# Bifurcation diagrams of a brass instrument model including nonlinear acoustic propagation

Filipe Soares, Vincent Fréour, Bruno Cochelin, Christophe Vergez, Keita Arimoto

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## To cite this version:

Filipe Soares, Vincent Fréour, Bruno Cochelin, Christophe Vergez, Keita Arimoto. Bifurcation diagrams of a brass instrument model including nonlinear acoustic propagation. International Symposium of Musical and Room Acoustics, Acoustical Society of America, May 2025, New Orleans, United States. pp.035026, 10.1121/2.0002161. hal-05242219

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Submitted on 27 Dec 2025

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# Bifurcation diagrams of a brass instrument model including nonlinear acoustic propagation

Filipe Soares Ⓔ; Vincent Freour; Bruno Cochelin; Christophe Vergez Ⓔ; Keita Arimoto

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Proc. Mtgs. Acoust. 58, 035026 (2025)

https://doi.org/10.1121/2.0002161

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ISMRA 2025
NEW ORLEANS
*Playing the Room*

**International Symposium on**
**Musical and Room Acoustics**

24-28 May 2025

Loyola University
New Orleans, Louisiana

**Musical Acoustics**

**Bifurcation diagrams of a brass instrument model including nonlinear**
**acoustic propagation**

**Filipe Soares**

LMA: Laboratoire de mecanique et d'acoustique, Marseille, Bouches-du-Rhone, FRANCE;
filipedcsoares@gmail.com

**Vincent Freour**

Department of Research and Development, Yamaha Corporation, Hamamatsu, Shizuoka, 430-0904, JAPAN;
vincent.freour@music.yamaha.com

**Bruno Cochelin and Christophe Vergez**

LMA: Laboratoire de mecanique et d'acoustique, Marseille, Bouches-du-Rhone, FRANCE;
bruno.cochelin@centrale-med.fr; vergez@lma.cnrs-mrs.fr

**Keita Arimoto**

Yamaha Corporation, Hamamatsu, Shizuoka, 430-0904, JAPAN; keita.arimoto@music.yamaha.com

In this work we use numerical continuation on a 2D model of a simplified brass instrument to assess the effects of nonlinear propagation on the character of its periodic solutions, as the blowing pressure increases. The horn is modelled using terms of its modes, computed from a 2D finite-element discretization of the linearized equations, including radiation and viscothermal losses. The proposed nonlinear acoustics model (Blackstock equation) is then projected onto the linear modal basis to generate a truncated set of nonlinear ODEs. These are then coupled to a simple 1-DoF lip model and the associated flow equation, leading to the final autonomous dynamical system. Bifurcation diagrams are used to assess the changes in periodic solutions as blowing pressure is increased. Results are compared to a linear acoustics model to highlight the effects of nonlinear acoustic propagation.
# 1. INTRODUCTION

In playing conditions, brass instruments are essentially composed by a set of vibrating lips coupled to an acoustic resonator. The musician's lips behave as an "outward-striking" valve¹ whereby an overpressure at the musician mouth will tend to open the lips while a positive acoustic feedback from the resonator will tend to close them. This coupling mechanism results in self-sustained oscillations, that can be studied using numerical continuation of simplified lip-resonator models.

Another defining characteristic of brass instruments is the development of nonlinear acoustic propagation inside the bore when played at large amplitudes². This results in spectral enrichment of the sustained sounds and gives brass instruments their particular timbral quality, their "brassiness". In previous studies, the modelling of nonlinear propagation was pursued using the Burger's equation, likely motivated by its simplicity³. However, this equation is based around directional (backward or forward travelling) waves and since standing waves develop inside the instrument, its validity is at least questionable.

Here we propose the use of a more generic model of nonlinear acoustics, namely the Blackstock equation (equivalent to the Kuznetsov equation), which contains second-order polynomial nonlinear terms and does not rely on any constraints related to the description of standing waves⁴. The acoustic model presented here describes the acoustics of the resonator in terms of its complex modes, stemming from a finite-element discretization of a linearized 3-D axisymmetric model. The linearized model describes radiation effects explicitly through the account of an exterior domain with a non-reflecting boundary. It also includes viscothermal losses occurring at the inner-walls through an impedance boundary condition⁵. Once a suitable complex modal basis is obtained, the Blackstock equation is then projected onto the modal basis leading to a set of truncated set of N nonlinear ODEs. Finally, these modal equations can be coupled to a 1-DoF lip model and associated flow equation to generate the final autonomous system, whose ensuing periodic solutions can be studied through numerical continuation.

# 2. MODEL DESCRIPTION

## A. NONLINEAR ACOUSTICS

For simplicity, we write here the nonlinear acoustics equations in a 1-D framework, even if in practice our model is a 3-D axisymmetric version. Additionally, acoustic perturbations are here modelled in terms of velocity potential $\phi(x, t)$ which has units of [m²s⁻¹]. More common acoustic variables like pressure and acoustic velocity can easily be obtain through either temporal or spatial derivatives of the potential

$$p(x,t) = -\rho \frac{\partial \phi(x,t)}{\partial t} \quad ; \quad v(x,t) = \frac{\partial \phi(x,t)}{\partial x} \qquad (1)$$

We use a second-order weakly-nonlinear acoustics equation known as the Blackstock equation

$$\ddot{\phi} - c^2 \frac{\partial^2 \phi}{\partial x^2} + 2 \left( \frac{\partial \dot{\phi}}{\partial x} \right) \left( \frac{\partial \phi}{\partial x} \right) + (\gamma - 1) \dot{\phi} \left( \frac{\partial^2 \phi}{\partial x^2} \right) = c^2 \frac{Q(t)}{S_0} \delta(x) \qquad (2)$$

where the right-hand-side contains a volume velocity source term, where $Q(t)$ is the flow rate (defined subsequently by the lip-valve equation); $S_0 = a_0^2\pi$ is the surface area of the mouthpiece, with $a_0$ the radius of the mouthpiece entry and $\delta(x)$ is the Dirac delta function, representing the localized nature of the source, at the mouthpiece entry, taken here as $x = 0$. With appropriate boundary conditions (including radiation and viscothermal losses), the linearized version of (2) can be solved for the complex modes of the system, defined by their complex eigenvalues $\lambda_n$ and complex eigenvectors $V_n(x)$. For brevity, we do not provide the details regarding the complex modal decomposition (which is performed in state-space), but these are described in a recent publication⁶. Eventually, we reach a complex modal expansion which reads
$$ \phi(x,t) = \sum_{n=1}^{N} [V_n(x)q_n(t) + \bar{V}_n(x)\bar{q}_n(t)] = 2 \operatorname{Re} \left[ \sum_{n=1}^{N} V_n(x)q_n(t) \right] \quad (3) $$

where the overbar denotes complex conjugation, $q_n(t)$ are the complex-valued modal coordinates for the velocity potential and $V_n(x)$ are the mode shapes normalized to $\max(|V_n(x)|) = 1$. Then, projection of the nonlinear acoustics equation onto a truncated set of $N$ complex modes leads to

$$ \dot{q}_n(t) - \lambda_n q_n(t) + \frac{1}{a_n} \sum_{k,m}^{2N} A_{kmn} \dot{q}_m(t) q_k(t) = c^2 \frac{Q(t) V_n(0)}{S_0 a_n} \quad \text{for } n = 1, 2, \dots, N \quad (4) $$

where $a_n$ is a complex normalizing factor with units of velocity [$\text{ms}^{-1}$] and the constant tensor associated with the nonlinear terms is given by

$$ A_{kmn} = \int_{\Omega} [2V'_{m}(x)V'_{k}(x) + (\gamma - 1)V_{m}(x)V''_{k}(x)]V_{n}(x)dx \quad \text{for} \quad \begin{cases} m, k = 1, 2, \dots, 2N \\ n = 1, 2, \dots, N \end{cases} \quad (5) $$

where $V_{m,k}(x) = [V_n(x) \bar{V}_n(x)]^T$ contains both elements of each conjugate pair of modes. Similarly, the modal coordinates inside the sum operator in (4) are $q_{m,k}(t) = [q_n(t) \bar{q}_n(t)]^T$ with $n = 1, 2, \dots, N$.

## B. 1-DOF LIP VIBRATION MODEL

The equation of motion for the simple vibrating lip model is given by

$$ m_l \ddot{y}(t) + \frac{m_l \omega_l}{Q_l} \dot{y}(t) + m_l \omega_l^2 (y(t) - y_l) = (P_l - p_0(t)) S_0 \quad (6) $$

where $m_l$, $\omega_l$ and $Q_l$ are the modal mass, angular frequency and quality factor of the lip-oscillator; $y_l$ is the static vertical position of the lips at rest (without flow); $y(t)$ is the lip's vertical displacement; $P_l$ is the (static) pressure at the mouth and $p_0(t)$ is the pressure inside the mouthpiece.

Defining a lip velocity as $v(t) = \dot{y}(t)$ we obtain the first-order version of the lip motion

$$ \left\{ \begin{aligned} v(t) &= \dot{y}(t) \\ \dot{v}(t) + \frac{\omega_l}{Q_l} v(t) + \omega_l^2 (y(t) - y_l) &= \frac{S_0}{m_l} (P_l - p_0(t)) \end{aligned} \right. \quad (7) $$

## C. FLOW THROUGH LIPS

For the flow through the lips, we consider a generic lip-opening area $S_l(t)$. For the simplest models, it can be assumed rectangular (with width $b_l$), hence $S_l(t) = b_l y(t)$. Then, the unsteady flow rate $Q(t)$ passing through the lips is given by a (quasi-static) Bernoulli relation

$$ Q(t) = S_l(t)u(t) = S_l(t)\sqrt{\frac{2(P_l - p_0(t))}{\rho}}\operatorname{sign}(P_l - p_0(t)) \quad (8) $$

where $u(t)$ is the average flow velocity passing through the lips and the $\operatorname{sign}(P_l - p_0(t))$ function is present to allow the possibility of reverse flow (when $p_0(t) > P_l$).

## D. FULL COUPLED SYSTEM

Finally, the fully coupled autonomous system is given by

$$ \left\{ \begin{aligned} \dot{q}_n(t) &= \lambda_n q_n(t) - \frac{1}{a_n} \sum_{k,m}^{2N} A_{kmn} \dot{q}_m(t) q_k(t) - c^2 \left( \frac{V_n(0)}{a_n} \right) \frac{Q(t)}{S_0} && \text{for } n = 1, 2, \dots, N \\ \dot{v}(t) &= v(t) \\ \dot{v}(t) &= -\frac{\omega_l}{Q_l} v(t) - \omega_l^2 (y(t) - y_l) + \frac{S_0}{m_l} (P_l - p_0(t)) \end{aligned} \right. \quad (9) $$
with

$$Q(t) = b_l y(t) \sqrt{\frac{2(P_l - p_0(t))}{\rho}} \text{sign}(P_l - p_0(t)) \quad (10)$$

In this work, the nonlinear dynamics of system above have been studied using the MANLAB software⁷, which combines the use of the harmonic balance method, for the discretization of periodic solutions, and the asymptotic numerical method, which is a high-order Taylor-based expansion method for the efficient numerical continuation of nonlinear systems.

## 3. ILLUSTRATIVE RESULTS

In this section we present some illustrative results of studies performed on a simplified trumpet made of 7 cylindrical/conical sections (including the mouthpiece), whose detailed geometry can be found in a recent publication⁸.

### E. COMPLEX MODAL BASIS IN 3D AXISYMMETRIC DOMAINS

Firstly, to give the reader a clearer perspective on the modal basis used to discretize the associated acoustics PDE, Figure 1 shows some examples of complex mode shapes of the simplified trumpet model. Note that, contrary to the motion of real modes where all points move in unison, complex modes contain phase shifts between different spatial regions. Hence, while real modes represent standing wave patterns, complex modes are a combination of standing and travelling waves. They are generally represented by their real Re[V_n(r, z)] and imaginary Im[V_n(r, z)] parts separately. Moreover, since complex modes are defined up to a multiplicative complex factor, it is often useful to “rotate” the complex mode by an angle β_n (i.e. multiply by e^(-iβ_n)) such that it maximizes its real part⁹. In this way, we can assess the “degree of complexity” of each mode.

From Figure 1, notice that lower order modes have a nearly null imaginary part, meaning that they represent almost perfect standing waves (i.e. real modes). Higher order modes on the other hand have much larger imaginary parts, representing their essentially radiative/dissipative character.

It is worth underlining as well that, contrary to more common 1-D models, here we calculate explicitly not only the acoustic oscillations inside the instrument, but also the radiated sound.

<figure><img src="image_7.png" /><figcaption>**Figure 1.** Some illustrative complex mode shapes V_n(r, z) of the simplified trumpet. The real and imaginary parts are represented in the top and bottom of each plot, respectively. Note: modes shapes were rotated such that their real part is maximized.</figcaption></figure>
## F. BIFURCATION DIAGRAMS OF THE COUPLED SYSTEM

Here we present some illustrative results of numerical continuation for a particular configuration, where the mouth pressure $P_l$ is taken as the bifurcation parameter. The fixed parameters in system (10) are the acoustic modal parameters $\lambda_n$, $V_n$ and $a_n$ (stemming from the 2D FE model) and the lip-oscillator parameters, taken as: $m_l = 0.4$ g, $\omega_l = 390 \times 2\pi$ rad·s$^{-1}$, $Q_l = 3$, $y_l = 0.1$ mm and $b_l = 5$ mm.

Firstly, the equilibrium solution, along with its stability, is sought in order to obtain the Hopf bifurcation point, i.e. the turning point at which if $P_l > P_l^*$ the system leads to self-sustained oscillations. Subsequently, the branch of periodic solutions stemming from the Hopf point can be calculated with respect to bifurcation parameter $P_l$. For all results shown here, modal truncation was set to $N = 50$ while the harmonic truncation (in the HBM) was set to $H = 15$.

To assess the effects of nonlinear propagation on the obtained periodic solutions, the above procedure was performed on an equivalent system with a linearized acoustics inside the instrument (i.e. $A_{kmn} = 0$) as well as for the fully nonlinear system. Figure 2 shows a bifurcation diagram for the two cases relating the control parameter, mouth pressure $P_l$, and the maximum absolute value of the pressure inside the mouthpiece $\max(|p_0(t)|)$ (left) as well as the peak radiated pressure (right).

<figure><img src="image_8.png" /><figcaption>**Figure 2.** Bifurcation diagram of a trumpet-lips model with and without including nonlinear propagation inside the instrument. The left plot shows the maximum absolute value of the pressure inside the mouthpiece $\max(|p_0(t)|)$ while the right plot shows the maximum absolute value of the radiated on-axis pressure, both against the mouth pressure $P_l$. Solid and dotted lines indicate stable and unstable solutions, respectively.</figcaption></figure>

The results in Figure 2 show that the nonlinear propagation effects come into play gradually as the associated pressure levels increase. In terms of the mouthpiece pressure, we note the nonlinear effects seem to restrain peak values. This is expected since nonlinear propagation will promote the transfer of energy from lower to higher frequencies, which will then be dissipated more effectively through radiation and viscothermal losses. On the other hand, we note that the peak of the radiated pressure increases exponentially when nonlinear propagation is included. It is perhaps more informative to detail the nature of the periodic oscillations in the two cases. To this end, Figure 3 shows a variety of indicative values (lip displacement, unsteady flow rate, radiated pressure, etc.) for the nonlinear and linearized model when the mouth pressure is fixed at $P_l = 13$ kPa.

In Figure 3 we get a better idea of the changes brought about by the addition of the nonlinear propagation terms. It seems that the nonlinear terms have little qualitative effect on the variables associated with the lip motion, as lip displacement $y(t)$, flow rate $Q(t)$ and mouthpiece pressure $p_0(t)$ remain qualitatively similar. However, we note a very pronounced change in the radiated on-axis pressure, with a much sharper peak in the response, in line with the typical behavior measured experimentally, with significant spectral enrichment. This is also seen in the calculated pressure fields which, in the nonlinear case, present larger derivatives (shaper wave fronts) particularly towards the end of the instrument.
<figure><img src="image_9.png" /><figcaption>**Figure 3.** Comparison of periodic solutions for the model with (red) and without (blue) nonlinear propagation for a mouth pressure of $P_l = 13 \text{ kPa}$. The plots on the left show the periodic motions of the lip displacement, flow rate, pressure at the mouthpiece and radiated pressure. The plots on the right show snapshots of the pressure fields inside (and outside) the instrument during one cycle.</figcaption></figure>

## 4. CONCLUSION

In this study, we show how the effects of nonlinear propagation can be included in typical acoustic models of brass instruments given in a modal framework. We illustrate how complex modal superposition not only allows us to model nonlinear acoustics with a reduced-order model but it also accounts for the localized effects of radiation and viscothermal losses. Contrary to the more commonly used Burger's equation, using the Blackstock equation, nonlinear acoustics can be calculated in standing wave scenarios. The bifurcation diagrams presented here show that, at large amplitudes, linear models for the acoustics are largely inadequate as pressure levels grow monotonically with increasing mouth pressure. The nonlinear terms act to both constrain excessively large pressure amplitudes and also result in spectral enrichment of the standing wave patterns, with sharper wave fronts (larger pressure derivatives) concentrated near the end of the instrument. Notably, the effects of nonlinear propagation are significant when examining the radiated pressure, with the generation of sharp peaks, consistent with behavior observed experimentally.

## REFERENCES

¹ A. Myers, J. Gilbert and M. Campbell, "The Science of Brass Instruments", Springer (2021).

² A. Myers, R. Pyle, J. Gilbert, M. Campbell, J. Chick and S. Logie, "Effects of nonlinear sound propagation on the characteristic timbres of brass instruments," J. Acoust. Soc. Am. vol. 131, no. 1, pp. 678-688, (2012).

³ S. Maugeais and J. Gilbert, "Nonlinear acoustic propagation applied to brassiness studies, a new simulation tool in the time domain," Acta Acustica (2017).

⁴ B. Kaltenbacher and R. Brunnhuber, "Nonlinear Acoustics," Snapshots of modern mathematics from Oberwolfach (2019).

⁵ M. Berggren, A. Bernland and D. Noreland, "Acoustic boundary layers as boundary conditions," *Journal of Computational Physics*, vol. 371, pp. 633-650 (2018).

⁶ F. Soares, B. Cochelin, V. Freour, C. Vergez, K. Arimoto, "Modal approaches for modelling nonlinear acoustics in 2D axisymmetric domains: application to brass instruments", Forum Acusticum, Malaga, Spain (2025).

⁷ B. Cochelin, C. Vergez, "A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions", J. Sound and Vib. (2009).

⁸ B. Cochelin, V. Freour and C. Vergez, "Exploiting the use of strong nonlinearity in dynamics and acoustics: the case of musical wind instruments", Springer Cham (2024).

⁹ P. Vacher, B. Jacquier and A. Bucharles, "Extensions of the MAC criterion to complex modes," in Proceedings of ISMA2010, Leuven, Belgium (2010).