Oscillation threshold of a Raman clarinet with localized nonlinear losses at the open end

Christophe Vergez (LMA); Micha\"el Jousserand; Nathan Szwarcberg (LMA); Tom Colinot

REVIEW 3 major objections 5 minor 24 references

Oscillation threshold of a Raman clarinet with localized nonlinear losses at the open end

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Localized nonlinear losses at a clarinet's open end raise the minimal blowing pressure needed to start a tone.

desk verdict A clean model paper that shows a new, plausible effect—localized nonlinear losses raise the oscillation threshold—but the magnitude rests on an unvalidated low-frequency v|v| law; deserves serious peer review. read the letter →

arxiv 2411.18185 v4 pith:A2RHN6BX submitted 2024-11-27 physics.class-ph

Nathan Szwarcberg (LMA) , Christophe Vergez (LMA) , Tom Colinot , Michaël Jousserand This is my paper · ORCID

classification physics.class-ph

keywords Ramanclarinetmodellocalizednonlinearlossesoscillationthresholdextinctioniteratedmapstabilitywindinstrumentacousticsboundaryconditionreflectionfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper works with a Raman clarinet, a minimal physical model in which an idealized reed is coupled to a cylindrical tube described by a reflection function, and extends it to include localized nonlinear losses at the open end. Its central result is that these losses raise the oscillation threshold: the minimal blowing pressure at which self-sustained oscillations start increases monotonically with the nonlinear-loss coefficient for all embouchures studied. The same losses also lower the extinction threshold, and do so about ten times more strongly. Because the two thresholds bound the playable pressure range, the model predicts that sharper edges at the open end make the instrument harder to start while also compressing the sound amplitude and preserving a direct onset bifurcation.

What carries the argument

The central object is the nonlinear reflection function $r_{nl}(\xi)=\lambda^2\xi\left(1-\tfrac{4}{1+\sqrt{1+\hat{K}_0|\xi|}}\right)$, obtained by inverting the quasi-steady loss law and composing it with one round trip through the tube. It converts the delay-line system into the iterated map $x_{n+1}=f(x_n)$, whose fixed points locate the equilibrium and multi-state oscillation regimes. Stability of a period-$n$ regime is decided by whether the derivative of the $n$-th iterate is less than 1 in modulus, and the boundaries of these stability regions define the oscillation and extinction thresholds.

What would settle it

Measure the onset blowing pressure of a cylindrical pipe with interchangeable open-end geometries, sharp-edged unflanged versus rounded or flanged, at fixed embouchure, using a slow linear pressure ramp and averaging over many trials to remove bifurcation delay; if the onset pressure does not rise monotonically with the inferred $C_{nl}$, or rises far more than the 4% predicted at $C_{nl}=0.7$, the central claim is refuted.

Watch

Extended reading notes

Core claim

For a cylindrical tube with a quasi-steady nonlinear boundary condition $p=\rho_0 C_{nl}v|v|$ at the open end, the paper derives a passive nonlinear reflection function and folds the entire instrument into a one-dimensional iterated map $x_{n+1}=f(x_n)$. Computing fixed points and their multipliers over a fine grid of blowing pressure $\gamma$, embouchure $\zeta$, and dimensionless loss coefficient $\hat{K}_0$, it finds that $\gamma_{\mathrm{osc}\nearrow}$ increases monotonically with $\hat{K}_0$ for all $\zeta\in[0,0.99]$. Concretely, at $\zeta=0.3$, the threshold rises from $0.378$ at $\hat{K}_0=0$ to $0.393$ at $\hat{K}_0=0.325$, a 4% increase, while the extinction threshold drops by 40% for the same parameters. For $\hat{K}_0>34$, the silent equilibrium coexists with stable oscillating regimes at high embouchure, and in the limit $\hat{K}_0\to\infty$ no stable oscillation exists below $\gamma=1$.

Load-bearing premise

The predicted rise of the oscillation threshold rests on modelling the open-end losses as a memoryless, quasi-steady relation $p=\rho_0 C_{nl} v|v|$ with a single constant $C_{nl}$; if the real losses are frequency dependent, hysteretic, or of another form, the monotonic increase may not survive.

Editorial extensions

If this is right

At a typical clarinet embouchure ($\zeta=0.3$), increasing $\hat{K}_0$ from 0 to the experimentally plausible value 0.325 raises the onset pressure by 4%, from $\gamma_{\mathrm{osc}\nearrow}=0.378$ to 0.393.
The same increase cuts the extinction threshold by 40%, so the blowing-pressure range that sustains a steady tone narrows substantially.
Nonlinear losses keep the onset bifurcation direct while flattening the amplitude curve near threshold, which the paper identifies as an aid for controlling soft dynamics; linear losses, by contrast, can make the bifurcation inverse.
For $\hat{K}_0>34$, the silent equilibrium coexists with stable oscillating regimes at high embouchure, and in the limit $\hat{K}_0\to\infty$ no oscillation occurs below $\gamma=1$.
The derived reflection function gives a simple way to include localized losses in waveguide models, which the paper suggests could inform side-hole undercutting in instrument design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A time-domain simulation using a frequency-dependent radiation impedance instead of the memoryless $v|v|$ law would test whether the monotonic threshold rise survives realistic end losses.
Because the predicted onset shift is only 4%, onset experiments must control the blowing-pressure ramp rate, since bifurcation delay can mask or exaggerate the effect.
The reflection-function form could be applied to tone holes, letting makers estimate how undercutting changes local loss and hence the playable range.
The model's amplitude-compressing behavior suggests end-loss tuning as a design lever for shaping loudness response somewhat independently of pitch.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a careful model paper with a new result. The authors extend the standard Raman clarinet iterated-map framework to include localized nonlinear losses at the open end, derive an explicit reflection function, and show by parametric stability analysis that increasing the nonlinear loss coefficient raises the oscillation threshold (the minimal blowing pressure to start oscillations) while lowering the extinction threshold. The threshold rise is new relative to the cited literature; the extinction effect was already known. The derivation is algebraically consistent and the stability classification follows the established iterated-map machinery. The paper is also honest: it states the quasi-static v|v| assumption and does not claim experimental validation of the threshold shift.

The soft spots are real but not fatal. First, the monotonic-increase claim across all zeta and the high-K0 behavior rest on a finite grid (Table 1) with no convergence checks, and no code or data are shipped. "Available upon reasonable request" is weaker than a public release. Second, as your stress-test note correctly points out, the threshold shift is controlled by the derivative of the nonlinear reflection function at the nonzero fixed-point pressure—i.e., by the slope of the v|v| law at the DC operating point. The coefficient Cnl was fitted from high-amplitude oscillatory measurements. If the real end termination has a different low-frequency or low-Reynolds impedance slope, the magnitude (or even sign) of the predicted shift could change. None of this is hidden; the paper is a model study and the undercutting implication is explicitly left as a future hypothesis. But readers should not take the 4% shift as a measured acoustic effect.

Bottom line: for people working on wind-instrument modeling, this is a useful and careful piece. It deserves a serious referee round. The referee should ask for the code and grid-convergence details and for a short discussion of the v|v| law's applicability at the operating point. I would cite it if I were writing in this area.

Referee Report

3 major / 5 minor

Summary. The paper studies a Raman clarinet model with localized nonlinear losses at the open end, represented by the quasi-steady pressure-velocity relation p = rho0*Cnl*v*|v|. The authors derive a closed-form nonlinear reflection function, embed it in an iterated map, and compute the stability of the equilibrium, two-state, and long-period regimes as functions of the blowing pressure gamma, the embouchure parameter zeta, and the nonlinear loss coefficient K0. The central result is that increasing localized nonlinear losses raises the oscillation threshold (the minimal blowing pressure for onset) by a few percent while strongly lowering the extinction threshold, and the paper also characterizes bifurcation types and multistability regions as the loss coefficient varies.

Significance. If the central prediction is correct, the paper provides a simple qualitative account of how end losses affect clarinet playability: stronger localized losses require higher mouth pressure to start the sound and reduce the maximum sustainable pressure, narrowing the dynamic range. The derivation is transparent and the iterated-map formulation is elegant; the nonlinear reflection function interpolates continuously between the open-tube and closed-tube limits, and the threshold shift is not obtained by fitting parameters in this paper, since Cnl is taken from earlier experimental studies. The predicted monotonic rise of the oscillation threshold is a falsifiable consequence of the model and could be tested experimentally. However, the quantitative monotonicity claim rests on finite numerical scans, and the physical extrapolation of a high-amplitude fitted loss law to the low-frequency/DC operating point is not validated in the manuscript.

major comments (3)

[Section 2.2, Eq. (4), and Section 5.1.2, Fig. 8] The predicted rise of gamma_osc_up with K0 is governed by the derivative of the nonlinear reflection function r_nl at the nonzero fixed point x* (Eqs. (7) and (11)), not by the large-amplitude behavior of Eq. (4). The coefficient Cnl = 0.7 is taken from high-amplitude oscillatory fits (Refs. 4 and 6), and the paper gives no evidence that the quasi-steady v|v| law with the same Cnl controls the linearized end impedance at the DC operating point that determines x*. If the real termination loss is frequency dependent, hysteretic, or has a different low-velocity slope, the reported 4% threshold increase could shrink, disappear, or reverse. Please either validate the low-frequency applicability of Eq. (4), show that a range of plausible low-velocity slopes preserves the monotone increase, or explicitly restrict the conclusion to the quasi-steady model.
[Section 5.1.2, Fig. 8, with Table 1] The claim that gamma_osc_up "increases monotonically with K0 for all zeta in [0, 0.99]" is a global statement drawn from a single finite grid (Delta_gamma = 1e-3, Delta_zeta = 5e-3, Delta_K0 = 5e-2) without a convergence check. Since f and its iterates have derivative discontinuities, narrow stability windows or small non-monotonic wiggles could be missed. Please provide a grid-refinement study, for example halving each step near the threshold, or give an analytic argument for monotonicity; if neither is available, soften the claim to the observed parameter range.
[Section 3] Fixed points are located by linear interpolation of zero-crossings on a uniform mesh whose bounds are chosen empirically (x_i in [-gamma - 0.1, gamma + 0.1]). The stability classification uses |f'(x*)| < 1, so the result is sensitive to the accuracy of x* and to the behavior of f at kinks. Please report the sensitivity of the R1/R2 boundaries to the mesh size and to the bracket choice, or use a root-finding method that handles non-smooth points.

minor comments (5)

[Introduction, first paragraph] The phrase "in can provide" should read "it can provide".
[Section 5.1.1, paragraph after Fig. 6] "Additionnally" should be "Additionally".
[Figure 6 caption] The caption does not state for which zeta the (lambda, gamma) plane is drawn, or whether it is a projection over zeta; please clarify.
[Eqs. (7) and (11)] The notation changes from r_nl and K0 in Eq. (7) to \hat r_nl and \hat K0 in Eq. (11); the relation \hat K0 = P_M K0 should be restated near Eq. (11) for readability.
[Section 4.2, Fig. 4] The phrase "the area where R2 is unstable shrinks towards high values of zeta" is ambiguous; if the unstable hole shrinks, state that explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the oscillation-threshold increase is a direct numerical consequence of an explicit iterated map, with no fitted parameter renamed as a prediction.

full rationale

The central claim (Section 5.1.2) is obtained by computing the stability of the fixed point of f(x)=gamma-X(gamma-2 rhat(x))-rhat(x) on a parameter grid. The nonlinear-loss reflection function rhat(xi)=lambda^2 xi(1-4/(1+sqrt(1+K0|xi|))) is derived algebraically from the quasi-steady boundary condition p=rho0 Cnl v|v| (Eq. 4); no parameter is fitted to the oscillation-threshold data being reported. The value Cnl=0.7 is imported from the independent experimental fits of Atig et al. (2004) and Dalmont and Frappe (2007), but the paper does not use those experiments as the source of the predicted monotonic increase; it computes the threshold shift from the model. The self-citations (refs. 9, 11, 24, and the Bergeot/Vergez/Karkar papers) are used for background statements (documented high-pressure behavior, multistability existence, bifurcation delay) and are not load-bearing for the derivation. The skeptical concern that the quasi-steady v|v| law may not characterize the real low-frequency end impedance is a modeling-validity or correctness risk, not a circularity: the paper's derivation chain does not assume the conclusion 'threshold increases' at any point. Therefore no reduction of the central result to its inputs can be exhibited.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are invented. The model reuses a known reed characteristic, a known nonlinear loss law, and a frequency-independent loss coefficient; the only new element is the explicit nonlinear reflection function and its use in the iterated map. The central qualitative result is obtained without fitting a target quantity.

free parameters (1)

Cnl (nonlinear loss coefficient) = 0.7 (best fit in Dalmont and Frappe 2007; theoretical maximum 0.5)
Dimensionless vena-contracta coefficient in Eq. (4). Calibrated to experiments in prior literature, not fit to the oscillation-threshold result; the paper also explores values up to 4.3 and K0 up to 100.

assumptions (5)

standard math Stability of a fixed point of f is governed by |f'(x*)| < 1, and stability of fixed points of iterates f(k) identifies stable regimes.
Invoked in Section 3 as the basis for all stability classification.
domain assumption The open end obeys the quasi-stationary turbulent loss relation p = rho0 * Cnl * v * |v| with constant Cnl.
Eq. (4), Section 2.2, taken from Atig et al. 2004 and Disselhorst and Van Wijngaarden 1980; it is the physical input that produces the nonlinear reflection function and the threshold shift.
domain assumption The reed channel follows the nonlinear characteristic Eq. (8) and its inverse from Taillard et al. 2010.
Section 2.3; standard Raman clarinet reed model.
domain assumption Plane-wave propagation with frequency-independent viscothermal losses represented by a constant lambda (Eq. 3).
Section 2.1; the Raman model simplification.
ad hoc to paper All fixed points relevant to regime stability lie in the sampled interval x_i in [-gamma - 0.1, gamma + 0.1], with boundaries chosen empirically.
Section 3; if a fixed point lies outside this range for some extreme parameter values, the stability regions and threshold curves could be incomplete.

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Cite this review

Pith. "Pith review of Oscillation threshold of a Raman clarinet with localized nonlinear losses at the open end." pith.science (2026). https://pith.science/paper/A2RHN6BX

@misc{pith2026241118185,
  author       = {Pith},
  title        = {Pith review of: Oscillation threshold of a Raman clarinet with localized nonlinear losses at the open end},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2RHN6BX}},
  note         = {Machine review of arXiv:2411.18185}
}

read the original abstract

Localized nonlinear losses are taken into account in a simple Raman clarinet model.The complete system is expressed as an iterated map, enabling to study the stability of the different playing regimes. A parametric study is carried out with respect to three major parameters: blowing pressure, embouchure and nonlinear losses coefficient.The model exhibits the well-known effect of reducing the maximum blowing pressure until the oscillations stop (extinction threshold) when nonlinear losses increase.Furthermore, the stability analysis also shows that increasing nonlinear losses increases the minimal blowing pressure for which the oscillations start (oscillation threshold).

Figures

Figures reproduced from arXiv: 2411.18185 by the authors.

**Figure 1.** Evolution of the shape of rˆnl when increasing Kˆ 0 (values are displayed on the figure), with λ = √ 0.95. with KL = 8Cnl/(ρ0c 2 0 ). The reflection function at x = 0 is deduced by substitution of Eq. (3) in Eq. (5): p −(0, t) = rnl[p +(0, t − τ )], where (6) rnl(ξ) = λ 2 ξ ·  1 − 4 1 + q 1 + K0|ξ|   , (7) with K0 = λKL and τ = 2L/c0. It can be checked that the reflection function is passive, i.e for all t > 0 a… view at source ↗

**Figure 2.** Evolution of f when increasing Kˆ 0 (values are displayed on the figure), with λ = √ 0.95, γ = 0.4 and ζ = 0.5. 3 Methods The properties of the iteration function are thoroughly described by Taillard et al. (2010).13 A few notions are recalled here to study the stability of the different regimes. Function f is an iterated function, meaning that xn+k = (f ◦ f ◦ ... ◦ f) | {z } k∈N∗ (xn) = f (k) (xn). For a given set … view at source ↗

**Figure 3.** Different views and cross-sections are given in the Supplementary [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Stability region of R2 in the (γ, ζ) plane, for different values of Kˆ 0 ∈ [0, 2] and λ = √ 0.95. The values of Kˆ 0 corresponding to the different colors are written on the figure. On the left panel, the color patches are superimposed in a fan shape from blue (Kˆ 0 = …

**Figure 5.** Figure 5: Evolution of the dynamic behavior of the clarinet with respect to the linear losses parameter [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Evolution of the playing range of the clarinet with respect to the linear losses parameter [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Same description as Figure [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Evolution of the amplitude of R2 (when stable) with respect to the blowing pressure γ, for ζ = 0.3. (a): variation of Kˆ 0, λ = √ 0.95. (b): variation of λ, Kˆ 0 = 0. 6 Conclusion The main aim of this study is to contribute to the understanding of nonlinear losses that…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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work page 1974

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