© The Author(s), Published by EDP Sciences, 2026

1 Introduction

The input impedance of wind instruments is a crucial quantity in musical acoustics, forming the basis for physical modeling approaches leading to sound synthesis, time-domain simulations, and automatic design [1, 2]. This quantity, denoted Z(f), is defined as the ratio of the acoustic pressure P(f) over the acoustic flow U(f) for each frequency f:

$$\begin{array}{c}\hfill Z(f)=\frac{P(f)}{U(f)}·\end{array}$$(1)

This is a complex number whose modulus represents the ratio of amplitude between these two quantities, and whose angle represents their phase shift. Research laboratories and wind instrument manufacturers now commonly use both experimental [3–5] and numerical [6] tools to estimate this input impedance, in order to study existing instruments [7], design new ones [8, 9], and control manufacturing processes [10]. However, the accuracy, reliability, and reproducibility of these tools are rarely questioned. Sources of inaccuracy vary between experimental and numerical setups. In measurements, variability can arise from sensor uncertainty ([11], Appendix B), climate variations, etc. In simulations, inaccuracies may stem from omitted physical phenomena (e.g., humidity), implementation errors, numerical approximations (e.g., rounding), and spatial discretization artifacts. Practical constraints in different domains often lead to trade-offs between time and accuracy. In practice, the final uncertainties in both cases are also due to constraints specific to each research or application domain (manufacturing control, real-time sound synthesis, prototyping, etc.).

The main objective of this work is to quantify the variability of impedance simulations and measurements under realistic conditions and evaluate the significance of various sources of inaccuracy. Unlike studies that seek to establish ideal measurement and simulation conditions, the approach presented here attempts to reflect real-world practices that can be carried out in craft workshops, laboratories, and instrument manufacturing facilities. It considers diverse motivations and scientific cultures, such as metrology, numerical analysis, industrial applications, and education. A secondary objective is to study the agreement between simulations and measurements while accounting for these uncertainties.

The presented work is inspired by the “Verification and Validation” process (V&V), first proposed by the Computational Fluid Dynamics (CFD) and Mechanical Engineering communities [12, 13] to ensure the accuracy and reliability of simulations based on physical models. Verification aims to ensure that the model accurately solves the underlying mathematical equations. Errors generally come from numerical methods (e.g., unconverged discretization) or implementation issues. This step typically involves comparisons with analytical solutions when available, or with results from independent numerical models in configurations where both are expected to be valid. Validation, on the other hand, focuses on assessing how well the model reproduces experimental data, accounting for uncertainties due to variability in geometry, material properties, boundary conditions, and measurement procedures. This V&V process is not well established in wind instrument modeling. Although comparisons between simulations and experiments are common (e.g., [14, 15]), the verification step is often omitted. Studies typically focus on validating new developments in configurations where existing models fail (e.g. [16]), without first verifying consistency between models on simpler, well-understood test cases. For chordophones, the V&V approach has been applied to validate finite-element models of guitar soundboards [17] or as a decision-making support tool for curators in cases of deep uncertainties [18].

To apply this V&V process to the input impedance of tubes, relatively simple test configurations were selected: open or closed cylindrical and conical pipes. This set enables rigorous comparison of data and the identification of potential sources of error. In addition, for these configurations, reference or analytical solutions exist and a relatively good agreement between simulations and measurements is expected.

To study these configurations, a consortium was formed, bringing together researchers from different institutions (academic laboratories, manufacturers, and a manufacturing school) with diverse scientific backgrounds (acoustics, signal processing, numerical analysis, etc.). The work was organized as a round-robin test, a structured benchmark in which the same configurations are independently studied by each participant. For the verification step, the input impedance of each configuration was computed by the researchers with their own independent methods and implementations, then compared. For the experimental part, a batch of tubes traveled between laboratories and was measured by the researchers with their own setups. The measured data were analyzed to estimate the experimental uncertainties and then compared to simulation results for the validation step. Modeling and experimental choices were not imposed uniformly. Each contributor employed their own tools and assumptions, thereby reflecting the diversity of practices in the field. One aim of this collaborative work was to initiate a discussion about these technical aspects, particularly by highlighting the strengths and limitations of each method, as well as the main factors influencing the obtained results. Another aim was to establish guidelines that will enhance the reliability of impedance data and ensure consistency across various methodologies.

The remainder of this paper is structured as follows. First, in Section 2, we describe the studied configurations and the different measurement and simulation techniques employed. Section 3 focuses on the verification step by comparing simulated data. In Section 4, the uncertainty of measuring the input impedance of a given pipe is studied through intra-specimen and inter-operator variability. The validation of the models is studied in Section 5 by comparing experiments with simulations while taking into account geometric uncertainties. Finally, Section 6 discusses key findings, proposes recommendations, and concludes with perspectives on enhancing model validation and measurement reliability in musical acoustics.

2 Material and methods

2.1 Geometries and radiation conditions

The seven configurations studied here are described in Table 1. The tubes were designed to have dimensions similar to the bore of common instruments. The short length (180 mm) was chosen to reduce the computation time of simulations for numerically heavy methods, but also to reduce manufacturing cost and allow the possibility of additive manufacturing (3D printing). The inner diameter of the cylindrical tubes was set to 14 mm to be close to those of some musical instruments (for a B♭ clarinet it is about 15 mm) and corresponds to a standard commercial dimension to facilitate the experimental part. The conical tube has a half-angle of about 2°, approximately corresponding to that of a soprano saxophone ([1], Chap. 7.4.4, p. 328). Three opening conditions are considered: closed, open unflanged (infinitely thin wall), and open with finite flange (finite wall thickness or circular flange following the nomenclature of [19]). The simulated data for the unflanged cases are not compared to measurements because this condition cannot be perfectly achieved experimentally.

2.2 Experimental samples

For the experimental part of this study, samples of tubes in different materials were fabricated using the dimensions of Table 1 as targets. Each sample consists of five specimens numbered 1–5, so that measurements can be compared with simulations, taking into account manufacturing repeatability and geometric measurement uncertainties, as detailed in Sections 2.4 and 4. The three materials chosen for the cylinders are brass (as provided by the supplier), boxwood (handmade by a craft instrument maker not connected to the authors), and Acrylonitrile Butadiene Styrene (ABS), a common polymer used in 3D-printing, printed by Fused Deposition Modeling (FDM) using a Zortax M200 3D-printer. The conical tubes are also 3D-printed in ABS with the same printer. The samples are shown in Figure 1.

Figure 1. Samples used for the experimental round-robin test, with the adapters and caps.

After manufacturing, the actual geometrical parameters were measured with a ruler for the length, a caliper for the external diameter, and a telescopic gauge plus caliper for the internal diameter (the wall thickness is deduced from internal and external diameters). The mean μ and standard deviation σ of each dimension are given for each type of sample in Table 2. The standard deviations are estimated from the measurement tools’ resolutions (0.5 mm for the ruler and 0.02 mm for the caliper) and the measured variability. The same pipes were used for the open (O) and closed (C) conditions, using 3D-printed caps for the closed conditions and grease to guarantee good sealing.

From Table 2, it appears that the brass tube sample is the most uniform. The boxwood sample, handcrafted using a wood lathe and gouges, has the largest length deviation, and the 3D-printed samples (ABS cylinders and cones) have the largest wall thickness variation (for cones, the small absolute value should be compared to the thin wall).

2.3 Observables

The impedances resulting from measurements and simulations were shared through three-column files (frequency, real part, imaginary part). In order to facilitate the comparison between simulated and measured data, all impedances were post-processed to be scaled by the lossless approximation of the characteristic impedance at the entrance of the pipe:

$$\begin{array}{c}\hfill Z_c=\frac{\rho c}{\pi R_0^2}\end{array}$$(2)

with ρ and c the air density and the speed of sound (imposed in simulation and estimated from the temperature and humidity in measurements [20]) and R₀ the entrance radius. From these scaled impedances, two types of observables were extracted as represented in Figure 2: resonance characteristics of both impedance and admittance, specifically adapted to musical acoustics, and the value of the reflection coefficient at specific frequencies, specifically adapted to compute distances between simulations. This study focuses on observations within the frequency range of interest: f ∈ [20, 5000] Hz.

Figure 2. Modulus (top) and angle (bottom) of an impedance with some observables (simulation of an unflanged open cylinder): the frequency $f_{\mathrm{res}}^{(2)}$ and amplitude $a_{\mathrm{res}}^{(2)}$ of the 2nd resonance, the frequency $f_{\mathrm{anti}}^{(3)}$ and amplitude $a_{\mathrm{anti}}^{(3)}$ of the 3rd anti-resonance (or admittance resonance), and the impedance at the observed frequencies Z(f obs), used to compute the distance between the reflection coefficient.

2.3.1 Resonance characteristics

From a musical perspective, it is interesting to analyze the frequency and amplitude of the acoustic resonances of the pipe. The methods of determining these characteristics have been chosen here to avoid, as much as possible, a complex algorithm including a minimization process (as in mode fitting). For simplicity, the impedance and the admittance peaks are referred to as resonance (index r e s) and anti-resonance (index a n t i) peaks, respectively.

For the ith (anti-)resonance peak, the frequency f ⁽ⁱ⁾ is defined as the frequency at which the impedance phase crosses zero (Fig. 2). For both simulated and measured data, this is estimated using a linear interpolation of the phase adjusted over a range of ±5 cents and at least 11 samples around the zero-crossing point. The amplitude a ⁽ⁱ⁾ is defined as the modulus of the impedance (or admittance) at that frequency (Fig. 2), estimated by using a quadratic interpolation of the logarithm of the modulus fitted over the same interval.

To facilitate comparison between resonances and between studied cases, these quantities are analyzed through their relative deviation with respect to a reference value for each peak. This is expressed in cents for frequencies:

$$\begin{array}{c}\hfill d_f^{(i)}=1200{log}_2\left(\frac{f^{(i)}}{f_{\mathrm{ref}}^{(i)}}\right)\end{array}$$(3)

and in decibels for amplitudes:

$$\begin{array}{c}\hfill d_a^{(i)}=20{log}_{10}\left(\frac{a^{(i)}}{a_{\mathrm{ref}}^{(i)}}\right),\end{array}$$(4)

with x ⁽ⁱ⁾ the characteristic (frequency or amplitude) of the ith impedance peak. The choice of these reference values is not always trivial and will be discussed in Sections 3 and 4 for each pipe configuration investigated. However, since the absolute values of these observables are less relevant than the deviations between the shared data, the choice of reference values is not critical.

2.3.2 Reflection coefficient

In numerical studies, it is common to analyze a norm of the deviations between two simulation results to quantify their agreement. This approach is adopted here for both simulations and measurement. In an impedance curve, the presence of peaks can lead to large deviations from small variations (e.g., a slight shift in a resonance peak). To mitigate this effect, the reflection coefficient is preferred here [8, 21]:

$$\begin{array}{c}\hfill \mathcal{R}(f)=\frac{Z(f)-1}{Z(f)+1}·\end{array}$$(5)

where Z(f) is the considered impedance. The distance between two reflection coefficients (ℛ _p , ℛ _q ) is then computed from the ℓ²-norm at the N arbitrarily imposed frequencies $f_{\text{obs}}^{(i)}$ (Fig. 2):

$$\begin{array}{c}\hfill d_{\mathcal{R}}^{(p,q)}=\frac{\sum _i^N{|{\mathcal{R}}_p(f_{\text{obs}}^{(i)})-{\mathcal{R}}_q(f_{\text{obs}}^{(i)})|}^2}{\sum _i^N{|{\mathcal{R}}_p(f_{\text{obs}}^{(i)})|}^2}·\end{array}$$(6)

For simulations, these N observed frequencies were decided at the beginning of the collaborative work (before starting the experimental step): f _obs,simu ∈ {20, 100, 500, 1000, 5000} Hz. For experiments, they were modified afterwards because some datasets were acquired only over a restricted frequency range: f _obs,meas ∈ {110, 200, 500, 1000, 2000, 3900} Hz. This distance $d_{\mathcal{R}}^{(p,q)}$ was computed for each pair (p, q) of impedance curves, giving a two-dimensional distance matrix per studied case. For numerical simulations, this distance should reach machine precision for two equivalent models and numerical methods. In the current study, with some data being shared in single precision, this distance should be lower than 10⁻⁶.

2.4 Measurement protocols

The presented experimental data come from different experimenters from five laboratories using their own measuring devices. These experimenter-sensor pairs are hereafter labeled as operators Oi ¹. Nevertheless, the devices used by operators O2, O3, O4 and O5 were similar and produced by Almacoustics in Le Mans [4, 22]. This device requires one calibration step (measurement of a cap with “infinite” impedance) and has a 16 mm output diameter. Operator O1 used a handmade sensor with a 4 mm output diameter² [21], designed from the two-microphone/three-calibration technique [3] and using resonance-free calibrations (“infinite”, “anechoic” and radiation impedance) [23].

All of these devices use a logarithmic sine sweep whose frequency range depends on the device and the operator: from 20 Hz to 100 Hz for the minimum frequency and from 4 kHz to 10 kHz for the maximum frequency. For the purpose of comparison, all measurements have been truncated to the range [100, 4000] Hz. To measure impedance, these devices must be connected hermetically to one end of the pipe. Most operators (O1 to O4) attached the tubes to the impedance head with a connector to facilitate alignment. Initially printed in polylactic acid (PLA, operator O1), these connectors were reprinted in a flexible material (thermoplastic polyurethane, TPU) for operators O2 to O4 to accommodate small external diameter variations between specimens. Operator O5 preferred to place the tube by hand. Because the quality of the impedance measurement strongly depends on air tightness at the contact between the impedance head and the sample, cork grease or clay was used at this junction.

For a batch of five tubes, the measurements were repeated five times for tube 1 to estimate the intra-specimen variability (labeled intra hereafter) for a given tube. Then, the four remaining tubes (2–5) were each measured once and, together with one of the measurements of tube 1, used to estimate inter-specimen variability (labeled inter), analyzed in Section 5. Each batch was measured this way for closed and open boundary conditions. Because the complete protocol is time consuming, some operators focused on selected configurations with, at least, the closed cylinders (Tab. 3). Each operator was free to redo some measurements using their own criteria (deviation between similar measurements or modulus of the reflection coefficient above one).

Both sound velocity and air density depend on air temperature T and relative humidity R H. This dependence affects the amplitude of the impedance and either stretches or compresses the frequency axis. The calibration steps naturally scale the obtained data by the lossless characteristic impedance Z _c (Eq. (2)), removing the main amplitude dependence. However, in order to compare the data, it is necessary to correct the effect on the frequency axis by applying conservation of wavelength:

$$\begin{array}{c}\hfill f_{\mathrm{comp}}=f_{\mathrm{meas}}\frac{c}{c_{\mathrm{meas}}(T,RH)}\end{array}$$(7)

where f _meas is the measured frequency axis, f _comp is the one used for comparison, c is the dry-air speed of sound (Tab. 4), and c_meas(T, RH) is the speed of sound during the measurement estimated with the expressions developed in [20], from the measured temperature T and relative humidity RH. This correction neglects the dependence of thermo-viscous effects on T and RH. The deviations of the observables induced by this approximation have been quantified through simulations (Appendix A). These deviations are negligible compared to other uncertainties (|Δf_res|< 0.2 cent and |Δa_res|< 0.2 dB for T ∈ [18, 32] °C). The sensors used to measure the temperature and the relative humidity generally have a precision of 0.1 °C and 2% RH; however, due to possible heterogeneity between the measured point and the air inside the tube, the following uncertainties are assumed: 0.5 °C and 5%.

2.5 Numerical models

Six researchers (partly overlapping with those involved in the experiment), labeled A to F, simulated the input impedance for the configurations listed in Table 1, using their own software or implementations of different models. In these simulations the wall surface was assumed perfectly smooth, rigid, and non-porous, neglecting the effect of the material. The values of the physical quantities were also imposed to focus on deviations due to models or numerical methods. They correspond to dry air at 25 °C, computed with formulae given by Chaigne and Kergomard [1], Chap. 5, p. 241. They are listed in Table 4 in SI units, with eight digits, in order to reach this precision in the comparisons.

Operators A to D used a 1D representation of propagation in the pipe and effective boundary conditions for the radiation of the open pipes. The propagation model is based on the telegrapher’s equation in the frequency domain [1]:

$$\begin{array}{c}\hfill \{\begin{array}{cc}{Z}_v(D(x),\omega )u(x,\omega )+\frac{\mathrm{d}p(x,\omega )}{\mathrm{d}x}=\hfill & 0\hfill \\ Y_t(D(x),\omega )p(x,\omega )+\frac{\mathrm{d}u(x,\omega )}{\mathrm{d}x}=\hfill & 0\hfill \end{array}\forall x\in [0,L]\end{array}$$(8)

where p(x, ω) is the average acoustic pressure across the tube cross-section at the location x and, u(x, ω) is the acoustic flow, defined as the integration of the acoustic velocity across the tube cross-section. The coefficients Z _v (D, ω),Y _t (D, ω) model the thermo-viscous effects (dispersion and losses) that depend on the inner diameter D and the angular frequency ω. The main expressions used are those obtained by Zwikker and Kosten (ZK) [24] for cylindrical pipes. Because these expressions involve Bessel functions, historically difficult to evaluate, Keefe [25] proposed a “wide-tube” approximation involving a development valid for radii much larger than the boundary-layer thickness. In a recent development, Thibault [26] proposed another expression for conical pipes.

For the radiation condition, the expressions proposed by Silva [27] to approximate the solution of the Levine and Schwinger model are mainly used. More details are given for each case in Sections 3.1, 3.2, and 3.3. Operators A and B used their own implementation of the transfer matrix method (TMM) to solve these 1D equations (8). This technique uses the analytical solution for cylinders and approximations for other geometries (cone, etc.). Operator D used the unmaintained version of the PAFI toolbox [28] based on the same approach, but without full control over all aspects of the model. Operator C used the software Openwind [29, 30] that offers the possibility to solve the equations with TMM or a 1D finite-element method (FEM) [31]. In these 1D simulations, the input impedance is simply defined as the pressure–flow ratio at the entrance of the pipe.

Operator E used their own implementation of the multimodal (MM) approach [32, 33]. The thermo-viscous effects are taken into account as a modification of each mode shape through the expressions proposed by Bruneau et al. [34]. For open pipes, the radiation domain is modeled using a larger pipe with Perfectly Matched Layers (PML) on walls surrounding the actual pipe of interest [35]. Operator F used the software Montjoie [36] to solve 3D acoustic propagation with FEM (or 2D axisymmetric). The thermo-viscous effects are taken into account using a two-step simulation (sequential linearized Navier–Stokes model, SLNS) [37], or effective boundary conditions based on the Cremer approach [38]. More details are given in [39]. For open pipes, the radiation domain is bounded by perfectly matched layers. These two methods, used by operators E and F, which incorporate 2D or 3D effects are especially interesting to study propagation in pipes with discontinuities or 3D folding or even conical pipes, in which the assumption that the acoustic energy is carried mainly by the first propagating mode is not satisfied. This is not the case in the cylinders; however, applying the V&V process to such simple cases may give confidence to use these models in much more complex configurations. In these higher-dimensional simulations, the input impedance is defined as the mean pressure across the input area over the integral of the velocity on the same domain³.

For the least computationally intensive calculation methods (e.g. 1D models), the recommendation was to use the following frequency axis for the simulations: [0.2 Hz, 20 kHz] with a 0.2 Hz step. For the simulations using methods with higher computation cost (e.g. MM or 2D and 3D FEM), it was recommended to use a multiple of the frequency step (e.g. 1 Hz). Only the reference frequencies needed for reflection-coefficient observables were imposed (f ∈ {20, 100, 500, 1000, 5000} Hz, cf. Sect. 2.3). The time needed to perform the entire set of computations varies strongly with the method. For TMM, it ranges from less than one second to a few seconds depending on the implementation, whereas 1D FEM requires about 1 min and the MM approach between 2 and 4 min. The 3D FEM requires dozens of seconds per frequency. It is important to note that this computation is particularly disadvantageous for FEM, as it requires a very fine mesh for the extremely high frequencies.

In order to study the practical use of simulation, each operator was responsible for choosing the spatial discretization, making their own compromise between accuracy and computation time.

3 Simulation and model verification

3.1 Closed cylinder

The first simulation case considered is the closed cylinder. Here, the “closed” condition is defined numerically by imposing a velocity normal to the cap wall equal to zero, which gives a zero-flow condition in 1D or an infinite impedance. For a cylinder, the telegrapher’s equation (Eq. (8)) can be solved analytically even with thermo-viscous effects. Together with the boundary condition just defined, the input impedance is simply:

$$\begin{array}{c}\hfill Z_{\text{closed cyl}}=Z_{\mathit{cc}}coth(\mathrm{\Gamma }L)\end{array}$$(9)

where L is the length of the cylinder, $Z_{\mathit{cc}}=\sqrt{Z_v/Y_t}$ and $\mathrm{\Gamma }=\sqrt{Z_v{Y}_t}$ are the complex characteristic impedance and the wave number including thermo-viscous effects. This impedance, using the original Zwikker and Kosten [24] (ZK) expressions for Z _{c c} and Γ, has been computed by operator C⁴ and is used as the reference. The obtained values for the observables are given in Table in Appendix A. The existence of this solution and the clear definition of the boundary condition make this case a good starting point for this verification stage. In particular, all the TMM simulations should give exactly the same results (apart from the choice of expressions for visco-thermal effects). Indeed, in this case, TMM is formally similar to the analytical solution of equation (9). For this configuration, operators A, B, and C used TMM with the ZK expressions; operator C also used 1D FEM and TMM with the first- and second-order Keefe approximations (Keefe 1 and 2), and operator D used the second-order Keefe approximation. Operator E used the MM model and operator F used 3D FEM with SLNS resolution (2D axisymmetric).

The deviations from the reference (Eq. (9)) of both the frequency (x-axis, Eq. (3)) and the amplitude (y-axis, Eq. (4)) of the resonance characteristics are represented in Figure 3. The markers represent the median value for the 10 peaks considered per simulation (5 resonances and 5 anti-resonances) and the error bars indicate the min–max ranges. The FEM-3D simulation is not represented as the frequency step was too wide to extract the resonance characteristics. From a practical perspective, all the simulations are consistent, with frequency deviations within ±0.3 cents and amplitude deviations within ±0.3 dB. Nevertheless, while certain results match the reference solution almost perfectly (in Fig. 3, markers from B and from C using ZK expressions or 2nd order Keefe approximation are superimposed on (0, 0)), others deviate slightly. The deviation of operator E using MM simulation (mainly 0.08 dB higher) is certainly due to the model of thermo-viscous effects retained. The deviation of operator D (−0.05 dB and −0.09 cents) cannot be explained only by their use of Keefe’s first-order approximation, because the results computed by operator C with the same expression are markedly closer to the reference values, particularly for the resonance frequency. This is certainly due to an implementation choice or the rounding of a numerical value (e.g., the physical quantities of Tab. 4), and similarly for the small deviation of operator A.

Figure 3. Closed cylinder, 18 cm long and 14 mm internal diameter. Resonance characteristics deviations from the mean value over all simulations for each resonance from equations (3), (4). The marker is the median value for a given simulation and the error bars the min–max range. The shapes of the markers correspond to the operators and the colors to the model and numerical method. Four markers are superimposed on (0, 0): Keefe 2 TMM C, ZK 1D-FEM C; ZK TMM B and ZK TMM C.

To quantify more precisely the similarity of the simulation results, Figure 4 displays log₁₀(d_ℛ), the matrix of distances of the reflection coefficient (Eq. (6)). For this observable, the 3D-FEM simulations are included. In addition, although d_ℛ does not rely on a reference value (Eq. (6)), the analytical solution from equation (9), used as the reference for d_f and d_a, is also provided for comparison. Theoretically, this relative distance between two equivalent models should reach machine precision (here log₁₀(d_ℛ)< −6, cf. Sect. 2.3.2). By this criterion, the TMM implementations of operator B and the two simulations provided by operator C (ZK–TMM and 1D FEM) are consistent, which confirms previous observations on resonances (Fig. 3). The TMM simulations from operator A, the 3D FEM implementation of operator F, and the second-order approximation of operator C are also very close with a distance log₁₀(d_ℛ)<  − 3.7 ≈ log₁₀(2 ⋅ 10⁻⁴). For the 3D FEM, this small distance can be due to a not fully converged simulation. This first simulation group points to a consensus on the closed-cylinder simulation.

Figure 4. Closed cylinder. Relative distance log10(d ℛ) of the reflection coefficient at reference frequencies (f ∈ {20, 100, 500, 1000, 5000} Hz). The symbol “=” corresponds to distances below the precision of shared data (log10(d ℛ)<  − 6). By construction, this matrix is almost symmetrical, as long as the two compared quantities are close enough. The color scale conveys the same information as the numerical values, and is common to all similar figures (Figs. 5c, 5d and 6b). It ranges from dark blue (low difference) to light yellow (high difference), here limited to light green.

The impact of the approximation on the thermo-viscous effects is evident here, particularly for the multimodal method (MM, log₁₀(d_ℛ)≈ − 2.4 ≈ log₁₀(4 ⋅ 10⁻³)) and for the first-order development (Keefe 1, log₁₀(d_ℛ)≈ − 3.5 ≈ log₁₀(3 ⋅ 10⁻⁴)). The implementation error in the software used by operator D is also clearly visible (log₁₀(d_ℛ)≈ − 3.2 ≈ log₁₀(7 ⋅ 10⁻⁴)).

3.2 Open cylinder

For the open cylinder, the same solution of the telegrapher’s equations can be used. However, the resulting impedance

$$\begin{array}{c}\hfill Z_{\text{open cyl}}=Z_{\mathit{cc}}tanh(\mathrm{\Gamma }L+{tanh}^{-1}(Z_{\text{rad}}/Z_{\mathit{cc}}))\end{array}$$(10)

features the radiation impedance Z _rad that models the radiation at the open boundary. In this study, three radiation configurations were simulated: assuming an infinitely thin wall (unflanged), taking into account the 7 mm or 2 mm width of the wall (flanged). Results for the 2 mm wall width are omitted here to avoid redundancy, as the 7 mm case exhibits similar but more pronounced behavior⁵.

Each operator was free to choose how to model the radiation. For the unflanged pipe, a model of this impedance was established by Levine and Schwinger, with several approximations proposed by Silva [27]. It can be directly used in 1D simulations. For the multimodal approach, which needs finite wall width, very thin walls were chosen (< 1 mm). Conversely, the finite flange can be modeled naturally in multimodal simulations, whereas in 1D simulations, the empirical model proposed by Dalmont [19], Eq. (42) was used, interpolating between unflanged and infinitely flanged conditions.

The values used as reference are given in Table in Appendix B. For the unflanged configuration, this is computed with equation (10) and the non-causal expression ([27], Eqs. (21), (22)) for Z _rad. For the flanged case (finite wall), without an analytical solution, the median value of each observable is used as reference. This choice affects the absolute values of the observables, but has little impact on the relative distances between operators and models, which are the focus of this analysis.

As for the closed cylinder (Sect. 3.1), the observables are compared in Figure 5. For the unflanged configuration (Unfl., Fig. 5b and 5d), the two simulations of operator C are similar to the reference (analytical solution). A very small deviation is observed for operator B, only visible in the ℓ²-norm (log₁₀(d_ℛ)≈ − 3.3). Similar deviation is observed between these two operators for the flanged configuration (Fl. Fig. 5a and 5c). Operator A is very close to the others for the unflanged configuration, but deviates substantially for the flanged one (d_f ≈ 0.1 cent, d_a ≈ 0.2 dB, log₁₀(d_ℛ)≈ − 2.6). This may result from a different choice of expressions for the unflanged and infinitely flanged conditions used in the radiation model of [19], or from an implementation error. The deviation between the 1D simulations and the multimodal ones remains of the same order of magnitude as for the closed cylinder (for the three configurations log₁₀(d_ℛ)≈ − 2.4), making it difficult to interpret whether this comes from the model of thermo-viscous effects or from the radiation model.

Figure 5. Open cylinder, with two radiation conditions simulated: with 7 mm wide wall (Fl.: flanged, (a) and (c)) and infinitely thin wall (Unfl.: unflanged, (b) and (d)). Deviation of the resonance characteristics from the reference: (a) and (b) (marker: median; error bar: min–max range). Relative distance matrix for the reflection coefficient: (c) and (d).

Analysis of these two configurations suggests that opening the pipe leads to additional modeling complexity, which increases the deviation between the results while keeping them within an acceptable range (d_f <  1 cent, d_a <  0.5 dB).

3.3 Conical pipe

Since the telegrapher’s equations with thermo-viscous effects have no analytical solution for conical pipes, the TMM approximates the thermo-viscous effects by computing them in an equivalent cylinder ([1], Chap. 7.4.5). By slicing the pipe, this approach slowly converges to the solution of the telegrapher’s equations [31]. This corresponds to a spatial discretization. As with other numerical methods, the spatial resolution is chosen by each operator according to their own habits regarding the balance between accuracy and computational time.

For a conical shape, there are several modeling or computational options available. This affects the shape of the wavefront, introducing an additional source of variation in the simulation results. In 1D, it is possible to impose plane wave or spherical wave by slightly changing the telegrapher’s equations ([1], Chap. 7.4, p. 322). Without instruction, operators A and C simulated plane wavefront (PW) and operator C and D spherical wavefront (SW). One of the advantages of multimodal, 2D or 3D finite element methods is that they do not impose the shape of the wavefront. However, the chosen shape of the source surface (and the radiating surface if the pipe is open), indirectly affect the wavefront. Operator E uses a plane boundary condition at the entrance, which is compatible with a plane wavefront. Operator F, on the other hand, uses a spherical cap, which is more compatible with a spherical wavefront. For simplicity, they are subsequently associated with the plane and spherical wavefront simulations, respectively.

Because of these two factors (discretization and wavefront) and the lack of analytical solution, the median values over all simulations of each observable are used as reference in equations (3) and (4) (Tab. B.1 in Appendix B). Here again, the choice of reference value mainly impacts absolute values, with negligible effect on the relative distances under study.

Only the results for the closed conical pipe are analyzed in detail. The resonance characteristics for this configuration are shown in Figure 6, where two groups of simulations are visible, corresponding to the wavefront shape. These two groups have similar amplitudes, but the spherical-wavefront simulations are about 1 cent flatter (Fig. 6a) because the characteristic length is the wall length (hypotenuse) and not the central axis. The distance between the reflection coefficients is about log₁₀(d_ℛ)≈ − 2 (Fig. 6b). However, within each assumption, the deviations are substantially higher than for the closed cylinder (Sect. 3.1).

Figure 6. Closed conical pipe. The notation “Sw” indicates simulations with spherical wavefronts or assimilated (empty markers) and “Pw” plane wavefronts (filled markers). (a) Deviation of the resonance characteristics from the reference. (b) Relative distance matrix for the reflection coefficient.

Within the plane-wavefront simulations (Pw.), the ℓ²-norm reaches log₁₀(d_ℛ)= − 1.9 (i.e. d_ℛ >  1%) (Fig. 6b) and, even if the medians are close, the deviations of resonance characteristics reach 5 cents and 2 dB for the MM simulation (Fig. 6a), the maximum of deviation occurring for the lowest frequency peaks. This suggests the existence of “non-plane” modes in such non-cylindrical pipes.

Within the spherical-wavefront simulations, the agreement is slightly better (a pairwise comparison gives: d_f <  1 cent, d_a <  0.8 dB, log₁₀(d_ℛ)<  − 2.7). For this conicity (2° half-angle), the use of the hydraulic radius in Zwikker and Kosten expressions, as proposed by Thibault [26] to better model thermo-viscous effects in conical tubes (“Sw+HR” in Fig. 6), has a negligible effect (log₁₀(d_ℛ)≈ − 5). The distance between TMM and 1D FEM computed by operator C (log₁₀(d_ℛ)≈ − 3) illustrates the importance of spatial discretization and suggests that the TMM simulation is not fully converged here.

3.4 Summary of the verification step

These observations are summarized in Table 5 by displaying the mean and maximal deviations obtained by pairwise comparison of the simulations for each configuration. The deviations d_f and d_a are estimated for each resonance peak and pair of simulations. Then, the RMS and maximum values are computed for each configuration. For the reflection coefficient, the mean and maximum values are computed directly from log₁₀(d_ℛ), i.e. the values shown in Figures 4, 5c, 5d and 6b. This table also includes the results for the open conical tube (Cone Unfl.), which were not presented previously, as this configuration combines elements already discussed for the open cylinder and the closed cone.

From a practical point of view, the deviations between all these simulations remain relatively small (mainly d _f  <  1 cent and d _a  <  0.5 dB) and those despite the diversity of methods used. This relative good agreement allows a comparison with experiments.

However, this table illustrates that adding complexity to the model (radiation, or pipe shape) induces wider deviations between numerical results. Indeed, the worst agreement is obtained for the closed conical pipe with plane waves, followed by the open conical pipe, etc. At this stage it is difficult to say whether this is due to a limitation in the models (e.g. wavefront shape or radiation condition in the 1D models) or numerical problems caused by the increasing complexity of the configurations (wall conditions, radiation, etc.).

4 Impedance measurement uncertainties

To assess the reliability and accuracy of measurements, it is essential to identify the sources of variability. This study focuses on four categories of variability associated to the specimen (i.e., a given pipe with a given boundary condition, such as one of the five closed brass tubes) and the operator (i.e., the experimenter–sensor pair):

• The intra-specimen/intra-operator variability refers to the variability observed in the outcomes when one operator (an experimenter using one device) repeats the measurement on one specimen (a given shape and boundary condition). This includes the sensor resolutions but also variations due to handling. It is possible to separate these effects by studying the variability of repeated measurements with or without removing the specimen between repetitions, and with or without recalibrating the sensor.

• The intra-specimen/inter-operator variability is that observed when different experimenters or different setups measure the same parameter on the same specimen. In this study, this includes the post-processing steps necessary to compute the impedance from the microphone signals.

• The inter-specimen/intra-operator variability refers to the variability observed in the results when one experimenter uses one device to repeat measurements on different specimens of the same shape and boundary condition (e.g., the five closed brass tubes). This includes variations in the intrinsic properties of the specimens, such as geometric variability. This is necessary for comparison with simulations.

• The inter-specimen/inter-operator variability is that observed when different experimenters or different setups evaluate the impedance on a batch of similar specimens.

A final source of deviation can arise in the post-processing step when quantities of interest are extracted from raw data. This aspect is not studied here, as all impedances have been processed together by the same operator to extract the observables. This section focuses on the intra-specimen/intra-operator variability and the intra-specimen/inter-operator variability, in order to assess the uncertainties associated with the impedance measurement of a given object. The inter-specimen variability is treated in Section 5 in the context of model validation.

The reference values against which the experimental observables are compared, are given in Appendix B (Tab. B.2). They come from simulations computed for the average geometry of each sample (Tab. 2) and with the values from Table 4 for the physical quantities. Analytical solutions are used for the cylinders (Eqs. (9), (10)) and a converged 1D FEM solution is computed for the conical pipes using Openwind [29]. For open tubes, the finite wall width radiation model was used [19]. As will be seen later, the variability of the measurements is much greater than the differences between the simulations in Section 3, so any model could be used as a reference without affecting the subsequent observations.

4.1 One case analysis

First, the variability of the resonance characteristics of a single specimen, the closed brass tube numbered 1, is analyzed in Figure 7. This shows the deviation, from the reference values, of the frequency (x-axis) and amplitude (y-axis) of the four resonances (first row) and the four anti-resonances (second row) for the five repeated measurements of the impedance of the same specimen and for the five operators.

Figure 7. Deviation from the reference values of the frequency (d f in cents) and amplitude (d a in dB) of the 4 resonances and 4 anti-resonances, for the 5 repeated measurements of a given closed brass cylinder. Each operator is associated with a marker shape and a color.

It appears clearly that the intra-specimen variability, both intra- and inter-operator, is much greater than the deviations observed between simulations in Table 5 (in simulations: d_f <  0.2 cent, d_a <  0.2 dB, compared to several cents and dB in measurements). In addition, the deviation between operators seems to be significantly greater than the deviation between repeated measurements by a single operator. These two observations will be confirmed further in the rest of Section 4 through the global analysis of all configurations.

Both these intra- and inter-operator variabilities do not seem to be particularly related to the absolute frequency of the peak. For most operators, the variability between repeated measurements (intra-specimen/intra-operator) and the deviation from the reference are similar for all resonances and for all anti-resonances. This is not true for the anti-resonances of operator O1, for whom the variability varies markedly with the observed peak. These variations, as well as the size of the deviations from the references, may be attributed to the cap used to close the tube (in PLA for O1 and in TPU for the others, Sect. 2.4). This might also be due to the significant difference in diameter between the sensor used by O1 (4 mm) and the tube (14 mm), which results in a poor signal-to-noise ratio, critical for anti-resonances (Sect. 2.4).

However, as this observation is specific to experimenter O1 and does not appear to be directly related to the absolute frequency, in the following parts of the study, for each measurements, the deviation of the four resonances will be analyzed together on one side, while the four anti-resonances will be treated on the other. This is similar to the analysis carried out in Section 3, but by separating resonance and anti-resonance.

4.2 Intra-specimen/intra-operator variability

For each operator and configuration, the standard deviation (std) of each observable is calculated across five repeated measurements on the same specimen, capturing intra-specimen/intra-operator variability. These standard deviations give a measure of the spread of an individual operator’s markers in the sub-graphs of Figure 7. As previously explained, these std values are averaged across all resonance and anti-resonance characteristics, yielding four values per operator and configuration: σ_f, res, σ_f, anti, σ_a, res, and σ_a, anti, as shown in Figure 8.

In most cases, these standard deviations remain small (σ _f  <  5 cents, σ _a  <  1 dB). Slightly higher values appear for closed pipes, particularly in amplitude (Fig. 8, first row), likely due to the added manipulation associated with cap placement, which may cause leakage. This effect is more pronounced in the conical pipe, whose thinner wall complicates airtight sealing.

Figure 8. Intra-specimen/intra-operator standard deviation (same specimen, several measurements) for each operator averaged across all the resonance peaks (dots) and anti-resonance peaks (crosses). The frequency deviations are expressed in cents relative to the reference values (Tab. ).

As already noted, operator O1 generally shows higher standard deviations, especially for the anti-resonances of closed brass and 3D-printed pipes, suggesting the cap is a key factor. Operator O2 shows an unusually high frequency deviation for the closed wood pipe (σ _f  ≈ 15 cents, Fig. 8, 1st row, 3rd column), due to an unidentified experimental issue.

For the reflection coefficient, the standard deviation is computed for each operator and each observed frequency f _obs. As it shows minimal dependence on frequency, it is averaged per operator and configuration, then expressed as a percentage in Table 6. As with resonance frequency and amplitude, intra-variability is higher for operator O1 under closed conditions, particularly for the 3D-printed cylinder and the conical pipe.

4.3 Intra-specimen/inter-operator variability

For each configuration and each observable, the intra-specimen/inter-operator variability can be estimated by computing the standard deviation across operators of the average value over the five measurements of the specimen 1. The obtained standard deviations, averaged per observable type, are reported in Table 7.

From Table 7, it can be seen that this inter-operator variability exceeds the intra-specimen/intra-operator variability (investigated in Sect. 4.2) for all observables and reaches more critical values: in nearly all cases σ _f  >  5 cents (up to 19 cents) and σ _a  >  1 dB (up to 3.4 dB). As before, variabilities are higher under closed conditions, emphasizing issues with the cap. For closed pipes, resonance and anti-resonance peaks show similar deviations. It can be noticed that, in open pipes, the standard deviation of anti-resonance frequencies remains comparable to intra-operator levels (σ_f, anti ≤ 6 cents, as in Fig. 8).

To give an overall picture, Table 8 reports average standard deviations across all configurations and operators. For intra-operator values, means are computed from 160 measurements for resonance characteristics (8 configurations, 4 peaks, 5 experimenters) and 240 for ℛ (8 configurations, 6 f_obs, 5 operators). For inter-operator values, 32 and 48 measurements are used respectively. These overall results reflect expected variability in tube measurements regardless of material or boundary condition. Intra-operator deviations (σ_f ≈ 3 cents, σ_a ≈ 0.5 dB) align with prior studies ([11], App. B), while inter-operator variability appears substantially larger and potentially problematic.

4.4 Origin of inter-operator variations

The high intra-specimen/inter-operator variability observed, reaching more than 15 cents for certain peaks, is potentially musically important and warrants investigation. One possible explanation is systematic differences in measurement protocols, such as variations in how the tubes are connected to the sensor, potentially introducing additional effective length. Assuming the frequency deviation arises solely from such a geometric offset, the associated length deviation can be estimated as:

$$\begin{array}{c}\hfill {\sigma }_L=L\frac{{\sigma }_f}{f},\end{array}$$(11)

Using the values from Table 8, this corresponds to σ _L  ∈ [0.4, 2] mm for resonances, which appears unlikely given the expected handling precision.

Another hypothesis concerns temperature estimation. The internal tube temperature may differ from ambient conditions due to handling, particularly in thermally conductive brass tubes. Assuming a similar proportionality:

$$\begin{array}{c}\hfill {\sigma }_T=2T\frac{{\sigma }_f}{f},\end{array}$$(12)

with T the absolute temperature in Kelvin. The observed deviations would require a temperature variation of σ _T  ∈ [1.5, 6] K, which again seems implausible.

This deviation could also come from the sensors. To test this hypothesis, one operator measured the same pipe using five different CTTM sensors. Although this experiment was conducted post hoc with a different tube (1 m long, 20 mm inner diameter), the resulting standard deviations, approximately 3 cents and 0.5 dB, are comparable to intra-specimen/intra-operator variability (Tab. 8), which suggests sensor differences are not the primary cause.

Finally, the sensor calibration process may contribute to variability. Operators typically calibrate only once per batch of tubes made from the same material. Since the calibration itself is a measurement, its uncertainty propagates as a systematic bias within each batch. If calibration were the dominant factor, intra-specimen/inter-operator variability would be expected to be of similar magnitude, which is not observed (cf. Sect. 5.1).

In conclusion, the inter-operator deviations most likely result from a combination of factors, including handling inconsistencies, environmental conditions, and calibration practices, rather than a single dominant source.

5 Model validation

A comparison of Sections 3 and 4 shows that the differences between simulations are significantly smaller than the intra-specimen variability in the measurements. This enables comparison between experimental and simulated data to validate or not the model, i.e., to determine the extent to which the model predictions agree with the measurements.

In addition to the experimental uncertainties discussed in Section 4, this validation of the models requires accounting for uncertainties in both geometry and physical quantities (e.g. temperature and relative humidity; see Sect. 2.4) within both the measurements and the simulations. These are accounted for in the models by performing Monte Carlo simulations (Sect. 5.3). The influence of these physical and geometric uncertainties on the measured observables is assessed by repeating data acquisition across multiple specimens, enabling estimation of inter-specimen variability (Sect. 5.1).

5.1 Inter-specimen variability

Inter-specimen variability is assessed by computing, for each operator, the standard deviation across five specimens for each configuration. This yields one value per variable (σ_{f⁽¹⁾, res}, σ_{f⁽²⁾, res}, etc.) and per operator. For clarity, these standard deviations are first averaged by observable type (σ_f, res, σ_f, anti, σ_a, res, σ_a, anti and σ_ℛ), and the median across operators is then reported in Table 9.

The values in this table directly reflect the geometric variability of the specimens, as summarized in Table 2. Variability is highest for wooden pipes and lowest for brass pipes. For the latter, inter- and intra-specimen variabilities are comparable (see Fig. 8 and Tab. 6). In conical pipes, inter-specimen variability is also close to the already elevated intra-specimen/intra-operator variability. In the case of the 3D-printed cylinders under closed conditions, the elevated amplitude variability is likely due to surface defects at the sealed end, introduced during the printing process.

5.2 Total variability

The total variability can be estimated by combining the sources of variability already discussed with the precision of the measuring device. As usual, a normal probability density function is assumed, with a standard deviation given by:

$$\begin{array}{c}\hfill {\sigma }_{\text{total}}=\sqrt{{\sigma }_{\text{intra-spec}}^2+{\sigma }_{\text{inter-spec}}^2+{\sigma }_{\text{inter-op}}^2+\frac{{(2m)}^2}{12}}\end{array}$$(13)

where σ _intra-spec, σ _inter-spec, and σ _inter-op denote the standard deviations due to intra-specimen, inter-specimen, and inter-operator variability, respectively. The term (2m)²/12 corresponds to the variance of a rectangular distribution with a width of 2m. This models the measurement uncertainty, where m represents the measurement resolution, taken as m = 0.1 Hz for frequencies, m = 0.1 dB for amplitudes, and m = 1% for the reflection coefficient ℛ. These terms are negligible compared to the other sources of uncertainty.

Table 10 reports the total standard deviations for each configuration and observable type, along with the overall median values. The results indicate that inter-operator variability (see Tab. 8) is the dominant contributor in most cases. For the wooden pipe, however, the high inter-specimen variability also has an important impact on the total. As previously observed, variability tends to be higher in the closed condition than in the open one, with the exception of cone resonances.

The frequency variability remains higher than 5 cents for most configurations, particularly in closed conditions. These values are significant from a musical point of view (by comparison, the auditory detection threshold is around 5 cents). These findings emphasize the need to reduce inter-operator variability to achieve reliable and musically acceptable measurements.

These values can be used to design an experiment a priori. The number of specimens n needed to validate a phenomenon inducing a deviation of about Δ on an observable with variability σ can be estimated by the expression proposed by Lehr [40]:

$$\begin{array}{c}\hfill n=16{\left(\frac{\sigma }{\mathrm{\Delta }}\right)}^2.\end{array}$$(14)

Let us imagine that a setup similar to the one presented here⁶ is used to validate the finite-flanged radiation model from [41], with respect to the unflanged model. For the 3D-printed pipe with 7 mm wall width, the expected frequency deviation from the models is about Δ _f  ≈ 10 cents (App. B, Tab. B.1) and the experimental variability is about σ _f  ≈ 9 cents (Tab. 10). This results in n = 12 specimens. For brass pipes, the experimental variability is lower (σ _f  ≈ 5 cents) but, due to a thinner wall (2 mm), the deviation between models is also reduced (Δ _f  ≈ 5 cents) leading to n = 10 specimens. If the aim is to determine whether the measurement conditions for the conical pipes correspond to plane or spherical wavefronts, this would necessitate about n = 1300 specimens, as the expected deviation is very small (Δ _f  ≈ 1 cent, Fig. 6) and the experimental variability relatively high (σ _f  ≈ 9 cents, Tab. 10).

5.3 Simulations vs. measurements

In comparison to the experimental variability reported in Table 10, all models investigated in Section 3 show consistent behavior. To evaluate the agreement between measurements and model predictions, simulations were conducted using the analytical solution when available (Eqs. (9), (10)) and a one-dimensional finite-element (1D FEM) model for conical tubes. These computationally efficient models allow for Monte Carlo simulations, with 1000 realizations per configuration completed within a few minutes of computation time.

The simulations incorporate uncertainties in geometry, temperature, and humidity by drawing random samples from normal distributions based on the dimensional tolerances provided in Table 2 and the estimated uncertainty ranges discussed in Sections 2.2 and 2.4. For open-end conditions, both unflanged and finite-flange radiation models are considered.

The mean values for each observable are close to the reference values computed for the mean geometries (App. B, Tab. B.2), and the corresponding standard deviations are reported in Table 11. For frequencies and the reflection coefficient, the simulated variability is of the same order of magnitude as the experimental inter-specimen standard deviation (Tab. 9). In contrast, the simulated standard deviation for amplitude is lower, highlighting that resonance frequencies can be measured with greater accuracy than peak amplitudes.

Figure 9 compares the distributions obtained from simulations with those measured by each operator. Inter-operator deviations are clearly visible, reaching up to 30 cents for frequency (e.g., between O1 and O2 for the closed and open wooden pipes) and 10 dB for amplitude (e.g., between O2 and O5 for the closed 3D-printed pipes). While no consistent bias is observed across all operators, some tendencies appear: operator O2 generally reports slightly lower frequencies, operator O1 often records lower resonance amplitudes, whereas operator O5 obtains higher amplitude, especially for closed configurations..

Figure 9. Statistical comparison of measured and simulated deviations in frequency and amplitude for resonances, across all configurations and operators. Markers represent the median over all resonance peaks and specimens (4 peaks × 5 specimens), with error bars indicating the interquartile range. Monte Carlo simulation results are shown as crosses. For open conditions, black crosses correspond to simulations using the finite-flange radiation model, while gray crosses indicate the ideal unflanged case.

In general, measured amplitudes tend to be lower than those predicted by simulations, especially under closed-end conditions (except for O5). Deviations are most pronounced for wooden pipes, likely due to their greater variability. Lastly, the radiation model accounting for wall thickness (finite flange) shows better agreement with experimental data than the idealized unflanged model.

To quantitatively assess model validity for each configuration, i.e., whether model predictions agree with the measurements or not, two statistical analyses are performed. First, a bilateral test dealing with heterogeneous variance (heteroscedastic) [42] was applied to the absolute values of each observable. Because results were consistent across configurations for a given observable type, the values were averaged accordingly and reported in Table 12. Low values (typically p-value < 0.01) indicate statistically significant differences between the simulated and measured distributions. No conclusions can be drawn for higher values.

In order to evaluate the agreement between simulations and measurements, the deviation

$$\begin{array}{c}\hfill \mathrm{\Delta }\mu ={\mu }_{\text{MC}}-{\mu }_{\text{exp}}\end{array}$$(15)

is computed for each observable, where μ _MC is the mean of the Monte Carlo simulations and μ _exp is the mean of the measurements. This deviation is compared to its associated uncertainty,

$$\begin{array}{c}\hfill {\sigma }_{\mathrm{\Delta }\mu }={\sigma }_{\text{MC}}+\frac{{\sigma }_{\text{total}}}{\sqrt{N_{\text{exp}}}},\end{array}$$(16)

obtained by propagation of uncertainty, where σ _MC is the standard deviation of the Monte Carlo simulations, σ _total is the experimental variability from equation (13), and N _exp is the number of measurements. The resulting values, averaged by observable type, are reported in Table 13. When the deviation is smaller than the uncertainty (Δμ ≤ σ_Δμ, in bold in Tab. 13), this supports model validity.

The combined results from Tables 12 and 13 suggest that the models accurately predict the resonance frequency of both ABS cylinders and cones, for open and closed configurations, as well as for the open brass pipe. For amplitude, none of the values in Table 13 meet the validation criterion. However, there is clearly better agreement between simulations and measurements for the open brass pipe than for the other configurations. Both tables also show that discrepancies are more pronounced for closed configurations (low p-values and high Δμ_a) than for open ones.

For the wooden pipes, both open and closed, the significance values in Table 12 are especially low, and the simulations exhibit significantly higher resonance amplitudes than the measurements as visible in Table 13, and in Figure 9. Taken together, these results show that the current models do not accurately reproduce the measured acoustic behaviour of wooden pipes. Aside from a possible unidentified bias, one explanation for this discrepancy is the presence of rough and porous inner surfaces, which enhance thermo-viscous losses by altering the boundary layer [43]. Recent advances [44] suggest modeling strategies that account for such surface effects and could improve simulations. The good agreement observed for open brass supports the validity of both the propagation model in smooth, non-porous geometries and the finite-flange radiation model [19].

For closed conditions, it remains unclear whether the observed discrepancies result from experimental limitations or from assumptions in the modeling approach, which considers the cap as a perfectly rigid (infinite impedance) termination. However, the experimental explanation appears more plausible given the practical challenges associated with measuring closed pipes. Sealing the cap after placing the tube on the sensor can increase internal pressure and slightly displace the tube, thereby degrading measurement quality. Notably, operator O5 employed an alternative capping method and obtained results that were in better agreement with the simulated amplitudes (see Fig. 9).

6 Conclusion

This document presents the results of a collaborative study conducted by a group of researchers from public and private research institutes. It is probably the first of its kind in the musical acoustics community. The study aims to compare the simulated and measured impedance of a set of acoustic resonators in the context of musical instrument manufacturing.

A key feature of the presented approach is the comparison of simulated and measured data obtained by different researchers using their own models and experimental setups within their shared validity range. This allows the verification and validation of these models and the study of the accuracy and reliability of impedance measurement in realistic conditions.

In addition, this work underlines the interest of performing such collaborative round-robin studies for both numerical and experimental aspects.

The first and main interest of this process is perhaps that it offers an environment for the authors to exchange around the technical aspects of their practices. For example, concerning the simulations, the presented data reflect the outcome of several iterative refinements, during which implementation errors and overly coarse approximations were identified and corrected through collaborative exchanges among contributors. Although only the final results are shown, the good agreement between simulations (Sect. 3) stems from multiple cycles of discussion and comparison. This process not only enhanced the accuracy and consistency of the results but also fostered valuable technical dialogue, reinforcing the robustness of the proposed best practices.

This work highlights several aspects where further effort would be valuable. Regarding the simulation part, the results are very close, but the thermo-viscous model used in the multimodal approach could be improved. Similarly, the lack of “three-dimensional” data (or 2D axisymmetric) for several configurations underlines the difficulty of including these thermo-viscous effects in some geometries. However, by focusing on the low-frequency range (f <  5 kHz and λ≫ pipe radius) and on simple geometries, this study avoids discussing the assumptions underlying the one-dimensional model (essentially the presence of a single low-frequency propagating mode). Multimodal and three-dimensional approaches do not suffer from this limitation and are therefore expected to provide more accurate results at relatively high frequencies, particularly for radiation effects and non-cylindrical pipes.

Regarding model validation, it seems necessary to include the surface condition of the walls in the model (porosity, roughness). However, a previous study [43] indicates that post-treatment of the wood of wind musical instruments (polishing and oiling) tends to eliminate this effect. Another conclusion of the measurement–simulation comparison, developed especially in a research report [20], is the importance of including the humidity effect in the estimation of the values of the physical quantities.

Regarding the measurements themselves, reducing inter-operator variability seems to be a priority. This could be improved by adopting good practices that emerged from the discussions among the authors. To achieve this, it seems important both to validate the calibration step and to assess correct placement of the measured sample (centering and air tightness). The calibration can be validated by measuring, for each measurement session, a reference tube whose geometry is perfectly known and for which the model is valid (e.g., a brass cylinder, open to avoid cap issues), and then checking the agreement between the measured and simulated data. To assess the absence of leakage or misplacement of the sample, it is useful to repeat the measurement, compare with a similar measurement or, if possible, compare with simulations. More practically, the discussion underlined the importance of allowing the sensor to reach thermal equilibrium with the room, especially for sensors with closed cavities; otherwise a drift can be observed. In this study, one type of device was predominantly used (four out of five sensors), making it difficult to generalize these conclusions to all impedance sensors. Repeating the proposed protocol with other devices would help overcome this limitation.

A good practice concerning simulations is to verify each new software (or code) by assessing the agreement of the results on simple geometries with known analytical solutions or valid simulation results from another software. In this sense, this article, and especially the numerical values from Table B.1, could be used. When a spatial discretization is used, it is advisable to estimate the precision of the results provided, for example by comparing with the calculation obtained with a refined spatial step, similarly to how measured data should be given with uncertainties. Finally, the use of common software and/or common file formats can also help reduce implementation errors and facilitate comparison between software.

To enable other community members to easily compare their simulations and/or measurements with the data presented here, the raw data [45] and the post-processing programs used to extract observables and to generate the presented tables and figures are available [46]. This allows anyone to share their own data and increase the representativeness of this round-robin study, for example, by including other type of impedance sensor.

This study concerns a very limited set of geometries that are simple compared with woodwind instruments. It could be very interesting to perform similar round-robin tests with horns, bent tubes, tubes with side holes, etc. However, as this process is time consuming and requires logistics between the participating laboratories, it is important to resolve the issues identified here before extending it to more complex configurations.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

The raw data associated with this article are available in Zenodo, under the reference [45]. The scripts used to analyze the data set and generate the figures are available in GitLab, under the reference [46].

References

Appendix A

Temperature correction

The compensation for the temperature difference by applying equation (7) to the measured frequency axis neglects the modification of the thermo-viscous effects. The bias induced by this simplification on the observables of Section 2.3 is studied here. The impedance of a closed cylinder with the dimensions from Table 1 is simulated with physical quantities estimated for dry air and temperature within [18, 32] °C using the expressions from [20]. Equation (7) is then applied to the frequency axis to compensate the variation of the speed of sound with respect to 25 °C. The deviations of the resonance frequency and peak amplitude from the reference values actually computed at 25 °C are represented in Figure A.1.

Figure A.1. Simulation of the deviation induced by the correction of the speed of sound with temperature on frequency of resonance in cents (top) and the impedance amplitude (dB) at this frequency.

It appears that the consequence of neglecting the thermo-viscous effects in this compensation is negligible for this wide temperature range with respect to the actual variability between measurements (Tab. 8). A similar conclusion holds for the humidity rate.

Appendix B

Reference values

The reference values of the observables used for the simulations in Section 3 are reported in Table B.1. For the conical pipe, the values correspond to the median over all shared simulated data. For the cylinders, analytical solutions were computed using the physical quantities from Table 4. For the closed cylinder, equation (9) was applied; for the open cylinder, equation (10) was applied with radiation impedance obtained either from the non-causal expression in [27] for unflanged radiation or from the expressions in [19, 27] for finite-flanged radiation.

The reference values of the observables used for the measurements in Sections 4 and 5 are reported in Table B.2. These values were obtained using the mean geometries from Table 2 and the physical quantities from Table 4. For closed pipes, equation (9) was used. For open pipes, equation (10) was used together with radiation impedance accounting for wall thickness, using the expressions of [19, 27]. For conical pipes, converged 1D FEM simulations were performed using the Openwind software [29].

All Tables

All Figures

	Figure 1. Samples used for the experimental round-robin test, with the adapters and caps.
In the text

Figure 2. Modulus (top) and angle (bottom) of an impedance with some observables (simulation of an unflanged open cylinder): the frequency $f_{\mathrm{res}}^{(2)}$ and amplitude $a_{\mathrm{res}}^{(2)}$ of the 2nd resonance, the frequency $f_{\mathrm{anti}}^{(3)}$ and amplitude $a_{\mathrm{anti}}^{(3)}$ of the 3rd anti-resonance (or admittance resonance), and the impedance at the observed frequencies Z(f obs), used to compute the distance between the reflection coefficient.

In the text

Figure 3. Closed cylinder, 18 cm long and 14 mm internal diameter. Resonance characteristics deviations from the mean value over all simulations for each resonance from equations (3), (4). The marker is the median value for a given simulation and the error bars the min–max range. The shapes of the markers correspond to the operators and the colors to the model and numerical method. Four markers are superimposed on (0, 0): Keefe 2 TMM C, ZK 1D-FEM C; ZK TMM B and ZK TMM C.

In the text

Figure 4. Closed cylinder. Relative distance log10(d ℛ) of the reflection coefficient at reference frequencies (f ∈ {20, 100, 500, 1000, 5000} Hz). The symbol “=” corresponds to distances below the precision of shared data (log10(d ℛ)<  − 6). By construction, this matrix is almost symmetrical, as long as the two compared quantities are close enough. The color scale conveys the same information as the numerical values, and is common to all similar figures (Figs. 5c, 5d and 6b). It ranges from dark blue (low difference) to light yellow (high difference), here limited to light green.

In the text

	Figure 5. Open cylinder, with two radiation conditions simulated: with 7 mm wide wall (Fl.: flanged, (a) and (c)) and infinitely thin wall (Unfl.: unflanged, (b) and (d)). Deviation of the resonance characteristics from the reference: (a) and (b) (marker: median; error bar: min–max range). Relative distance matrix for the reflection coefficient: (c) and (d).
In the text

	Figure 6. Closed conical pipe. The notation “Sw” indicates simulations with spherical wavefronts or assimilated (empty markers) and “Pw” plane wavefronts (filled markers). (a) Deviation of the resonance characteristics from the reference. (b) Relative distance matrix for the reflection coefficient.
In the text

	Figure 7. Deviation from the reference values of the frequency (d f in cents) and amplitude (d a in dB) of the 4 resonances and 4 anti-resonances, for the 5 repeated measurements of a given closed brass cylinder. Each operator is associated with a marker shape and a color.
In the text

	Figure 8. Intra-specimen/intra-operator standard deviation (same specimen, several measurements) for each operator averaged across all the resonance peaks (dots) and anti-resonance peaks (crosses). The frequency deviations are expressed in cents relative to the reference values (Tab. ).
In the text

Figure 9. Statistical comparison of measured and simulated deviations in frequency and amplitude for resonances, across all configurations and operators. Markers represent the median over all resonance peaks and specimens (4 peaks × 5 specimens), with error bars indicating the interquartile range. Monte Carlo simulation results are shown as crosses. For open conditions, black crosses correspond to simulations using the finite-flange radiation model, while gray crosses indicate the ideal unflanged case.

In the text

	Figure A.1. Simulation of the deviation induced by the correction of the speed of sound with temperature on frequency of resonance in cents (top) and the impedance amplitude (dB) at this frequency.
In the text