Laboratory of the Royal Institute of Technology in Stockholm worked with me in Cleveland on a detailed study of air columns that are useful for musical instruments. This work, which was both theoretical and experimental, dealt with trumpet, trombone, and French horn bells. We unearthed a number of subtle relationships between our experiments and calculations that we could not clarify immediately. It is only recently that it has been possible to prepare complete reports on our results.¹³

An excellent source of information on brass instrument acoustics is to be found in the text and bibliography of the 1972 doctoral dissertation submitted by Klaus Wogram to the Technischen Universität in Braunschweig.¹⁴ Of particular interest here is the extensive reference to European research, which is, unhappily, unfamiliar to many in the English-speaking parts of the world.

This completes our overall survey of the historical side of air column acoustics, so that we are in a position to return to the interrupted account of the way we have come to understand the means whereby the air column governs the player's lips to produce a tone.

Trumpet Acoustics
Acoustical Preliminaries
The "Water Trumpet"-- An Analog to What Happens inside a Trumpet
The Function of the Player's Lips
The Function of the Pipe and Bell--Inside the Air Column
The Cooperation Needed for Musical Results
The Baroque Trumpet
The 'Internal' Spectrum of the Modern Trumpet
The 'Internal' Spectrum of the Baroque Trumpet
Relation of Internal to External Tone Color Spectrum
The Menke Trumpet
The Problem of Clean Attack
Mahillon in Retrospect
Conclusion
Bibliographic Notes