A pressure-based transfer matrix method and measurement technique for studying resonances in flutes and other open-input resonators Open Access

A fundamental question in musical acoustics is the relationship between the resonances of an instrument and its geometry. Much work has been done to develop models and numerical methods that are simple enough to use yet accurate enough to be useful.^1–8 

This paper presents an alternative, pressure-based formulation of a transfer matrix method (TMM)^1,2,4,7,8 that has long been used in an impedance-based formulation to model the input impedance and passive resonances of wind instruments. Also presented in this paper is an indirect experimental method for determining the input impedance of open-input resonators that relies on the present pressure-based TMM analysis for its implementation.

Both the original impedance-based TMM formulation and the present pressure-based formulation solve the wave equation for sound in a tubular system by utilizing products of one or more 2 × 2 matrices to relate the values of two physical quantities at the instrument's input to the same quantities at the instrument's output (or other reference point). In the impedance-based formulation, the quantities being evaluated are the acoustic pressure P and the acoustic flow U; in the pressure-based formulation, the quantities being evaluated are the amplitudes of the forward-going and backward-going waves, $P+$ and $P−$⁠, respectively.

Since the two approaches are fundamentally equivalent, one might question the need for an alternative formulation, especially since the wave amplitude ratios $P+/P−$ and impedance $P/U$ are related to each other in a way that allows one to be determined if the other is known. A natural preference for the impedance-based formulation might also be expected given that impedance measurements comprise the bulk of extant wind instrument data. However, it might be argued that the dominance of impedance-based techniques over the pressure-based alternatives is due more to historical accident than to any considered analysis of relative merit for a particular application.

The selection of input impedance as the quantity to be measured experimentally is not always the ideal choice⁶ in open-input instruments like the flute, where the instrument resonances occur at input impedance and pressure minima. In these cases, the use of impedance as the measured quantity gives one the worst signal-to-noise ratio precisely in the spectral regions of most interest. Another difficulty with input impedance measurements in open-input resonators is that the impedance spectrometer is typically connected to the resonator in a way that obstructs the input orifice, unavoidably perturbing the resonator geometry. For these cases, one might argue that an internal acoustic pressure measurement makes more sense, especially if it can be coupled with a model that allows the pressure spectra to be fit directly.

In the experimental method presented here, approximately planar sound waves from a localized source are directed into the resonator's open input orifice while the resulting acoustic pressure is measured by a microphone at a position x inside the resonator. The resonator's power spectrum, denoted as I_x, is obtained by normalizing the Fourier power spectrum of the pressure data recorded at the measurement position by the Fourier power spectrum of the pressure data recorded just outside the source. The present pressure-based TMM formulation is then used to reproduce the I_x spectrum (a simple function of the $P+$ and $P−$ calculated at the measurement position x) and the input impedance (a simple function of the $P+$ and $P−$ calculated at the resonator's input).

The above-described measurement technique has two novel features relative to conventional impedance measurements: the way that the system is driven and the location at which the pressure is measured. In conventional impedance measurements, the system is driven with a closed acoustic current source at the input end of the resonator, and the pressure is measured at the same location. In contrast, the present system is driven by near-field radiation (allowing a more natural input aperture geometry), and the measurement position is inside the resonator (at a location where there is never a minimum at an interesting frequency). This offers several advantages for open-input resonators such as the flute. First, the embouchure hole remains open during the measurement, allowing the flute's resonance spectrum to be measured as a function of lip coverage. Second, the microphone can be placed at the flute's head joint cork without perturbing the instrument's acoustics or interfering with its playability, a feature expected to be especially useful in cases where cork intensity data are being acquired while the instrument is actually being played.

Perhaps the strongest argument for a pressure-based TMM formulation is its intuitive appeal: there is no need for the electrical engineering vocabulary of inertance, compliance, series resistance, and shunt terms common to impedance-centric methods. Resonator components, such as input, output, and tonehole orifices, are no longer described in terms of their respective impedances but rather are characterized by complex reflection coefficients comprising physically meaningful radiative loss and end-correction terms. Deemphasizing the traditional reliance on impedance concepts can have pedagogical benefits as well.

The remainder of this paper is organized as follows. The present pressure-based TMM model is described in Sec. II (where it should be kept in mind that the approximations and assumptions of the traditional impedance-based formulations apply to the present formulation as well). The experiment setup, measurement procedures, and resonators studied (ranging from simple non-branched, one-end-closed straight tubes and chimney pipes to several configurations of a modern flute) are detailed in Sec. III. Data and fits are presented and discussed in Sec. IV. Although the resonance spectra for the one-end-closed straight tubes (Sec. IV A) appear to require unphysically high wall losses for the shorter tube lengths, the good agreement seen for both one-end-closed polyvinyl chloride (PVC) chimney pipes (Sec. IV B) and a modern flute set up with a variety of lip and keywork and open/closed tone hole configurations (Sec. IV C) suggests no major problems with either the theory or measurement method. Input impedance spectra for a few of these resonators are then presented in Sec. IV D and compared to the corresponding I_x fits from which they were derived.

The pressure-based TMM model of this paper is an adaptation and expansion of a matrix method developed by Heavens⁹ and used by others^10,11 to compute optical reflection from and transmission through a multilayer stack of lossy dielectric thin films such as those shown schematically in Fig. 1. This stack comprises N layers of index j, thickness $Dj$⁠, and complex refractive index $nj=nj−ikj$⁠, where $nj$ is the real part of the refractive index and $kj$ is the extinction coefficient. In the optical TMM method, one computes the amplitudes of the electric fields associated with the forward-going and backward-going waves within each thin film layer and uses the Fresnel coefficients computed from the films' complex refractive indices to match the electric fields on either side of each thin film interface.

In the acoustic analog of the Heavens optical method, the dielectric layers in the stack of Fig. 1 are replaced by lossy-wall tubular segments of length $Lj$ and complex cross-sectional area $Sj$ to produce the generic tubular structure of Fig. 2. However, two additional issues must be addressed before the model can satisfactorily model pressure waves in resonators complicated enough to be interesting (e.g., wind instrument bores). The first is what to use for $S0$ and $SN+1$⁠, the effective areas for the regions just outside the input and output ends of the tube, analogous to $n0$ and $nN+1$⁠, the refractive indices of external ambients 0 (the medium of the incident wave) and N + 1 (the medium into which the transmitted wave exits). This is handled in Sec. II B, where a length correction and effective $Sj$ are extracted by matching the model's reflection coefficients (functions of $S0$ and $S1$ for the tube input and $SN$ and $SN+1$ for the tube output) to literature^12–16 expressions for the acoustic reflection coefficients. The second issue is how to introduce the tube branches potentially needed for the embouchure hole, cork cavity, and tone holes, situations for which an optical-acoustic analog is less apparent. This is addressed in Sec. II C, where it is shown that a solution developed to model quantum wires with attached resonators¹⁷ can be directly incorporated into the present adaptation of the Heavens matrix method. The model description concludes with Sec. II D, where it is shown how input impedance may be extracted from the computed pressure amplitudes for comparison with the literature.

For the acoustic version of the Heavens method, we start with the generic tubular structure of Fig. 2 and the wave equations for sound waves in a lossless cylindrical tube of cross-sectional area $S$⁠:

where $p$ is the acoustic pressure (i.e., the local pressure relative to the pressure of the undisturbed medium), u is the acoustic flow (i.e., the volume velocity), $ρ$ is the gas density, $χ$ is the isentropic gas compressibility, and $c$ is the speed of sound in the medium external to the tube. However, these equations must be modified if they are to account for the effects of thermal conduction and viscosity. For cases in which the thermal and viscous boundary layers are small compared to the tube radius (as is the case for most woodwind bores) and a harmonic time dependence, Chaigne and Kergomard¹⁸ provide the following equations and solutions (modified by K.L.S. to apply to a jth tube segment rather than to just a single tube):

where $Pj$ and $Uj$ are the complex amplitudes of the acoustic pressure and volume velocity,

and

where $Zcj$ is the characteristic impedance of the jth tube segment, $rvj$ is the dimensionless ratio of the jth tube segment's radius to the thickness of its viscous boundary layer (where the subscript v is used to indicate that the ratio is to the thickness of the viscous boundary layer), $rtj$ is the dimensionless ratio of the jth tube segment's radius to the thickness of its thermal boundary layer (where the subscript $t$ is used to indicate that the ratio is to the thickness of the thermal boundary layer), $ΓCK$ is the propagation constant of Ref. 18, and $α1$ and $α2$ are functions of the dimensionless quantity known as the Prandtl number. The Prandtl number Pr is equal to $Cp(η(T)/κ(T)) ≡ ν2$⁠, where $Cp$ is the specific heat of air at constant pressure, and $η(T)$ and $κ(T)$ are, respectively, the temperature-dependent values for viscosity and thermal conductivity of air. Pr has a value of ∼0.71 and is nearly independent of temperature; since $rtj=ν □ rvj$⁠, the use of $Pr=ν2$ allows the thermal term $rtj$ to be eliminated from the expression for the propagation constant in Eq. (9). The values of $α1$ and $α2$ are given by

and

where $γ$ is the ratio of the specific heats of air at constant pressure and constant volume, i.e., $Cp/Cv$ = 1.402. For the relatively large tube diameters being considered here, the values of $rvj$ are given by^18,19

where $aj$ is the tube radius of segment j (in cm), $f$ is the frequency in Hz, and $ΔT$ is the temperature deviation from 300 K. The computer code used to implement the present model allows the $1/rvj$ term (proportional to the thickness of the viscous boundary layer) to be multiplied by a “loss multiplication” scaling factor $∝loss$ (typically having a value between 1 and 2) to take into account that wall losses in real musical instruments are usually larger than those calculated for perfectly smooth tubes.^7,20

Applied to the branchless N-segment structure of Fig. 2, the solution to the above equations has the form

and

where $Pj+$ is the amplitude of the forward-going pressure wave in tube segment j, $Pj−$ is the amplitude of the backward-going pressure wave in tube segment j, and the complex cross-sectional area $Sj$ is defined by

where

and

The assumption of continuity requires that the values of $Pj−1 and Pj$ given by Eq. (14) match at each j − 1/j interface, i.e.,

and

where $Lj$ is the length of the jth tube segment. Equations (19) and (20) can be rearranged to give the recurrence relations

where $rj$ and $tj$ are given, respectively, by

and

It should be noted that $rj$ and $tj$ are the acoustic analogs of the Fresnel coefficients commonly used in optics to describe reflection and transmission, respectively, at the interface between two dielectrics (and are, in fact, identical to the optical Fresnel coefficients with the substitution of complex refractive index $nj$ for complex cross-sectional area $Sj$⁠).

From Eq. (21), one can obtain $P0 ±$(the forward- and backward-going wave amplitudes in the ambient just outside the tubular system's entrance) in terms of $PN+1±$(the forward- and backward-going wave amplitudes in the ambient just outside the tubular system's exit):

where

Equation (24) can be used to get the tubular system's reflectance and transmission coefficients R (⁠$=P0−/P0+$⁠) and T (⁠$=PN+1+/P0+$⁠), respectively, once it is realized that $P0+$⁠, the forward-going wave incident on the tube's opening, can be presumed to be unity in some arbitrary units and $PN+1−$⁠, the amplitude of the backward-going wave in medium N + 1 just outside the exit end of the tube, is zero (because the exiting wave undergoes no further reflections). Writing the matrix product

one finds that

and

The reflectance and transmittance intensities for the system are then given by

and

where the superscript “*” denotes the complex conjugate, and the $SN+1/S0$ factor in the expression for $T$ originates from the fact that the energies of the reflected and transmitted waves are $S0RR*$ and $SN+1TT*$⁠, respectively.

The above equations may also be used to find the magnitude of the pressure amplitude squared $|P|2$ at any tube segment boundary (or at any point along a tube segment, since an existing segment can be split into two segments, one upstream from the point and one downstream from the point), once it is realized that

where $Pm$ is the acoustic pressure at the boundary between tube segments m − 1 and m, and the same iterative procedure used to produce Eq. (24) from Eq. (21) can provide the following expression for $Pm±$⁠:

Alternatively, one can determine the magnitude of the pressure amplitude squared at a point within the mth tube segment at a distance x from the m − 1/m boundary using Eq. (14) in combination with the values of $Pm +$ and $Pm−$ obtained from Eq. (32). This gives the values of $Pm +(x)$ and $Pm −(x)$ needed to compute $|Pm(x)|2$⁠.

This section provides expressions for $S0$ and $SN+1$⁠, the effective cross-sectional areas for the ambient external to the tube ends at the input and output, respectively. The cross-sectional area for an ambient external to a closed tube end is taken to be zero. For open tube ends, the reflection coefficients provided by the Heavens method (which are functions of the as-yet-to-be-determined $S0$ and $SN+1$⁠) are matched to literature expressions for the complex, frequency-dependent acoustical reflection coefficient at the end of a pipe or duct

where $k=ω/c$⁠, and $Lend$ is the end correction. If the magnitude of the reflection coefficient $|Rend|$ is known, it is a simple matter to extract cross-sectional area values for the spatial regions external to the tube ends (⁠$S0$ and $SN+1$⁠, both real) from Eq. (22). Using the subscript end_N (or, equivalently, end_out) to denote that the tube end being considered is the output end of the tube and that j − 1 = N (the index of the last tube segment), one obtains

Ignoring viscosity and thermal conduction effects (i.e., considering only $SN$⁠, the real part of $SN$⁠) and assuming $SN<SN+1$ (which is always the case since N + 1 is the index of the ambient just outside the end of the tube), Eq. (34) may be rearranged to obtain

The analog to Eq. (35) for the input end of the tube, using the subscript end_1 (or, equivalently, end_in) to denote that the tube end being considered is the input end of the tube, is

where $|Rend_1|$ is found from the same computation used to find $|Rend_N|,$ but with the parameters appropriate for the input end of the tubular system.

Values for the reflection coefficient magnitudes $|Rend|$ are fairly well established for infinitely flanged and unflanged tubes having circular cross sections^12,13 and less well established for rectangular orifices,^15,16 circular holes terminating in circular^14,15 or cylindrical flanges,¹⁴ and round tone holes partially obstructed by overlying keys.¹⁴ Dalmont et al.¹⁴ provide $|Rend|$ for an unflanged circular aperture, accurate for $ka<3.5$⁠:

(though the same authors suggest that it is marginally more accurate to extract $|Rend0|$ from Caussé's expression⁴ for the output impedance of an unflanged pipe, as was done here). The corresponding Dalmont et al.¹⁴ expression for an infinitely flanged aperture is

The $e2ikLend$ component of the Eq. (33) reflection coefficient is incorporated into the present matrix model as an end correction. Tube segments of length $Lend_1$ and $Lend_N$ with respective cross-sectional areas of $S1$ and $SN$ are added to the input and output ends of the tube, introducing a phase change in the reflected wave equal to $2ikLend$⁠, where the factor of 2 comes from the fact that the reflected wave has to make a round trip through the end correction segment/layer.

For unflanged apertures, Dalmont et al.¹⁴ provide the following end correction, accurate for $ka<3.5$⁠:

where the $sin2(2ka)$ term is included only for $ka<1.5$⁠. The corresponding correction for an infinitely flanged aperture is

For completeness, it is noted that the $Lend$ and $|Rend|$ values given by Eqs. (37)–(40) for $ka≪1$ are approximately $[0.6133a]$ and $[1−(ka)2/2]$ for an unflanged aperture and $[0.8216a]$ and $[1−(ka)2]$ for an infinitely flanged aperture.

This section shows how the tube branches needed for modeling woodwind instruments may be incorporated into the formalism of the Heavens matrix method. (For the case of the flute, branches are required for the embouchure hole, the cork cavity—the volume between the embouchure hole and the stoppered end of the head joint—and the tone holes.) The present solution is an adaptation of one developed by Shao et al.¹⁷ to model quantum wires with attached resonators. Consider the generic junction of Fig. 3, which has been inserted between two adjacent tube segments (for example, between the mth and m + 1th segments of Fig. 2), where the subscripts L and R refer to the segments of the original tube to the immediate left (L) and right (R) of the junction, and the subscript B refers to the newly added branch. Three equations follow directly from the constraints that the pressure at the junction end of all three branches (L, R, and B) must be equal and that the flow coming into the junction must equal the flow coming out of the junction:

and, making use of Eq. (15),

where the $S$ values indicate the complex cross-sectional area of the L, R, and B segments adjacent to the junction. Following the analysis of Shao et al.,¹⁷ Eqs. (41)–(43) are first solved to get a 3 × 3 scattering matrix relating $PL−$⁠, $PR+$⁠, and $PB+$ to $PL+$⁠, $PR−$⁠, and $PB−$⁠, respectively,

where the $rX$ terms are the reflection coefficients for the sound wave impinging on the junction from segment X (with X = L, R, or B), and the $tXY$ terms are the transmission coefficients of the sound waves passing through the junction from segment X into segment Y. By expressing the amplitudes $PB+$ and $PB−$ as the ratio $λ=PB+/PB−$⁠, Eq. (44) can be reduced to a 2 × 2 matrix relating $PL−$ and $PR+$ to $PL+$ and $PR−$⁠:

where

and

and where the values of $PB+$ and $PB−$ needed for $λ$ may be determined from Eq. (32) evaluated for the segments comprising the branch. Equation (45) may then be rearranged to produce a Heavens-matrix-method-compatible 2 × 2 transfer matrix:

where the $tjct$ and $Cjct$ terms are analogous to the $tm$ and $Cm$ terms, respectively, used in Eqs. (24) and (32) for the tube segments. The expression in Eq. (50) may now be inserted into Eq. (32) at the position of the junction (i.e., between the $Cm/tm$ and $Cm+1/tm+1$ terms corresponding to the mth and m + 1th tube segments) without further modification. However, seamless integration into the existing matrices requires the insertion of a dummy tube segment of zero length and cross-sectional area $Sm$ between the right side of the mth segment and the left side of the junction in order to eliminate the need for any consideration of the mth segment length in the $PL+$ and $PL−$ values being returned by the junction computation. Junction branches may be closed (as would be the case when modeling the cork cavity) or open (as would be the case for open tone holes), and they may be positioned along the main tube as desired.

The pressure amplitudes $Pm±$ and intensities $|Pm|2$ can now be computed at any point along the main tube. However, it should be noted that additional calculation is required to obtain these quantities if the position of interest is in a junction branch (for example, at the position of a microphone situated at the stopped end of the cork cavity). For such cases, the pressure amplitudes are recalculated with branch and right arms of the relevant junction flipped (using the already calculated $PR±$ and $PB±$ values) so that the position of interest lies on the main tube, and values returned are those for the end of the junction rather than those for the end of the main tube.

The present pressure-based TMM formulation allows one to compute the power spectrum (⁠$|P++P−|2$ vs frequency) expected at any point inside the resonator (e.g., in a flute head joint at the position of the cork). Since wind instruments are more commonly characterized by their input impedance spectra, it should be noted that $Zin$ is simply related to the already-computed values of the pressure amplitudes $P1+$ and $P1−$ (or, equivalently, the reflection coefficient $R=P1−/P1+$⁠) just inside the embouchure hole:

where $Zc (=ρc/S$⁠) is the characteristic impedance, i.e., the input impedance of an infinitely long tube having the same cross-sectional area as the resonator's input. This conveniently allows a comparison of the present impedance results with those in the literature (see, for example, Refs. 6 and 7).

A pressure-based experimental method that allows a direct check of the model is described next. Since it lacks the established pedigree of impedance-based methods, good agreement between the measurement and model will help validate the measurement technique as well.

A resonator containing an embedded microphone is excited by a broadband sound source positioned in front of (and a short distance away from) the resonator's input orifice. The acoustic pressure at the microphone position is recorded and Fourier analyzed to produce a raw power spectrum. A reference power spectrum is obtained in the same manner after removing the resonator and replacing it by a reference microphone. The resonator's power spectrum, denoted I_x, is obtained by dividing the raw power spectrum by the reference power spectrum. The subscript x refers to the microphone's location in the resonator (i.e., at the tube end for the non-branched resonators and at the head joint cork position for the flutes).

The setup, shown schematically in Fig. 4, has three basic components: the test object (an open-input resonator containing an embedded microphone, shown here as a flute), a computer-driven audio speaker and “gasline” delivery system for providing a localized sound source near the test object's input orifice, and a laptop computer for data acquisition and analysis. For sound isolation purposes, the speaker and laptop are in one room and the test objects were in another room. The speaker, a Universal Serial Bus (USB)-rechargeable iHome minispeaker model IHM60, ∼5.7 cm in diameter (SDI Technologies, Rahway, NJ), was connected to the input end of a long (∼3 m) flexible hose [inside diameter (ID) ∼ 1.9 cm] via a set of PVC reducing bushings and adapter tubes. The output end of the hose was connected to a ∼15-cm length of 1/2 in. copper-tube-size (CTS) chlorinated polyvinyl chloride (CPVC) pipe (ID, 1.2 cm) terminating in an output aperture directed at the resonator's input orifice. The output aperture used for the data in the present paper was the bare end of the CPVC pipe, positioned 1.0 cm in front of the resonator (although, as discussed in Sec. IV A, a 2-cm source-to-resonator spacing was sometimes used for diagnostic purposes).

The speaker at the gasline's input played an Audacity-programmed sound clip consisting of 150 s of pink noise.²¹ Acoustic pressure was measured with a 10-mm-diameter pre-polarized, cardioid-pattern condenser microphone (Audix ADX10-FLP, Audix, Wilsonville, OR) in combination with an XLR-to-USB adapter (Shure X2U, Shure Inc., Niles, IL). The microphone was embedded in a pair of rubber grommets and squeezed into place to make a planar airtight seal at the closed end of the resonator. (Note that the grommets also sealed the microphone's side vents, arguably transforming it from a free field microphone into a pressure microphone.) Data for the same resonators were also collected using chirp-based sound clips having a better spectral uniformity (since chirp power could be programmed to least partially compensate for gasline resonances), but the pink noise source was preferred for its simplicity.

Calibration runs were performed by replacing the resonator with a reference microphone assembly comprising the same grommet-embedded microphone (or, in the case of the flute resonators, a nominally identical microphone corrected for differences in frequency response) squeezed into a 1/2 in. PVC holder tube [ID ∼1.5 cm, outer diameter (OD) ∼2.1 cm]. In all cases, the microphone face was flush with the end of the holder tube and positioned to be the same (1.0 cm) distance away from the sound source as the resonator.

Microphone signals were collected by a laptop computer at a 44.1 kHz sample rate and saved in wav format. Spectral analysis on the acquired wav files was done by Audacity's fast Fourier transform (FFT) algorithm using a Hann window on contiguous sample segments 16 384 points in length (resulting in a frequency resolution of 2.7 Hz). A fresh reference spectrum was typically collected within an hour or two of every measurement with every effort made to perform both data and reference runs at the same temperature [measured by a PDQ400 Comark/Fluke digital thermometer (Comark Instruments, Beaverton, OR) near the resonator].

Representative reference spectra resulting from pink noise excitation are plotted in the supplemental material, Sec. A,²² Fig. S1, for two gasline terminations: the normal 1/2 in. CPVC gasline termination and an alternative three-dimensional (3D)-printed rounded-rectangle sometimes used for comparison purposes. While the normalized resonator spectra obtained with the two aperture types were nearly identical, the raw spectrum delivered by the CPVC aperture was preferred for its better spectral uniformity over the spectral range of interest.

The non-branched plain tube and chimney pipe resonators examined in this paper were all open at their inputs and closed at their far ends. As shown schematically in Fig. 5, the plain tubes examined had IDs of either 1.9 cm [the brass tube of Fig. 5(a)] or 2.58 cm [the 1 in. Schedule 40 PVC tube of Fig. 5(b)]. Each plain tube type was examined at three lengths: short (∼8 cm), medium (∼23 cm), and long (∼70 cm). The chimney pipes, shown in Figs. 5(c) (full view) and 5(d) (cut-away view), consisted of a narrow chimney section secured to a wider body section. The body tubes were made from the same three pieces of 1 in. Schedule 40 PVC used for the plain tube measurements; the chimney section was formed from a fixed-length (∼10 cm) piece of 1/2 in. CPVC tubing embedded in an adapter plug designed to extend into and make a snug fit with the open end of the body tube. The tube termination, consisting of grommet-encased microphone centered in a PVC fixture can be seen in Fig. 5(d) (at the end of the tube).

The C flute studied in these experiments was a circa 1980 solid silver, open-hole, B-foot Muramatsu (Muramatsu America, Royal Oak, MI). The cork stopper in the original head joint was replaced with a grommet-embedded microphone, an arrangement with advantages previously described by Coltman.²³ Open-hole (perforated French-style) keys were plugged with plastic key plugs (Yamaha-style cups, Yamaha Corporation of America, Buena Park, CA), and the desired keys (French, plateau, and/or register) were clamped closed with flute key clips (Ferree's Tools, Inc., Battle Creek, MI). The assembled flute was supported by a set of cradle clamp flute blocks (J. L. Smith Co., Charlotte, NC).

Measurements were mostly performed with a “no lips” configuration of the embouchure hole (i.e., just the bare hole). However, some data were also collected for a “with lips” configuration to mimic real playing conditions in which the embouchure hole is partially covered. For these experiments, half the embouchure hole was covered with a piece of ∼3-mm-thick self-stick foam weather-strip tape clamped in place with the same type of key clip used to hold the keys closed. This left a slight protrusion of the tape into the embouchure hole and reduced the areal dimensions of the embouchure hole opening from ∼1.2 × 1.0 cm² to ∼1.2 × 0.5 cm².

Several configurations of the flute's low G (fundamental at ∼392 Hz) were investigated, ranging from the conventional fingering of Fig. 6(c) (see the supplemental material, Sec. B,²² Table S-I, for details) with the normal tonehole lattice (bottom eight holes open) to the no-tonehole lattice arrangement of Fig. 6(a) in which the flute body and footjoint were replaced by a 1.9-cm ID (∼0.05 cm-wall) nickel-silver tube leaving the flute with a cylindrical bore body of the same acoustic length. Intermediate configurations included a low G with a truncated tone hole lattice [the four open holes left after removing the footjoint, as shown in Fig. 6(b)] as well as conventional and no-foot low G's with the keywork above the open holes removed.

Data were also collected for a regularly fingered low B (all-the holes closed, fundamental at ∼247 Hz), two made-up fingerings with just one of the flute's lower body holes open, and a regularly fingered low D# (lowest four holes open, fundamental at ∼311 Hz) with and without the keywork.

Fourier-analysis-derived power spectra from measurements on three types of open-input resonators (plain closed-end tubes, closed-end chimney pipes, and several configurations of a modern flute) will now be compared to pressure amplitude squared vs frequency calculations from the model.

Experimental I_end data, calculated fits, and difference spectra (ratio of fit to measurement) are shown in Fig. 7 for three lengths of the brass tube and in the supplemental material, Sec. C,²² Fig. S3, for three lengths of the 1 in.-PVC pipe. Resonator dimensions and fitting parameters for the non-branched resonators are listed in Table I. The fits were optimized manually; the loss multiplication factors $∝loss$ were unconstrained, and the tube lengths were allowed to vary within approximately ±1 mm of the physical lengths.

As expected, the I_end resonance peaks occur at odd-integer multiples of a fundamental whose wavelength is given by ∼$4L$⁠, where $L$ is the tube length. Also expected is the gradual reduction in spectral contrast (i.e., the dB spread between adjacent minima and maxima) as frequency increases, a consequence of wall and radiation losses.

The fits are fairly good overall, although the difference spectra (ratio of fit to measurement) typically show small blips at each resonance and a gradually rising baseline. This gradually rising baseline, seen in the difference spectra for all resonators examined, is attributed to the difference in microphone environment (pressure field vs free field) for the resonator and reference measurements, an issue that is discussed in more detail in the supplemental material, Sec. D.²² 

Loss multiplication factors $∝loss$ of ∼1.2–1.3 worked well for the long-tube fits. These values are somewhat greater than the value of unity that would be expected from first-principles calculations but are comparable to those observed in wind instrument bores and attributed to wall roughness.^5,20 An explanation for the apparent increase in the $∝loss$ values with decreasing tube length (with $∝loss$ values as high as 3–5 for the tubes with $L ∼$ 8 cm) was more elusive; as discussed in the supplemental material, Sec. D,²² losses from the microphone and/or the wall at the pipe's closed termination²⁴ seem unlikely, and losses from geometrical and environmental effects at resonator's input orifice^25–31 seem possible, despite the insensitivity of the spectra to source-to-resonator spacing.

Experimental I_end data and fits are shown in Fig. 8 for the three chimney pipes. The discontinuity in resonator diameter clearly introduces some additional spectral complexity but fits are very good, especially if one discounts the slightly upward sloping baseline in the difference spectra just discussed. Two features bear special mention. First is that the fitted $∝loss$ values are all identical to those found for the long plain tubes. Second is the somewhat surprising result that the fits are good despite the absence of correction terms often considered necessary when modeling resonators with abrupt changes in cross-sectional areas (in this case, a factor of ∼5), a topic discussed from several viewpoints in Ref. 18.

The flute is a more challenging structure to model since it requires the incorporation of side branches. As illustrated in Figs. 9(a) and 9(b), the input aperture (embouchure hole) of the flute is positioned in the flute's head joint, a tapered tube that is stoppered at its narrow end and connected to a cylindrical body tube containing keywork and tone holes at its wide end. Under normal playing conditions, the embouchure hole is open (although somewhat reduced in size as it is partially covered by the player's bottom lip, an effect discussed in more detail by Ernoult et al.³²) and is the entry point for the air jet used to excite the flute tones.

The head joint bore was approximated as a stepped-diameter tube comprised of short, fixed-diameter cylindrical segments. The rounded-rectangle profile of the tapered embouchure hole was approximated as a sequence of cylindrical segments of the same cross-sectional area. An expanded view of the tapered embouchure hole and cork cavity (the volume between the embouchure hole and the cork stopper) is shown in Fig. 9(c), and the approximation to this geometry that is used in the model (with the cork cavity and embouchure hole flipped) is shown in Fig. 9(d). A closed branch is used to model the cork cavity; open branches are used to model the open tone holes.

The present fits used the actual physical dimensions for head joint/body tube lengths and tone hole spacing. However, the effective length and diameter of the toneholes and the cork cavity length were varied empirically to give the best fits.

Full-sized closed tone holes were treated as ultrashort (0.001-cm-long) branches attached to an associated segment of the main tube having a length equal to the tonehole diameter and a cross section (listed in the supplemental material, Sec. B,²² Table S-I) that was computed by evenly distributing the extra tonehole volume³³ over the segment length. Due to their small volume, the three smaller toneholes were either ignored (in the case of the two trill keys) or treated as full-sized toneholes with zero excess volume (in the case of the register hole).

Experimental I_cork spectra are shown in Fig. 10(a) for two lip configurations of the Fig. 6(a) no-lattice low G. Increasing embouchure hole coverage from no lips to with lips markedly increases the spectral contrast of the low frequency I_cork resonances and (consistent with the literature²² and player experience) reduces the frequency of the fundamental. Relative to the no-lips case, the with-lips fundamental is flattened by about 44 cents (or ∼2.5%, from ∼406.3 Hz to ∼396.0 Hz). Lip coverage effects gradually diminish with increasing frequency; above 3000 Hz, the spectra for the two lip coverages are nearly identical. Interestingly, lip coverage appears to have very little effect on the frequencies of the I_cork minima.

The embouchure hole input was initially modeled using the infinite-flange expressions of Eqs. (38) and (40) for $|Rend_in|$ and $Lend_in$⁠. However, the resulting fits systematically underestimated the frequencies of the resonances between 1500 and 2500 Hz, a fact attributed to the inadequacies of treating the rounded-rectangle embouchure hole/curved lip plate assembly as a round pipe opening onto an infinite flange. To improve the fits, $Lend_in∞$ was replaced with $Lend_inembouchure$⁠, a lip-coverage-dependent hybrid of the no-flange $Lend0$ and infinite-flange $Lend∞$ (see the supplemental material, Sec. E,²² for details), constructed to be shorter (i.e., closer to $Lend0$⁠) in the frequency range where the computed peak positions need to be at slightly higher frequencies. This embouchure hole end correction term was used for all the fits of this paper (unless otherwise noted), although it had relatively little effect when fitting flute geometries with multiple open tone holes.

Fits for the no-lattice low G (using the hybrid $Lend_inembouchure$ expression) are shown in Figs. 10(b) and 10(c), along with the difference spectra. Minor mismatches in extrema position and magnitude still remain even with the hybrid expression, but the overall fit is not bad given the uncertainties in the true values of $Lend_inembouchure$ (which primarily affects the peak positions) and the use of $|Rend_in∞|$ (which governs the spectral contrast) for $|Rend_inembouchure|$⁠.

Data and fits for the case of an all-the-holes-closed B-foot low B are shown in the supplemental material, Sec. F,²² Fig. S7. The fitting quality was similar to that obtained for the no-lattice low G data of Fig. 10, providing bore segments containing tone holes included the tonehole volume correction. Interestingly, as shown in the supplemental material, Sec. F,²² Fig. S8, modeling the closed tonehole lattice as a uniform tube of slightly expanded cross section also produced an excellent fit, whereas modeling the flute body/footjoint assembly as a straight tube did not.

The no-lips B-foot low D# was investigated as a prelude to modeling the B-foot low G because the four open toneholes of the D# comprise the lower half of the eight open toneholes in the G. However, to better understand how the tonehole fitting parameters and reflection coefficients were affected by tonehole geometry, I_cork data and fits were obtained for both the normally open holes (i.e., with keypads a few mm above the tonehole) and bare holes without the overlying keywork (i.e., footjoint keys removed). Fitting was simplified by the fact that the four open toneholes within each keywork case (i.e., with keywork or without) were nominally identical, conveniently introducing the constraint that the same fitting parameters be used for each open tonehole in a given fit.

I_cork data and best-parameter fits for the two low D# cases are shown in Fig. 11; a selection of alternate fits for the same data is shown in the supplemental material, Sec. G,²² Figs. S7 and S8. It was found that spectral contrast matching was better with the infinite flange model for toneholes with keywork and better with the no-flange model for bare toneholes without keywork.

I_cork data and fits for the (full-lattice) B-foot low G normal keywork are shown in Fig. 12. Below ∼2000 Hz, the spectra are similar to those of Fig. 10 for the no-lattice low G; above ∼2000 Hz, the resonances become noticeably more irregular in spacing and magnitude due to tonehole lattice effects. Data and fits for cases between these two extremes are shown in the supplemental material, Sec. H,²² Fig. S10; as expected,^34,35 tonehole lattice effects can be somewhat mitigated with a shorter tonehole lattice (e.g., no footjoint) and/or larger tonehole conductance (e.g., no overlying keywork).

An additional check of the tonehole fitting parameters used for the open body toneholes of the B-foot low G is provided by the I_cork data and fits of the supplemental material, Sec. I,²² Fig. S11, for two made-up flute notes, each a “forked fingering” with just a single body hole open [i.e., tonehole 8 or 9 in the tonehole diagram of Fig. 6(d)]. Again, agreement between data is very good.

The same parameters used to fit the I_end and I_cork data are now used to compute the resonator's input impedance and its inverse, the input admittance. Figure 13 shows the phase and magnitude of the input admittance $Zc/Zin$ derived from Eq. (51) along with the corresponding I_end spectra for the 70-cm brass tube and chimney tube spectra of Figs. 7(a) and 8(a). The frequencies of the maxima in the |$Zc/Zin$| and corresponding I_end spectra are very nearly coincident with agreement better than 0.1 cent for all the peaks of the 70-cm brass tube spectra shown. Agreement for the chimney tube geometry is slightly worse, although the average absolute value of the discrepancies (averaged over the 28 resonances shown) is below 0.3 cents if the resonances in the impedance spectra are defined as the average positions of the maxima in |$Zc/Zin$| and zero crossings in the phase of $Zc/Zin$⁠, a point discussed in more detail in the supplemental material, Sec. J.²² 

Impedance plots analogous to Fig. 13 are shown in Fig. 14 for the full-lattice B-foot low G fits of Fig. 12 and in the supplemental material, Sec. J,²² Fig. S11, for the no-lattice low G fits of Fig. 10. The |$Zc/Zin$| maxima are again nearly coincident with the corresponding maxima in the I_cork spectra, allowing a near-immediate identification of the resonances just from the I_cork data alone. However, as expected from the analysis of Coltman,²² there are small systematic discrepancies. In particular, the I_cork resonances below ∼2000 Hz are consistently lower in frequency—by about 1 Hz for the fundamental (at ∼400 Hz) and about 2–3 cents otherwise. It can also be seen that the reduction in spectral contrast at high frequency is much less pronounced in the I_cork spectra, a potential advantage for an I_cork-based measurement if the higher frequency resonances are of interest.

The present impedance results for the flute's conventionally fingered low G (with the keywork) are in good qualitative agreement with those directly measured by others,^6,7 although exact comparisons are difficult given the differences in how the embouchure hole coverage is taken into account. In addition, both approaches seem to show comparably small mismatches between the calculated fits and the experimental data (I_cork in the present paper vs $Zin$ in Ref. 7). Overall, this is a strong argument for the validity of the present pressure-based TMM approach for modeling and the associated experimental technique for resonator characterization.

A pressure-based formulation of a TMM was developed to model pressure-amplitude-squared (⁠$|P++P−|2$⁠) vs frequency spectra recorded at positions inside open-input resonators such as chimney pipes and flutes. Fits to the data generated all the parameters needed to determine resonator input impedance. Although some refinements in embouchure hole and tonehole modeling would probably be beneficial for the flutes, the agreement between model and experiment was good enough to suggest that the present pressure-based formulation may be a viable alternative to experimental methods and models based directly on impedance.