A. H. Benade

On the Propagation of Sound Waves in a Cylindrical Conduit

The series impedance and shunt admittance of an acoustic line is calculated from the linearized acoustic equations. Exact and limiting formulas for small and large tubes are provided for R, L, G, C, the real and imaginary parts of the characteristic impedance Z 0, as well as the phase velocity v and attenuation constant α. All results are presented in convenient form for quick computation on the basis of tables and graphs. A self‐consistent set of molecular data is presented. Accuracies of formulas and of the data are discussed in detail.

Equivalent circuits for conical waveguides

The acoustical properties of a cone of length L and small‐end radius a0 are shown to be representable via an equivalent circuit involving a pair of inertances, a transformer, and a nontapered duct that has the same length and small‐end radius as the cone to be represented. This is shunted at the small end by an inertance proportional to x0, which is the distance from this end to the (projected) apex of the cone. At the large end, there is a similar inertance proportional to the apical distance xe=x0+L. This inertance has a negative sign. The nature and correctness of the representation are demonstrated and a number of illustrative examples are presented, including a transformation of a clarinet (which is cylindrical) by means of a vent hole that gives it the basic playing properties of a saxophone (which is conical). The equivalent circuits are easy to calculate with and speak well to the intuition to suggest properties of the cone that are not otherwise very apparent.

Survey of impedance methods and a new piezo‐disk‐driven impedance head for air columns

The six major techniques used heretofore for the swept‐sinusoid measurement of air column input impedances are first reviewed, after which the basic theory of a new, impedance head design that requires no servo mechanism is outlined. In this simple device, a low‐cost piezoelectric disk is used as the primary flow excitation source, and the resulting response pressure is detected by a similarly inexpensive miniature electret microphone. Detailed experimental tests of the device are presented. Analysis of these tests shows that, up to about 5000 Hz, essentially no calibration corrections are needed; furthermore, the driver impedance is so high that its perturbations of the air column are experimentally negligible. A detailed mathematical analysis of the major finite‐impedance, displaced‐microphone, and higher‐mode perturbations associated with the driving disk itself are set forth, and an outline of general procedures for obtaining the absolute calibration of any air column impedance head is given.

Two‐ear correlation in the statistical sound fields of rooms

The normalized correlation function R( f ) connecting the signals at the left and right ears was measured in a reverberant room as a function of frequency. Head diffraction effects are made evident by comparing these data to the correlation function predicted and observed for two microphones separated by a headwidth (15 cm) of space and moving in the same room. In contrast to the familiar two‐mike sinc(δ) correlation (δ=kd), head diffraction gives a two‐ear correlation function which is well described by sinc(αδ)/[1+(βδ)4]1/2, where α=2.2 and β=0.5. The dominant effect of head diffraction is to lower  fD, the decoupling frequency [first zero of R( f )] to 500 Hz, about half the no‐head value. The denominator in the fitted correlation function represents the near total independence of the signals at the two ears above  fD. Thus with head diffraction the variance in a paired‐mike room averaged pressure amplitude estimate at around 500 Hz is already as small as that obtained sans head only at frequencies abo...

The clarinet spectrum: Theory and experiment

While a great deal is known about the oscillation and radiation mechanisms of the clarinet, an overall account has not been provided of the manner in which the tones of a clarinet are generated and then transmitted into the listening room. This article presents such an account for the clarinet in general. Room‐averaged spectra measurements are presented that are shown to be consistent with the implications of theory, with both measurement and theory showing a great deal of spectral regularity over the playing range. The fact that the tonehole lattice cuttoff frequency is essentially constant over the scale is of particular importance, making it possible to define a single semiempirical formula for the odd harmonics of all the notes in the scale and a closely related one for the even harmonics. Both of these formulas are fitted to the data by way of the same three adjustable parameters.

On Woodwind Instrument Bores

The properties of horns that are suitable for use in woodwinds are deduced from first principles. The cylindrical pipe and complete cone are shown to be the only shapes which satisfy these requirements exactly. The behavior of nearly perfect cylinders and almost complete cones is described, the influence of closed finger holes on the effective bore of an instrument is discussed, and the effect of the mouthpiece cavity is analyzed. Damping of the normal modes by the walls of the bore is shown to play a dominant role in the playing behavior and tone color of woodwinds, and various consequences are deduced.

Wave propagation in strongly curved ducts

A theoretical and experimental investigation has been made of wave propagation in the long‐wavelength limit in curved ducts of both rectangular and circular cross section. The two‐dimensional solution of the wave equation for propagation in curved rectangular bends is adapted for the case of a circular cross section. The wave admittance and phase velocity are computed in terms of integrals over the cross section of the duct, in which there is a radial variation in flow. Experiments to measure the wave admittance and phase velocity are carried out with semicircular segments from a baritone horn musical instrument tuning slide. Three air column arrangements are used; the curved duct assembly is placed in the center of the air column between segments of straight tubing, or antisymmetrically placed near the open or closed ends of the air column. The first‐mode resonance frequencies of the air column are measured, and the frequency shifts relative to a straight duct are used to compute the wave‐admittance and ...

The saxophone spectrum

The geometrical nature of the saxophone air column is outlined, along with a sketch of the theory of sound production and radiation of saxophones. The discussion is based upon a similar but somewhat more detailed discussion of the clarinet and its spectrum published recently by Benade and Kouzoupis [J. Acoust. Soc. Am. 83, 292–304 (1988)]. Room‐averaged spectra were measured for essentially all notes in the low and second registers of a B‐flat tenor and an E‐flat alto saxophone played at a mezzoforte level (so that the reed was beating steadily). It is shown that the observed spectra have a great regularity of pattern that is in good agreement with theoretical expectation. All amplitudes of spectral components of all notes fit the same spectrum envelope function, E(x)=[Nx/(1+x7)]1/2. Here, N is a normalizing constant and x=f/fb, where fb is a break frequency characteristic of the instrument. This break frequency is closely related to the instrument’s tonehole lattice cutoff frequency fc. In addition to th...

Measured End Corrections for Woodwind Tone‐holes

Impedance parameters of a tone‐hole system are expressible in terms of geometrical hole lengths L plus added end corrections. These were estimated from frequency measurements on resonators. Account was taken of boundary‐layer effects, which are small above 300 Hz. For a hole or tube of radius b, each correction is of the form ΔL=b⋅E, where E depends on the end geometry. For a tube whose flange has an outer radius (b+w), the measured end correction is Et=0.821−0.13[(w/b)+0.42]−64. A disk of radius r placed normal to the tube axis at a distance h from its end (e.g., a pad poised over its tone hole) gives Ed=0.61(r/b)0.18(b/h)0.39 for 1≤(r/b)≤4, provided Ed≥0.65. For a hole of radius b drilled through the wall of a pipe whose outside radius is R, the outer‐surface correction is E0=0.64[1+0.32ln(0.3R/b)] for 1.5≤(R/b)≤7. The inner‐surface correction for this hole is found using a doubly closed resonator tube of inside radius a and of variable length, into whose side is inserted an open side‐branch of variable...

Impedance measurement source and microphone proximity effects

The input impedance Zin as normally defined is the air column pressure response to a unit plane‐wave flow excitation. The experimental measurement of Zin for an arbitrary air column is affected by the nearfield interactions between an excitory source (capillary, annulus, membrane, etc.) and microphone. In addition to the plane‐wave component, the source produces a local disturbance (due to the evanescent modes) at the microphone. The net microphone pressure pm can be written in terms of the source strength us, the wave impedance R0 and radius a of the air column entryway, and wavenumber k: pm = usR0[(Zin/ R0)+jkaE]. The evanescent mode factor E is real and independent of frequency for ka<1. Its value depends on the source and microphone geometry. Measured values agree with detailed calculation. Placement of one transducer along the center axis of the air column, with the other at R = 0.6a from the center eliminates all azimuthal mode coupling (Jm Bessel functions, m≳0), and minimizes the coupling to the l...