Rémi Mattéoli

Joël Gilbert

Christophe Vergez

Jean-Pierre Dalmont

Sylvain Maugeais

Soizic Terrien

Frédéric Ablitzer

Minimal blowing pressure allowing periodic oscillations in a model of bass brass instruments

In this study, an acoustic resonator -a bass brass instrument -with multiple resonances coupled to an exciter -the player's lips -with one resonance is modelled by a multidimensional dynamical system, and studied using a continuation and bifurcation software. Bifurcation diagrams are explored with respect to the blowing pressure, in particular with focus on the minimal blowing pressure allowing stable periodic oscillations and the associated frequency.

.

Cases are highlighted where periodic solutions in the bifurcation diagrams are reached for blowing pressures below the value given by the linear stability analysis. Thus, bifurcation diagrams allow a more in-depth analysis. Particular attention is devoted to the first playing regime of bass brass instruments (the pedal note and the ghost note of a tuba in particular), whose behaviour qualitatively differs from a trombone to a euphonium for instance.

Introduction

One main goal of the acoustics of wind instruments is to describe and quantify the intonation and ease of playing of an instrument. From the physics viewpoint, it is interesting to model the coupled system formed by the musician and the instrument. Of particular interest is the influence of the musician's control param-eters on the oscillation frequency (linked to the intonation), and the minimum mouth pressure required to achieve auto-oscillations (related to the ease of playing). Indeed, it is assumed here that the musician's feeling of ease of playing partly relies on the threshold blowing pressure: the higher the latter, the higher the physical effort the player has to make to play a note [START_REF] Campbell | The Science of Brass Instruments[END_REF]. In practice, the musician can play several notes without depressing any valves in the case of a tuba, or moving the slide in the case of a trombone. These playing regimes are called the natural notes (B 1, B 2, F3, B 3, D4, F4,... in the case of a trombone or a euphonium for instance), and their frequencies are close to the resonance frequencies of the instrument as a whole, except for the lowest note playable (B 1).

Several tools are available to infer the oscillation frequencies and the threshold blowing pressures from a model of the system. Linear stability analysis allows to predict the behaviour of a dynamical system in the vicinity of its equilibrium solutions. The method consists in linearising the system and analysing its eigenvalues to determine whether or not the equilibrium solutions are stable. This method has already been applied to physical models of musical instrumentsbrass instruments -in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF]. Linear stability analysis is, in essence, only valid close to the equilibrium solutions. As such, this method alone might not explain the existence of periodic regimes, as highlighted for instance in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF] for the first regime of a saxhorn.

Alternatively, bifurcation analysis gives access to an extensive knowledge of permanent regimes (namely equilibrium and periodic regimes) of the dynamical system, as well as their stability when relevant. Bifurcation diagrams ideally represent all the families of equilibrium and periodic solutions with respect to one parameter of interest. In this paper, we investigate bifurcation diagrams of a physical model of a brass instrument to understand some aspects of its behaviour.

We consider here a simple model: the player's lips are modeled through the 'outward-striking valve' [START_REF] Campbell | The Science of Brass Instruments[END_REF], Cullen et al., 2000, Elliott and Bowsher, 1982], a one degree-of-freedom system.

Also, the nonlinear sound propagation in the instrument's bore which accounts for the 'brassy sounds' [START_REF] Campbell | The Science of Brass Instruments[END_REF] at high sound levels is neglected [START_REF] Msallam | Physical Model of the Trombone Including Nonlinear Effects. Application to the Sound Synthesis of Loud Tones[END_REF], Berjamin et al., 2017]. The airflow blown by the musician into the instrument is described by a nonlinear algebraic equation. Some of the functions in this equation being nonsmooth, a regularised form of this equation is used for numerical reasons.

In section 2, we describe the considered model and numerical methods, namely linear stability analysis and numerical continuation. In sections 3 and 4, we use linear stability analysis and bifurcation diagrams, respectively, to investigate the minimal blowing pressure and playing frequencies of the trombone. Eventually, we focus in section 5 on the first regime of two bass brass instruments (the trombone and the euphonium), mainly thanks to bifurcation diagrams. This highlights a major difference in the behaviour of these two instruments.

Theoretical background

Generic brass model

This subsection details the brass-instrument model considered throughout the article. Brass instruments as a whole can be described through both linear and nonlinear mechanisms. More precisely, a localised nonlinear element (the lips' valve effect, i.e. the airflow modulation caused by the lips' vibration) excites a passive linear acoustic multimode element (the musical instrument, usually characterised by its input impedance in the frequency domain). The latter acoustic resonator exerts, in turn, a retroaction on the former mechanical resonator. Such musical instruments are self-sustained oscillators: they generate an oscillating acoustic pressure (the note played) from a static overpressure in the player's mouth (the blowing pressure).

Figure 1: Schematic representation of the player's mouth coupled to the mouthpiece of a brass instrument.

The brass instrument coupled to the player is then described by a system of three equations. Because it relies on major simplifications [Campbell et al., 2021, Elliott and[START_REF] Elliott | [END_REF], this model is often referred to as 'elementary'. More precisely, the three equations link the lip-opening height h(t), the pressure in the mouthpiece p(t) and the volume airflow entering the instrument u(t), which will be the three independent variables of interest in this paper. They are all represented schematically in figure 1. Several control parameters (i.e controlled by the musician) are also involved: this includes in particular the blowing mouth pressure p m and the lips' resonance frequency f .

First, the vibrating lips of the musician are described by a one-degree-of-freedom damped oscillator:

ḧ + ω Q ḣ + ω 2 (h -H) = p m -p µ , (1)
where ω = 2πf and Q are the angular resonance frequency and the quality factor of the lips, respectively, µ is the lips' mass per unit area, and H is the lip-opening height at rest. The Bernoulli's theorem applied between the mouth and the mouthpiece and taking into account the pressure drop caused by the presence of turbulence in the mouthpiece leads to the following equation:

u(t) = wh + (t) sgn (p m -p(t)) 2 |p m -p(t)| ρ , (2)
with w the lip-opening width (considered as constant) and ρ the air density. Here, h + = max(h, 0) accounts for the fact that the lips cannot physically interpenetrate: as soon as the lips touch (h = 0), the volume flow is forced to zero. The sign function sgn accounts for the possibility of air flowing from the instrument into the player's mouth.

Finally, the acoustic input impedance Z(ω) of the resonator is described in the Fourier domain as the ratio between the pressure and the volume flow at the input of the instrument. This provides a last link between the mouth pressure and the volume airflow:

P (ω) = Z(ω)U (ω), (3)
with ω the angular frequency. Figure 2 shows the modulus and phase of Z with respect to the angular frequency ω, for a bass trombone (A. Courtois Legend AC502). Several limitations of this generic model have already been pointed out in previous works [START_REF] Campbell | The Science of Brass Instruments[END_REF], Velut et al., 2017]. Firstly, it is worth noting that the instrument is described with a linear model of its input impedance: the nonlinear propagation of sound in the instrument itself is not taken into account, preventing any description of the characteristic 'brassy sound' of a brass instrument at high sound levels. However, since we focus here on the emergence of auto-oscillations, this effect can be neglected. Furthermore, the harmonics created by the nonlinear distortion of the wavefront propagating in the instrument are known not to be reflected back to the player when they reach the bell [START_REF] Hirschberg | Shock Waves in Trombones[END_REF].

Secondly, lips are described as a one-degree-offreedom damped oscillator, which is also a strong assumption. As a matter of fact, both experiments using artificial lips [START_REF] Cullen | Brass Instruments: Linear Stability Analysis and Experiments with an Artificial Mouth[END_REF] and in-vivo measurements [START_REF] Newton | Mechanical response measurements of real and artificial brass players lips[END_REF] demonstrated that the lips' oscillation mechanism relies mainly on a pair of lips' resonance frequencies. Furthermore, only the first resonance of the lips reflects the 'outward-striking swinging door' mechanism described in [START_REF] Campbell | The Science of Brass Instruments[END_REF], hence the single degree of freedom. In any case, the one-degree-offreedom damped oscillator prevents any description of the 'buzzing phenomenon', which corresponds to the lips' vibration without the instrument.

Thirdly, the lips' opening area S in the airflow (equation 2) is assumed to be proportional to h, and is thus written S = w × h with w a constant. This assumption was only partially validated in [START_REF] Bromage | Open Areas of Vibrating Lips in Trombone Playing[END_REF], depending on which regime is studied as well as on the sound level. In particular, a law S ∝ h 2 would be more adequate when studying the first regimes like the pedal note.

Numerical considerations

In this paper, we aim at determining the minimal blowing pressures for which each periodic regime (natural notes B 1, B 2, F3, B 3, D4, F4,...) is observable, as well as the corresponding playing frequencies.

In this subsection, the numerical methods allowing for the investigation of the periodic solutions of model {(1) ∪ (2) ∪ (3)} are presented, as well as the modal representation of the input impedance required for the practical implementation of these methods.

Numerical methods

Several numerical methods are available to investigate the influence of a control parameter -such as p m or f , as they are the most obvious parameters the player can change -on the dynamics of the system. Linear stability analysis [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF] consists in studying the stability of the equilibrium solutions to small perturbations, obtained by zeroing all time derivatives. This allows one to determine the threshold value of the control parameter at which the equilibrium solution destabilises, which means that at least one of the eigenvalues of the system exhibits a positive real part, and where a small-amplitude periodic regime (stable or unstable) can emerge. When a small-amplitude periodic regime indeed emerges, such a change in the system's behaviour is called a Hopf bifurcation [Kuznetsov, 2004].

In the case of brass instruments, both p m and f can be considered as control parameters. Figure 4 (blue curve) represents the threshold pressure predicted by the linear stability analysis with respect to f . This analysis also provides the threshold frequency, that is to say the frequency of the periodic regime that emerges when the equilibrium solution destabilises. This is represented in the bottom plot in figure 4.

One of the main advantages of this method is its straight-forward implementation. However, it gives very little information on the oscillation regime of the model far from the so-called Hopf bifurcation point at which the equilibrium loses its stability. This method has already been used in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF], and is applied here to the trombone (see blue curves in figure 4).

To know more about the oscillating solution arbitrarily far from the bifurcation point (i.e the point at which the equilibrium solution becomes unstable), another approach consists in numerically solving the whole system, thanks to an ODE solver [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF]. In doing so, both the transient and the stationary parts of the solution are obtained for any value of the control parameter. Nevertheless, this approach becomes tedious and unsuitable in the context of the systematic investigation of the influence of the control parameter on the oscillation regime.

Continuation methods, on the other hand, are more suitable to gain access to a more extensive view of all the oscillating regimes of the system. It consists in computing the waveform of the oscillating solution for successive values of the control parameter. The waveform corresponding to a new value of the control parameter is then deduced from previously computed waveforms, through a predictor/corrector algorithm. The behaviour of an oscillatory solution of the system with respect to a control parameter such as p m can then be assessed by plotting bifurcation diagrams, which are shown in section 4. This approach is implemented in several softwares, such as AUTO [START_REF] Doedel | AUTO 97: Continuation And Bifurcation Software For Ordinary Differential Equations[END_REF], Gilbert et al., 2020, Cochelin and Vergez, 2009, Karkar et al., 2013, Fréour et al., 2020] which is used in this publication. However, continuation methods require the system to be written in the form dX dt = F (X), with certain smoothness properties on F . Therefore, some work has yet to be done on the equations presented in section 2.1, which is done in the following.

Input impedance

Since the continuation method requires the equations to be written in the time domain, equation (3) cannot be used as it is. An analytical form of the input impedance Z is therefore required to perform an in-verse Fourier transform of (3). This quantity can be quite easily measured, and is represented in figure 2 (blue curve). This measured impedance can then be numerically fitted, in the frequency domain, by a sum of N individual acoustical resonance modes of the following form [Ablitzer, 2021] :

Z fit (ω) = N n=1 jωA n ω 2 n -ω 2 + 2jξ n ω n ω . (4)
Here, (A n , ω n , ξ n ) ∈ R 3 are the modal parameters of the modal decomposition. For numerical reasons, it has been chosen to convert this set of parameters (A n , ω n , ξ n ) into another one (C n , s n ) so that the fitted input impedance writes ( z * is the complex conjugate of z):

Z fit (ω) = Z c N n=1 C n jω -s n + C * n jω -s * n , (5)
where Z c is the characteristic input impedance of the resonator defined as Z c = c Se with S e the input crosssectional area, and (C n , s n ) ∈ C 2 are defined as follows:

     C n = An 2 1 + j ξn √ 1-ξ 2 n , s n = ω n -ξ n + j 1 -ξ 2 n . (6)
The new coefficients (C n , s n ) also verify the relation Re (C n s * n ) = 0, since one switched from a 3-real parameters description with (A n , ω n , ξ n ) to a 2-complex parameters description with (C n , s n ).

The fitted impedance Z fit (ω) is plotted (orange dashed curve) in figure 2.

Reinjecting equation ( 5) in equation ( 3) and applying an inverse Fourier transform (for details, see appendix A) leads to the following expression in the time domain [START_REF] Fréour | Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons[END_REF], Silva et al., 2014]:

ṗn = Z c C n u + s n p n , n ∈ 1, N . (7)
The mouthpiece pressure p can then be written as p = 2

N n=1 Re (p n ). Eventually, the system {(1) ∪ (2) ∪ (7)} is now written in the form dX dt = F (X), with: X = {X m } m∈ 1,2(N +1) = h; ḣ; {Re(p n )} n∈ 1,N ; {Im(p n )} n∈ 1,N (8)
the state vector, so that h = X 1 , ḣ = X 2 , and p = 2

N n=1 Re (p n ) = 2 N +2 n=3 X n .
Taking the real and imaginary part of the N equations ( 7) yields 2N real equations, so that the nonlinear function F is defined as:

F : X →                    X 2 -ω Q X 2 -ω 2 (X 1 -H) + pm-2 N +2 n=3 Xn µ Re [s 1 (X 3 + jX N +3 ) + Z c C 1 u(X)] Re [s 2 (X 4 + jX N +4 ) + Z c C 2 u(X)] . . . Re s N X N +2 + jX 2(N +1) + Z c C N u(X) Im [s 1 (X 3 + jX N +3 ) + Z c C 1 u(X)] Im [s 2 (X 4 + jX N +4 ) + Z c C 2 u(X)] . . . Im s N X N +2 + jX 2(N +1) + Z c C N u(X)                    , (9)
with u :

X → wX + 1 sgn p m -2 N +2 n=3 X n × 2 p m -2 N +2 n=3 X n .

Regularisation of the volume airflow

For the continuation method implemented in AUTO to work correctly, F has to be a C 1 vector function.

In this respect, noting that sgn(x) = x/ |x| for x = 0 and x + = max (x, 0) = 1 2 (x + |x|), equation ( 2) is regularised based on the regularisation |x| ∼ η→0 =

x 2 + η used in section 2 of [START_REF] Colinot | Influence of the "Ghost Reed" Simplification on the Bifurcation Diagram of a Saxophone Model[END_REF]:

u ∼ η→0 w × h + h 0 h h0 2 + η 2 × p m -p √ p 0 4 pm-p p0 2 + η , (10) or equivalently u ∼ η→0 w × X 1 + h 0 X1 h0 2 + η 2 × p m -2 N +2 n=3 X n √ p 0 4 pm-2 N +2 n=3 Xn p0 2 + η (11)
in terms of the components of the state vector X, with η the regularisation parameter, which is fixed to 10 -6 in the following. p 0 is defined in a similar way as the closure pressure for woodwind instruments [START_REF] Gilbert | Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation[END_REF]: p 0 = µω 2 0 h 0 , except that ω 0 is chosen close to the first resonance frequency of the resonator. Indeed, the choice of ω generally considered for woodwind instruments is not suitable in the case of brass instruments where ω is no longer constant. In practice, the choice is not ω 0 = ω res,1 (where {ω res,n } n∈ 1,N are the resonance angular frequencies of the resonator), but rather ω 0 = ω res,4 /4, because the fourth resonance of instruments having the same tube length (a trombone and a euphonium for instance) appears to be quite constant, contrary to the first resonance frequency which can vary up to 8 semitones between a trombone and euphonium.

In the following, the system equations (9) with u(X) given by equation ( 11) are processed numerically in a dimensionless form, which is detailed in appendix B. Typical bifurcation diagrams will be shown in section 4.

Lips' parameters and linear stability analysis

The results of the linear stability analysis detailed in subsection 3.1 suggest that the lips' parameter f can be tuned so as to obtain a given desired oscillatory regime. The values of the other lips' parameters are difficult to measure and thus hardly found in the literature. In a first subsection, we consider the same parameters as in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF], which are detailed in table 1. In order to be more consistent with experimental results, we introduce, in a second subsection; a frequency-dependent lip-opening at rest, based on experimental data from [START_REF] Elliott | Regeneration in brass wind instruments[END_REF].

Constant lip-opening height at rest

In the case of constant lips' parameters, the results of the linear stability analysis of the system are represented in figure 4 by the blue U-shaped patterns. As described in subsection 2.2, these results give the values of the mouth pressure p m at which the equilibrium solution destabilises (the threshold mouth pressures), as well as the related frequency at which the oscillating solution emerges (the threshold frequencies).

As a matter of fact, the multiple U-shaped patterns on the top plot of figure 4 reflect the fact that for a given configuration of the resonator (i.e without moving the slide nor activating any trigger), the trombone player can play several notes called 'natural notes' just by changing the lips' resonance frequency (see for instance [START_REF] Campbell | The Science of Brass Instruments[END_REF]) represented in figure 5, from the B 1 (58 Hz) to B 4 (466 Hz) approximately.

For each natural note or regime n to emerge, there is an optimal lips' resonance frequency f opt,eq

,n which corresponds to the minimum of a U-shaped pattern. This is associated with an optimal threshold mouth pressure p opt,eq thresh,n and an optimal threshold frequency f opt,eq thresh,n . The notation f opt,eq

,n

has been chosen to be as consistent as possible with [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF]: in this article, the equilibrium optimal threshold value (i.e. the threshold inferred from the linear stability analysis) of a quantity q in the n th regime was written q opt thresh,n . Here, the threshold quantities are obtained either using linear stability analysis or through a bifurcation analysis, which will be defined and addressed in section 4. Therefore, it has been chosen to add an extra superscript 'eq' for 'equilibrium', referring to the linear stability analysis, or 'per' for 'periodic solutions', referring to the analysis of bifurcation diagrams. Furthermore, it can be noticed that the patterns on the bottom plot of figure 4 are always above the line whose equation is f thresh = f , which is a characteristic of the outward-striking valve model: the instrument always plays a note slightly above the lips' resonance frequency.

Frequency-dependent lip-opening height at rest

In order to assess the validity of the results presented above, a comparison is drawn with the minimal threshold mouth pressures estimated experimentally in [START_REF] Gilbert | L'analyse de stabilité linéaire pour évaluer la facilité d'émission des cuivres -étude comparative de trombones ténor et basse[END_REF], for the exact same trombone as the one considered in this paper. In figure 4, the comparison between the minima of the blue U-shaped patterns and the red dots (representing the experimental measures of the minimal threshold mouth pressures) highlights significant differences. In particular, the results of the linear stability analysis, on the one hand, and of the experiment, on the other hand, are quite different: compared to the ex-perimental data, threshold pressures computed numerically are too low for the first regimes, and too high for the last regimes. To overcome this issue, additional information coming from experiments in [START_REF] Elliott | Regeneration in brass wind instruments[END_REF] is considered. Indeed, the lips' opening height at rest H is measured to decrease monotonously with f . The experimental points are plotted in black on the top plot of figure 3, together with two regression functions which aim at fitting these points. The inverse function H = k/f (k is given in table 1) appears here to better fit the experimental points than the linear regression H = af + b. Therefore, the former H-profile will be retained hereafter, even if the case H = h 0 is still given alongside for sake of comparison until the end of section 4. Figure 3: Considered H-profiles vs. lips' resonance frequency. Blue line corresponds to the constant value of H = h 0 from [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF], black points correspond to experimental data from [START_REF] Elliott | Regeneration in brass wind instruments[END_REF], and the orange and green curves correspond to the regression functions chosen to fit the experimental points, respectively H = af +b and H ∝ 1/f .

h 0 [m] w [m] Q 1 µ [m 2 kg -1 ] 5.0 × 10 -4 1.2 × 10 -2 7.0 1.1 × 10 -1 [kg m -3 ] c 0 [m s -1 ] k [m Hz] 1.2 3.4 × 10 2 7.4 × 10 -2
Table 1: Lips' parameters used in the present publication, mostly taken from [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF].

Bifurcation diagrams and minimal blowing pressure

In this section several typical bifurcation diagrams of a trombone are shown and discussed in detail. The additional information obtained with the continuation method on the threshold mouth pressure is also discussed, leading to new conclusions compared to the case of the linear stability analysis.

Case of the third regime

In this section the example of the third regime of a trombone is considered (F3, third U-shaped pattern on top plot of figure 4), as it shows a variety of different behaviours of the system.

The left plot of figure 6 shows the bifurcation diagrams obtained by continuation for a constant Hprofile H = h 0 and for three different but close values of f . For each branch of periodic solution, there is a global minimum of the mouth pressure (identified by the vertical dotted lines) below which the regime does not exist anymore. This minimum can be lower than or equal to the linear threshold mouth pressure given by the linear stability analysis in section 3. As for this latter method, an optimal threshold mouth pressure p opt,per thresh,n1 which can be lower than or equal to p opt,eq thresh,n is introduced, as well as the associated optimal threshold frequency f opt,per thresh,n and the associated optimal lips' resonance frequency f opt,per ,n

. For instance, in the case f = 175.09 Hz (green curve) the observed bifurcation is a direct Hopf [Kuznetsov, 2004] and the minimal blowing pressure is the same as the one given by the linear stability analysis. On the contrary, in the case f = f opt,eq ,3 (blue curve) and f = f opt,per ,3 (orange curve), a small stable portion at the very beginning of the branch is noticed, thus rigorously classifying the bifurcation as a direct Hopf. However, since this stable portion only exists for a narrow range of p m it would be very difficult to observe it experimentally, hence we choose to neglect it. In the following, this configuration will be referred to as 'almost-inverse Hopf' instead. In the case of this almost-inverse Hopf bifurcation, the solution emerging from the equilibrium is unstable (if we neglect the small stable portion at the very beginning) and stabilises further on the branch at a value of threshold mouth pressure lower than at the equilibrium. For f = f opt,per ,3

, the actual minimal blowing pressure is not only below the linear threshold pressure but also below the minimal blowing pressure given by the case f = f opt,eq ,3 .

However, it is worth noting that the linear stability analysis provides a rather accurate first guess of the note which is the easiest to play. Indeed, the frequency interval between the easiest note according to the linear stability analysis (corresponding to the minimal blowing pressure of the blue curve on figure 6) and according to the bifurcation diagrams (corresponding to the minimal blowing pressure of the orange curve on figure 6) is less than a semitone.

Eventually, it is reassuring to note that even if the variable H-profile H ∝ 1/f has a significant impact on the threshold pressure of the system, it has nonetheless qualitatively no influence on its threshold frequencies as shown on bottom plot of figure 4.

For the interested reader, an animation showing the evolution of the bifurcation diagrams as a function of the lips' resonance frequency f is provided online2 . It especially shows how the bifurcation can go from almost-inverse Hopf to direct Hopf through the merging of the two stable portions of the branch into a single one.

Minimal blowing pressures

The investigation performed in subsection 4.1 for two values of f is repeated for a wide range of lips' resonance frequency. For each value of f , the global minimal blowing pressure is extracted from the bifurcation diagrams, and these threshold values are compared with the results of linear stability analysis. The results of both methods are represented in figure 7, which clearly highlights the added value of the continuation approach. In particular, the fact the first regimes (B 1, B 2 and F3) emerge through inverse Hopf bifurcations results in an actual threshold pressure which is lower than the one predicted by the linear stability analysis. Nevertheless, it is worth noting that the system only exhibit direct Hopf bifurcations for regimes ≥ 4. This means that the linear stability analysis is sufficient to characterise the threshold mouth pressures for these regimes. Nevertheless, it is worth noting that similar investigations performed for a trumpet (which is very similar to a trombone considering its bore, except that its pitch is an octave higher) in [START_REF] Fréour | Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons[END_REF] highlighted an inverse Hopf bifurcation for the fourth regime. Also, the behaviour observed here for the trombone has also been observed in the case of the euphonium or the saxhorn, except for the first regime which will be addressed in section 5.

As in subsection 4.1, it is reassuring here to note that the variable H-profile H ∝ 1/f has qualitatively no influence on the system's behaviour discussed above, in particular its threshold frequencies as shown on bottom plots of figure 7.

Among all the regimes that can be played by the musician according to figure 7 and figure 5, the first one is especially worth focusing on. Indeed, it exhibits a very different behaviour depending on whether the instrument has a predominantly-cylindrical bore profile (case of the trombone or the trumpet for instance) or a predominantly-expanding bore profile (case of the tubas or the flugelhorn); see for instance figure 7.34 of [START_REF] Campbell | The Science of Brass Instruments[END_REF]. In this respect, the last section of this paper focuses on the study of the first regime of the trombone (predominantly-cylindrical bore profile) and the euphonium (predominantlyexpanding bore profile).

5 Bifurcation diagrams, case of the first regime

Case of the trombone

Similarly to the case of the third regime (see subsection 4.1), the bifurcation diagrams of the trombone show that the first regime generally emerges through an inverse Hopf bifurcations (see figure 8). As already discussed in subsection 4.2, this leads to an optimal threshold pressure p opt,per thresh,1 smaller than p opt,eq thresh,1 . = 164.00 Hz (left) or 165.17 Hz (right); green curves: case f = 175.09 Hz, which corresponds to an arbitrary f value at which the Hopf bifurcations are direct. The line thickness indicates whether the branch portion is stable (thick line) or unstable (thin line). The vertical dashed lines locate the value of the minimal threshold mouth pressure of each curve. However, it is worth noting that, similarly to other regimes, only one note is accessible to the musician since only one branch of the bifurcation diagrams is stable. The corresponding note is called the 'pedal note' (B 1, lowest note playable in first position and roughly one octave below the second regime B 2, see also figure 5).

The optimal playing frequency f opt,per thresh,1 of the pedal note predicted by the bifurcation diagrams is about a semitone lower than the optimal linear threshold frequency f opt,eq thresh,1 given by the linear stability analysis used in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF]. Therefore, the analysis of the bifurcation diagrams adds a significant information as regards the playing frequency, even though -as previously discussed in subsection 4.2 -the linear stability analysis already provides a rather good estimation of the playing frequency.

Case of a bass brass instruments with predominantly-expanding bore profile

For the regimes 2 to 8, the bifurcation diagrams of the euphonium (not shown here) are very similar to the trombone's ones, in terms of the direct or inverse nature of the Hopf bifurcations. However, the eupho-nium exhibits a specific behaviour for the first regime, which is qualitatively different from the one observed for the trombone.

In the case of the euphonium, the left part of figure 10 shows typical bifurcation diagrams, obtained for a value of lips' resonance frequency f in the range of the first regime. Contrary to the trombone, these diagrams display two stable regimes corresponding to two distinct periodic solution branches, thus resulting in two different playing regimes accessible to the musician. The note with the lowest oscillation frequency is called -as for the trombone -the 'pedal note' (B 1, lowest note playable without any valve pushed, see also figure 9), whereas the other one is referred to as the 'ghost note'3 (see figure 9).

On the one hand, the pedal note (red note on figure 9) is a natural note, which is as familiar as any other regime for both trombone and euphonium players. On the other hand, the ghost note (green note on figure 9) is a regime exclusively accessible to bass brass instruments with predominantly-expanding bore profile as a whole, such as the euphonium. Even though the ghost note is not much used in a musical context Figure 7: Top and bottom plots represent respectively the threshold pressures and threshold frequencies given by the linear stability analysis (blue) and the continuation method (orange) vs. the lips' frequency. Left: case H = h 0 ; right: case H ∝ 1/f . The circles identify the value of f corresponding to a local minimum of blowing pressure. Horizontal dotted lines on the bottom plots locate the values of the acoustical resonances of the resonator; the black dashed line on the bottom plots represents f thresh = f . by euphonium players, it can be played by any tuba player and was brought to light for the first time in [START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF] using linear stability analysis. In particular, it was the only note playable for values of f close to the first acoustic resonance of the resonator according to this method. As a matter of fact, since the branch corresponding to the pedal note (see the left plots of figure 10) does not emerge from the equilibrium solution (whereas the branch of the ghost note does), this regime could not be found by linear stability analysis which gives trustworthy information only in the vicinity of an equilibrium. Furthermore, the ghost note can be linked to one of the 'factitious notes' mentioned in [Herbert et al., 2019], where it is referred to as a 'phenomenon whereby a player can sound a note intermediate between notes low in the series of natural notes' and sounds a perfect fourth above the pedal note in the case of an E -tuba.

To illustrate the uncommon behaviour of the bifurcation diagrams shown in figure 10 (left), figure 10 (right) shows results of time-domain simulations. The first simulation (left) corresponds to the blue point on the ghost note branch of the bifurcation diagrams. Its instantaneous frequency is 69.5 Hz, which is coherent with the oscillation frequency displayed on the top right plot in figure 10. The second time-domain simulation (right) corresponds to the orange point on the pedal note branch of the bifurcation diagrams. Its instantaneous frequency is 60.6 Hz, which is coherent with the oscillation frequency displayed on the top left bottom in figure 10.

In the case of the pedal note, it is worth noting that over one period, p(t) becomes higher than p m (horizontal dotted line), which is physically questionable as it was never observed experimentally. This phenomenon might be considered as a weakness of the model used, caused by the raw modelling of the contact between the lips in equation 2. To overcome this limitation, a contact force between the lips -currently unavailable in literature -could be added in this equation. However, it is worth mentioning that such a contact force has already been introduced in the case of reed instruments [Munoz Arancon et al., 2013].

Assessment of the frequency interval between pedal note and ghost note

It is possible to qualitatively check the relevance, in terms of intonation, of these two regimes predicted by the bifurcation analysis. In particular, one can compare the frequency intervals between the 'easiest-toplay' notes predicted by the bifurcation diagrams on the one hand, and by analysing recordings of a euphonium player on the other hand. In the case of a euphonium and considering the U-shaped patterns in figure 11, the 'easiest-to-play' notes are those with the lowest threshold mouth pressure. The corresponding oscillation frequencies are f opt,per thresh,PN = 60.6 Hz for the pedal note, f opt,eq thresh,GN = 75.6 Hz for the ghost note, and f opt,per thresh,2 = 120.4 Hz for the B 2 (in the bifurcation diagrams, we have actually f opt,per thresh,2 < f opt,eq thresh,2 ). Therefore, it has been found that the frequency interval between the pedal note and the ghost note inferred from the bifurcation analysis (75.6/60.6 ≈ 1.25 = 5/4) is about 2 tones (major third), whereas the recording of a euphonium player 4 leads to a high major third between the pedal note and the ghost note, which is 4 Recording of a euphonium player at http://perso. univ-lemans.fr/~rmatte/bb-euphonium_reg1-6.mp4 and of an E -tuba player at http://perso.univ-lemans.fr/~rmatte/ eb-tuba_reg1-6.mp4 playing successively an ascending series and a descending series of the six first natural notes, going through the ghost note when descending.

reasonably close. Furthermore, the frequency interval between the pedal note and the second regime obtained in the bifurcation diagrams is found to be close to 12 semitones (octave), as one would expect between a B 1 and a B 2.

Eventually, it is worth highlighting that the ghost note does exist for other tuba types than the euphonium, such as the E -tuba for instance 4 .

Conclusion

Most results in this study highlight the usefulness of linear stability analysis to understand various nearthreshold behaviours of a complete nonlinear model of brass instrument applied to trombone and euphonium, as well as the ability of bifurcation diagrams to quickly give valuable information about their periodic solutions.

The ease of playing of an instrument is chosen to be assessed based on the minimal threshold pressures. A good estimate can be obtained using linear stability analysis. At the same time, the associated threshold oscillation frequency is inferred, which is directly linked to the pitch of the note. For the considered elementary model of brass instrument, this method provides information on all the periodic regimes accessible to a trombone player for a given position of the slide, and on almost all periodic regimes accessible for a given fingering of a euphonium.

However, linear stability analysis turns out to be unable to describe the pedal note of the euphonium. In this respect, the analysis of bifurcation diagrams are conveniently introduced. It allows to illustrate the existence of the pedal regime in the form of a branch of periodic solutions far from equilibrium (hence the failure of linear stability analysis to describe it) separated from the one describing the ghost note, which itself emerges from the destabilisation of the equilibrium solution. Moreover, compared to the linear stability analysis, bifurcation diagrams give a more precise prediction of the threshold mouth pressures in the event of an inverse Hopf bifurcation. However, it is worth highlighting that the minimal threshold mouth pressure of each periodic regimes for a given slide position or fingering in the case of a trombone or a euphonium -except the pedal note of the lattergiven by the bifurcation diagrams is qualitatively the same as given by the linear stability analysis.

If the ghost note seems to exist on every model of tubas, the interval between the ghost note and the first regime appears to vary from a minor third to a perfect fourth depending on the tuba's pitch. Future work include a more quantitative study on a wide range of tubas going from the B -contrabass tuba to the B -baritone horn (an octave higher), in order to determine whether or not the frequency interval between the pedal note and the ghost note remains con- ,PN (Pedal Note) = 49 Hz. Left plots represent the maximum amplitude of the periodic oscillation branches vs. the blowing pressure (top) and the frequency of the corresponding periodic solutions vs. the blowing pressure (bottom). The line thickness indicates whether the branch portion is stable (thick line) or unstable (thin line). Right plots represent the time evolution of the acoustic pressure at two points of the bifurcation diagrams above, given by the intersection of the vertical dotted line and the branches. Top: time-domain simulation of the ghost note (blue point on the bifurcation diagrams); bottom: time-domain simulation of the pedal note (orange point on the bifurcation diagrams). The horizontal dotted line represents the value of p m . stant regardless of the considered bass brass instrument with predominantly-expanding bore profile. Moreover, it is worth noting that the pedal note also exhibits different behaviours for the high brass instruments, depending on the bore profile of the instrument. For example, the pedal note appears to be more 'slotted' (meaning that the player has the sensation of a well-defined pitch) on a flugelhorn (predominantly-expanding bore profile) than on a trumpet (predominantly-cylindrical bore profile).

One important limitation of the present study lies in the difficulty to estimate the lips' parameters used in the model. Indeed, it would be reasonable to think that lips' parameters would vary from playing a low note to playing a high note, not to mention from playing a contrabass tuba to playing an alto trombone for instance. Yet, these lips' parameters being difficult to measure both on an artificial mouth or directly on the player, they were kept the same regardless the instrument or the note played. Even though the obtained results look reasonable, i.e consistent with the musicians' experience, in vivo measurements of lips parameters during musical performance would be very valuable. Furthermore, even if the system dynamics does not seem to depend sensitively on the model's parameters such as the lip-opening height at rest H, it may be worth checking the impact of a finer modelling of various elements on the ease of playing and pitch: for instance, considering a mouth section proportional to H 2 (instead of only H in the present study), taking a second resonance frequency of the lips into account, or adding a contact force between the lips when they are about to touch each other.

Acknowledgements

We wish to thank Emmanuel Brasseur and Julia Mourier for useful discussions and for their help in this work.

A Inverse Fourier transform of the input impedance

This appendix is based on the demonstration found in section 6.3 of [Silva, 2009]. Quantities in the Fourier domain are written using capital letters, whereas quantities in the time domain Figure 11: Top and bottom plots represent respectively the threshold pressures and threshold frequencies given by the linear stability analysis (blue) and the continuation method (orange) vs. the lips' resonance frequency in the case of a euphonium. The circles identify the value of f corresponding to a local minimum of blowing pressure. In the case of the second regime, the easiest note playable is given by the minimum of the continuation method (orange curve), since its minimal threshold mouth pressure is lower than the one given by the linear stability analysis (blue curve). The first two regimes on the left of each plot represent respectively the pedal note and the ghost note. are written using small letters.

Starting from equation (5), we have P (ω) = Z fit (ω)U (ω) by definition of the input impedance. Since Z fit represents the impulse response of the resonator, the inverse Fourier transform of Z fit has to be real. Thus, Z fit is hermitian-symmetric, that is to say Z fit (-ω) = Z * fit (ω). The pressure in the mouthpiece in the time domain p can then be inferred by taking the inverse Fourier transform of P :

B Dimensionless variables and equations

From a numerical point of view, it is reasonable to make equations dimensionless in order to prevent the variables from varying on a too-large scale, especially when using the continuation method in AUTO. In all that follows, most dimensionless quantity are written with a tilde '∼' unless otherwise stated. The dimensionless variables of the system h, p, u , the dimensionless time t, the dimensionless mouth pressure γ, the dimensionless lips' resonance frequency θ and the dimensionless modal parameters (κ n , σ n ) are defined as follows :

h = h h 0 ; p = p p 0 ; u = Z c p 0 u ; t = ω 0 t ; γ = p m p 0 ; θ = ω ω 0 ; κ n = C n ω 0 ; σ n = s n ω 0 ,
where h 0 is given in table 1, ω 0 = Im (s 4 ) /4 is the quarter of the fourth resonance angular frequency of the resonator (chosen as so because the fourth resonance frequency appears to be quite similar between instruments of the same tube length, such as a trombone and a euphonium for instance), so that p 0 can be defined in a similar way as the closure pressure for woodwind instruments [START_REF] Colinot | Influence of the "Ghost Reed" Simplification on the Bifurcation Diagram of a Saxophone Model[END_REF]: p 0 = µω 2 0 h 0 . Given the new set of variables, the state vector is now:

X = X m m∈ 1,2(N +1)

Figure 2 :

 2 Figure 2: Modulus (top) and phase (bottom) of the input impedance of a bass trombone (A. Courtois Legend AC502) with respect to frequency. Blue curve: measured impedance; dashed orange curve: modal fit function with N = 15 modes; green curve: error between measured and fitted impedances, defined by ||Z fit | -|Z exp || and arg (Z fit )arg (Z exp ) respectively as regards the module and phase.

Figure 4 :

 4 Figure 4: Results of the linear stability analysis with the constant, linear and inverse H-profile described on figure 3. Top and bottom plots represent respectively the threshold mouth pressure and the threshold frequency vs. the lips' resonance frequency. Blue: H = h 0 , orange: H = af + b, green: H ∝ 1/f . Circles point out the minima of each U-shaped pattern on the top plot. Red points represent the experimental values of minimal threshold pressures given by[START_REF] Gilbert | L'analyse de stabilité linéaire pour évaluer la facilité d'émission des cuivres -étude comparative de trombones ténor et basse[END_REF]. The black dashed line on the bottom plot represents f thresh = f .

Figure 5 :

 5 Figure 5: Natural notes playable by a trombone, i.e without moving the slide nor activating any trigger. From left to right: B 1 (pedal note), B 2, F3, B 3, D4, F4, A 4 (naturally a bit flat compared to an equal tempered scale), B 4.

Figure 6 :

 6 Figure6: Top and bottom plots represent respectively the maximum amplitude of the periodic oscillation branches and their oscillation frequency vs. the blowing pressure. Left: case H = h 0 ; right: case H ∝ 1/f . Blue curves: case f = f opt,eq ,3

Figure 8 :

 8 Figure 8: Top and bottom plots represent respectively the maximum amplitude of the periodic solutions branches and their oscillation frequency vs. the blowing pressure in the case of a trombone, for f = f opt,per ,1 = 54.86 Hz. The line thickness indicates whether the branch portion is stable (thick line) or unstable (thin line).

Figure 9 :

 9 Figure 9: Natural notes playable by a euphonium, i.e without depressing any valves. From left to right: B 1 (pedal note, red), D2 (ghost note, green), B 2, F3, B 3, D4, F4, A 4 (naturally a bit flat compared to an equal tempered scale), B 4.

Figure 10 :

 10 Figure 10: Typical bifurcation diagrams (left) and time-domain simulations (right) in the case of the euphonium's first regime, obtained for p m = 1.8 kPa and f = f opt,per,PN (Pedal Note) = 49 Hz. Left plots represent the maximum amplitude of the periodic oscillation branches vs. the blowing pressure (top) and the frequency of the corresponding periodic solutions vs. the blowing pressure (bottom). The line thickness indicates whether the branch portion is stable (thick line) or unstable (thin line). Right plots represent the time evolution of the acoustic pressure at two points of the bifurcation diagrams above, given by the intersection of the vertical dotted line and the branches. Top: time-domain simulation of the ghost note (blue point on the bifurcation diagrams); bottom: time-domain simulation of the pedal note (orange point on the bifurcation diagrams). The horizontal dotted line represents the value of p m .

  ω )e -jω t dω , with ω = -ω. U being also hermitian-symmetric, we have: n (t)) , where p n is the complex component of the pressure in the mouthpiece p, whose dynamical behaviour is linked to C n and s n :P n (ω) = C n jωs n U (ω) ⇒ ṗn (t)s n p n (t) = C n u(t).

see subsection 3.1 for details about this notation.

Animation showing the evolution of the bifurcation diagrams as a function of the lips' resonance frequency f increasing from 134.2 Hz to 182.1 Hz in the case of the constant Hprofile H = h 0 at http://perso.univ-lemans.fr/~rmatte/bd_ dyn_AC_basse_reg_reg3.avi or in the case of the inverse Hprofile H ∝ 1/f at http://perso.univ-lemans.fr/~rmatte/ bd_dyn_AC_basse_Hinv_reg_reg3.avi.

We chose here to use the term employed by[START_REF] Velut | How well can linear stability analysis predict the behavior of an outward valve brass instrument model?[END_REF] to qualify this note being between a minor third and a perfect fourth higher than the pedal note.

with u now written as:

,

the dimensionless lip-opening height at rest, defined similarly to [START_REF] Gilbert | Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation[END_REF].


Adrien Gandolfo
email: ad.gandolfo@gmail.com

Hui Chen

Alexander Kukui

Amandine Durand

Brice Temime- Roussel

Jörg Kleffmann

Henri Wortham

Sasho Gligorovski

Impact of photocatalytic paints on indoor VOCs in a full-scale study: focus on regulated and reactive compounds

Keywords: VOCs, Photocatalysis, PTR-ToF-MS, TiO2, formaldehyde

Introduction

Volatile organic compounds (VOCs) are ubiquitous indoor, emitted from various indoor sources. In novel airtight and mechanically controlled ventilation equipped buildings, the VOC concentration is expected to rise, enhancing, therefore, public health risks. Investigators suggest that photocatalytic paints could reduce VOC levels indoor, but some studies showed that photocatalytic degradation of the paint organic binder could be an additional VOC source. While several studies report photocatalytic material limits in laboratory conditions1, none have been conducted in a full-scale study. Here we present the first real-scale study of photocatalytic paint impacts on indoor air quality through measurement of regulated indoor VOCs like formaldehyde, acetaldehyde, and benzene and also with insight into VOCs formed under typically highly oxidative conditions as peroxide and multi-oxygenated bearing compounds.

Materials/Methods

Indoor time-resolved measurements of VOCs held in an unfurnished building office in Martigues, situated in the southeast of France during October 2016. A PTR-ToF-MS (Ionicon Analytik GmbH, Innsbruck, Austria) was used to record VOC levels at low concentrations (as low as a few pptv) under high mass resolution. The identification of the compounds is based on fragmentation patterns and literature revue. Specific rate constants were used for quantification, and relative humidity corrections were applied. Two kinds of paint were studied: one standard white paint which serves as a reference and one white photocatalytic paint (3.5 % w/w, PC500 nanoTiO2). Both paint formulations, based on an organic binder, are identical except nanoTiO2 content. To avoid solvent emission during the campaign, plasterboards were covered by paints 6 weeks before starting measurements and stored in controlled environmental conditions (RH: ~50%, Temperature: ~20°C, in the complete absence of light). During experiments, indoor temperature shows a quasi-semblable day-by-day profile with a minimum (~20 °C) in the morning just before room isolation and reaches a maximum of around 30 °C in the afternoon. Relative Humidity was constant around the typical indoor level (45%). No mechanical ventilation has been used during the campaign, thus the air exchange rate occurred only through building leaks at a rate between 0.3 and 0.5 h-1.

Results and Discussion

Observed trends in VOC concentrations clearly state an accumulation mode from the moment the room is isolated from outdoor. VOC levels increased reaching a plateau 3-5 hours later depending on the air exchange rate and paint emission rates intensity. Figure 1 59 ions have been attributed to one of the 7 groups from m/z 21.022 to 373.075. Total ppb levels raised from about 200 at the isolation time to 1000 ppb in the afternoon. The 1 ppm mixing ratio is the maximum observed during all the campaigns. For example, in quite similar environmental and experimental conditions but in presence of reference paint, the background VOC total mixing ratio is 190 ppb and the VOC level reaches up a maximum of around 400 ppb being 60 % lesser than in presence of photocatalytic paints. On figure 1, isolation held a few minutes before 10h00. At this point, carbonyl compounds represent almost 48% of total concentration mainly dominated by formaldehyde and acetaldehyde. The acid group is the second major contributor to VOC levels followed by hydrocarbon (HC) group and alcohol group with respectively 21, 15, and 8.5 %. At 15h30, the contribution of carbonyls, HC, and acids increased 4, 1, and 1 % respectively while alcohol drop by 4 %. HC group gather compounds with CxHy chemical formula. Under our instrumental conditions, it is important to note that aldehyde above C3 tends to lose and their H2O pattern and respond as HCs in the PTR-MS instrument. Thus, here, we can argue that the increase of HC levels and contribution is supported by the increase of C4 and above aldehyde compounds. While their levels are low, it is noticeable to observe a contribution increase from 1.4 to 1.9 % of the oxygenated group. It regroups all the compounds bearing 2 or more oxygenated atoms in their formula which has not been identified as acid compounds. Among those, unique observation of ions CH4O2H+ (m/z 49.

Conclusions

The emerged outcomes of this comprehensive study suggest the need to use more adequate binders in photocatalytic paints to avoid additional VOC emissions. The presence of TiO2 as a photocatalyst in paints leads to a more toxic and reactive indoor environment not allowing yet to state the benefits of photocatalytic air remediation technology.

  depicts the experiment conducted in the presence of photocatalytic paint.

Figure 1 :

 1 Figure 1: VOCs measurements conducted in an unfurnished office room in presence of photocatalytic paints and UV-Transparent windows. Dash line shows the room isolation time. Pie charts represent 5 minutes measurements represented by the black rectangular. Measurement are alternatively conducted indoor (10 mins) and outdoor (5 mins), designing the crenel aspect of the figure)

  028), C2H6O2H+ (m/z 63.044), CH4O3H+ (m/z 65.024) C3H8O2H+ (m/z 77.059), which have been ascribed here as peroxide compounds, respectively namely methyl hydroperoxide, dimethyl peroxide, hydroxymethyl hydroperoxide, and isopropyl hydroperoxide occurred. Peroxide2 compounds are formed through a radical termination reaction route HO2 being peroxy, and hydroperoxy radicals. In addition, ion, C3H4O2H+ at m/z 77.023, which have been associated with PeroxyAcyl Nitrate (PAN)3 reach 1.2 ppb during the experiments showed in Figure 1. The formation of that compound is the mark of an important indoor oxidative capacity. Radical compounds measurements during this experiment support it showing OH and RO concentration higher by 50 and 200 % in presence of photocatalytic paint in comparison of reference paint (ref).

Acknowledgement

This work is a contribution from the LABEX SERENADE (no ANR-11-LABX-0064) funded by the ''Investissements d'Avenir'', French Government program of the French National Research Agency (ANR) through the A*Midex project (No. ANR-11-IDEX0001-02). Adrien Gandolfo received a doctoral grant from the French Environment and Energy Management Agency (ADEME) and the Region Provence-Alpes-Côte d'Azur.


Adrien Gandolfo
email: ad.gandolfo@gmail.com

Sabrine Tlili

Hui Chen

Alexander Kukui

Amandine Durand

Brice Temime-Roussel

Jörg Kleffmann

Henri Wortham

Sasho Gligorovski

Measurements and modeling of the spectral properties of sunlight at the irradiated volume and surface of an indoor environment: The impact on the oxidation capacity

Keywords: Architectural model, Irradiated volume, HONO photolysis frequencies, photochemistry, oxidative capacity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The ultraviolet (UV) and visible (VIS) fraction of the sunlight initiates photochemistry in the atmosphere by photolyzing certain molecules into highly reactive species such as the hydroxyl radical (OH). In indoor environments, several field campaigns demonstrated the variability of the actinic flux (and the corresponding photolysis frequencies) depending on windows cut off wavelength of ca. 330-390 1-3 nm pointing to the need for direct measurements. In addition, indoor, sunlight irradiation is spatially and temporally variable. A recent model study has illustrated the spatial heterogeneity of compounds as OH, HO2 and oxidation products under direct irradiation or in the shade 4 . It clearly appears that correctly and accurately assess indoor irradiated volume is of primary importance. To do so, the use of an architectural model can greatly help indoor scientific community 1 . Here, we present for the first time, a comprehensive architectural model in combination with direct solar flux measurements to predict the sun irradiation variability in an office indoor environment under various light conditions, different window types and season.

Materials/Methods

Time-resolved measurements of key indoor species and actinic flux have been conducted during October 2016, in a building office in Martigues, situated in the southeast of France. Two kinds of glass were studied: one commercial double glass safety window, SP510 Verrissima ® (Anti-UV) and one simple borosilicate glass (UV-transparent). To model the sunlight beam penetration within the experimental room, we performed a combination of two software: Revit 2018 ® and Autocad 2016 ® . The first was used to construct the building model in his virtual environment and to determine the sun pathway in the shy depending on the time of the day and the day of the year. The second was used to calculate the irradiated surface and volume within the room. The model outputs provide hourly data from 9 am to 5 pm.

Results and Discussion

The variation of the irradiated volume within the modeled room is displayed in Figure 1. For the of measurements campaign days (set as the 30 th of October), the irradiated volume ranges 0.9 to 6.6 m 3 with a maximum at noon and a minimum in the late afternoon (17h). Thus, between ~3 to 20 % of the total volume of the room undergoes photochemistry involved by direct sunbeam irradiation. Because of the high solar zenithal angle, December is the period that depicted the largest irradiated volume (the chosen modeled day is December the 21 st and represents the days subset in the winter period). The maximum is 9.7 m 3 at noon and represents almost 30 % of the total volume of the room. At the opposite, in June, the room is not directly irradiated by sunbeams due to the low solar zenithal angle (20° at noon in Martigues). Between both extremes, in March and in September at noon the irradiated volume is 1.7m 3 (~5% of the total volume). Indoor cutoff by windows occurs at 320 and 390 nm for the UV-transparent and Anti-UV glasses respectively. Around noon under direct irradiation, the transmittance by the windows, defined as the ratio between the outdoor TUV modeled and indoor measured actinic fluxes, are on average in the range of 0.6 % and 36 % (300<λ<400 nm) and 72 % and 83 % (400<λ<660 nm) and for the anti-UV and the UV-transparent windows, respectively. The difference of transmittance observed in the visible range between the two glass windows can be ascribed to the quality and the thickness of the glass 3 . In the near-UV range, the difference between the transmittance values is much more pronounced due to the UV treatment of the Anti-UV window. The photolysis frequencies associated with the measured actinic flux are summarized in table 1. Compounds with a large absorption band in the UV region (HONO, H2O2, O3 or HCHO) present differences in photolysis frequencies between both glass types of 2 to 3 orders of magnitudes. NO2 and HOCl which both absorb in the UV and in the VIS are less impacted by the UV treatment of the Anti-UV window (a factor of 10 and 5.5 is measured between photolysis frequencies with both glass windows, respectively). Nitrate radicals which are VIS sensitive are de facto (quasi) not impacted by the glass changing.

Accordingly to the mixing ratio levels observed indoor 2 , HONO depicted the higher OH production rate (5.9•10 7 molecules.cm -3 .s -1 ) while HOCl photolysis presents intensive OH and Cl radicals production rate events linked to specific cleaning activities with bleach (3.1•10 8 molecules.cm -3 .s -1 ).

Figure 1 :

 1 Figure 1: irradiated volume model outputs in Martigues (43°25՛0՛՛N,5°2՛35՛՛E). December, March/September and June are representative of the winter, spring/fall and summer periods. October is representative of the period in wich measurements have been performed

  Photolysis frequencies(s -1 ) UV-trans Anti-UV J(HONO) (HONO → HO+NO)4.8•10 -4 4.9•10 -7

Table 1 :

 1 photolysis frequencies of indoor key species under direct sunlight irradiation as a function of window types around noon

ConclusionsWe demonstrate that in a south-oriented room in the middle latitude of the northern hemisphere, the integrated irradiated volume is the largest in December. Since under direct irradiation, indoor actinic fluxes are not affected by the season 1 and the distance from the windows 2 , the winter season is the period when the indoor oxidative capacity is the greatest, the latter being mainly drive through HONO photodissociation.

Acknowledgment


Ingmar Schumacher
email: ingmar.schumacher@polytechnique.edu

Endogenous Discounting via Wealth, Twin-Peaks and the Role of Technology

Keywords: JEL classification: D90, C61, O41 endogenous discounting, Twin-Peaks of economic growth, multiple equilibria

We endogenize the discount rate via wealth and provide evidence that wealth affects the discount rate negatively. We give a necessary and sufficient condition for endogenous discounting to lead to the Twin-Peaks of economic growth and show that improvements in technology help avoid them.

Introduction

We are becoming more and more aware that convergence between country's income levels is far from obvious. Empirical observations have forcefully shown that rich countries are becoming richer, while poor ones are not able to catch up. Jones [START_REF] Charles | On the evolution of the world income distribution[END_REF], Azariadis [START_REF] Azariadis | The economics of poverty traps part one: Complete markets[END_REF] and Quah [START_REF] Quah | Twin peaks: Growth and convergence in models of distribution dynamics[END_REF] find that the world income distribution changed from a somewhat normal distribution shape toward a twin-peak shape. We provide new answers to the following two questions: One, what can be a potential source of these twin-peaks? Two, why were some of the countries that were believed to belong to the group of low steady state countries (like Taiwan, South Korea) able to reach a convergence path which led them to a high steady state?

Theoretical models have attempted to draw some conclusions on the possible sources of these twin-peaks of economic growth. These models have either built on non-convexities in technologies to arrive at multiple steady states (Azariadis and Drazen [START_REF] Azariadis | Threshold Externalities in Economic Development[END_REF]) or on incomplete markets (Durlauf [8]; Quah [START_REF] Quah | Twin peaks: Growth and convergence in models of distribution dynamics[END_REF]).

In this article we propose endogenous discounting as another potential source of the twin-peaks of economic growth. Specifically, we argue that wealth affects the discount rate by proposing that this is a shortcut for suggesting that wealthier countries have better health standards (Pritchett [START_REF] Pritchett | Wealthier is Healthier[END_REF]) and better insurance markets (Carroll [START_REF] Christopher | Buffer-stock saving and the life cycle/permanent income hypothesis[END_REF]), all of which generally affects the agent's discount rate negatively (Frederick et al. [START_REF] Frederick | Time discounting and time preference: A critical review[END_REF]).

There exist now an increasing number of endogenous discounting models, which nearly exclusively have consumption as the source of endogenity (e.g. Epstein and Hynes [START_REF] Larry | The rate of time preference and dynamic economic analysis[END_REF]; Obstfeld [START_REF] Obstfeld | Intertemporal dependence, impatience, and dynamics[END_REF]; Das [START_REF] Das | Optimal growth with decreasing marginal impatience[END_REF]). In a recent working paper, Drugeon [START_REF] Drugeon | A Propelling Role for Impatience in Long-Run Growth[END_REF] allows for consumption and capital to influence the discount rate, but focuses on increasing returns in the production function and does not provide a thorough analysis of multiple steady states.

Our treatment of wealth here will be based upon a broad view of wealth combining physical capital and human capital (see Mankiw [START_REF] Mankiw | The growth of nations[END_REF]). Fielding and Torres [START_REF] Fielding | Health, Wealth, Fertility, Education and Inequality[END_REF] estimate the relationship between wealth, health and education for 41 developing countries. They find that improvements in physical and human wealth lead to lower mortality rates. Their results are robust even across countries, pointing at a uniform effect of the variables in question. Similar results are due to Grossman [START_REF] Grossman | Education and nonmarket outcomes[END_REF], Feldman et al. [START_REF] Feldman | National trends in educational differentials in mortality[END_REF], Pappas et al. [START_REF] Pappas | The increasing disparity in mortality between socioeconomic groups in the united states, 1960 and 1986[END_REF], Preston and Elo [START_REF] Preston | Are Educational Differentials In Mortality Increasing in the United States?[END_REF], Richards and Barry [START_REF] Richards | US Life Tables for 1990 by Sex, Race, and Education[END_REF]. Wealth as a source of endogenous discounting has also been proposed by Becker and Mulligan [START_REF] Gary | The endogenous determination of time preference[END_REF] as well as Deaton and Paxson [START_REF] Deaton | Intertemporal Choice and Consumption Inequality[END_REF], who show that financial assets and human capital inequality grow as cohorts age. These results are interpreted by Becker and Mulligan [START_REF] Gary | The endogenous determination of time preference[END_REF] as a potential consequence of endogenous time preference.

We derive a necessary and sufficient condition for the existence of multiple steady states and show that discounting endogenized via wealth can provide another explanation for the development of the twin-peaks of economic growth. We believe this is a particular useful extension as existing contributions to the literature on twin-peaks exhaustively focus on the twin-peaks as a result of assumptions on technologies or uncertainty. Here, however, the twin-peaks arise through the preferences of the agents.

We answer our second question by demonstrating how improvements in technology can help avoid the twinpeaks. This result ought to be particularly interesting for the case of developing countries.

The Model

The model is based on an infinitely-lived agent approach where the agent obtains utility u(•) from consumption c(t). In addition, his wealth k(t) affects the discount rate ρ(k(t)) negatively. Wealth can be accumulated by investing but is reduced by consumption and constant depreciation δ > 0. The infinitely-lived agent then solves the following problem.

max {c(t)} U (c, k) = ∞ t=0 u(c(t))e -θ(t) dt s.t.        k(t) = f (k(t)) -c(t) -δk(t), ∀t θ(t) = ρ(k(t)), ∀t k(t) ≥ 0, c(t) ≥ 0, ∀t, with k(0) given. (1)
We make use of the following assumptions. (A1) We impose that the production function f : R + → R + follows standard assumptions of concavity, such that f (k) ≥ 0, f (0) = 0, f (k) > 0, f (k) < 0, with lim k→0 f (k) = ∞, and lim k→∞ f (k) = 0. We define k as the level of k which solves f (k) = δk, and k as the level of k that solves f (k) = δ. (A2) The utility function u : R + → R + is at least twice continuously differentiable with u (c) > 0, u (c) < 0, ∀c, and lim c→0 u (c) = ∞. (A3) We take ρ(k) : R + → R + with ρ (k) < 0 and ρ (k) > 0, where we impose lim k→∞ ρ(k) > 0.

This model generalizes the neoclassical Ramsey-Caas-Koopmans model by introducing a discount rate endogenized via wealth. It reduces to the classical model if we assume that ρ(k(t)) = r, ∀t. Integrating θ = ρ(k(t)) from t ∈ [0, τ ] then leads to θ(τ ) = rτ . Substituting this into (1) gives the classical model.

Solving the Model

We define the value function

V (k) = max {c} U (c, k) s.t. k = f (k) -c -δk, k(0) > 0 given, (2)
which leads to the following Bellman equation

0 = max c u(c) -ρ(k)V (k) + V (k) k . (3)
The first order conditions are

u (c) = V (k). (4)
By equation ( 4) we can write c(k), substituting this in (3) gives

0 = -ρ (k)V (k) + V (k) k + V (k)(f (k) -δ -ρ(k)). (5)
2

We know that V (k) = λ, with λ being the shadow price of capital. Also, V (k) k = ∂V (k) ∂k ∂k ∂t , implying V (k) k = λ. V (k) is the prospective optimal value of utility. Rewriting leads to the following system

ċ = - u (c) u (c) f (k) -δ -ρ(k) - ρ (k) ρ(k) u(c) u (c) + k , (6)
k = f (k) -c -δk. (7)
The last term of equation ( 6) is obtained by making use of the correspondence between first-order conditions from the Hamiltonian method and the Bellman method as well as using Michel's [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF] transversality condition. That this correspondence indeed holds is because the conditions from Stokey and Lucas [START_REF] Stokey | Recursive methods in economic dynamics[END_REF], are satisfied. For ρ (k) = 0 we see that equation ( 6) reduces to the Keynes-Ramsey rule of optimal consumption. We define m(k

) = ρ (k)ρ(k)-ρ (k) 2 ρ(k) 2
with m(k) > < 0, and n(k

) = u (c) 2 -u (c)u(c) u (c) 2 > 0.
Proposition 1 A necessary and sufficient condition for multiple steady states1 is ∃k > 0 such that

f (k) > (1 + n(c))ρ (k) + m(k) u(c) u (c) + ρ (k) 2 ρ(k) 2 n(c) u(c) u (c) , ( 8
)
with c = f (k) -δk.
In a standard Ramsey-Caas-Koopmans model with a concave production function this condition will never be satisfied since the right-hand side of equation ( 8) will be equal to zero. We denote the steady states as

0 < k l < k m (< k) < k h (< k).
When one linearizes the dynamical system around the non-trivial steady states, {c l , k l },

Figure 2 {c m , k m } and {c h , k h }, one gets the following differential system:

J =   ρ(k) + δ -f (k) -u (c) u (c) f (k) -ρ (k) -m(k) u(c) u (c) -(f (k) -δ) ρ (k) ρ(k) -1 f (k) -δ   . ( 9
)
As the trace is Tr(J) = λ 1 + λ 2 , and the determinant is Det(J) = λ 1 λ 2 , with λ 1 and λ 2 denoting the eigenvalues of the Jacobian, saddle-path stability occurs if the trace is positive and the determinant is negative. The trace of this matrix is given by Tr(J) = ρ(k) > 0, while the determinant is negative if

f (k) < ρ (k) + m(k) u(c) u (c) + ρ (k) ρ(k) (f (k) -δ)n(k). (10)
From the conditions for multiple equilibria we know that this condition does not hold for the middle steady state, which is thus instable. It holds for the low and high steady states, which implies saddle path stability. Complex dynamics arise if Tr(J) 2 < 4Det(J). As Det(J) < 0 for the low and high steady state, this excludes the possibility of complex dynamics for these steady states. However, for the medium steady state complex dynamics occur if

ρ(k) 2 < -4 u (c) u (c) f (k) -ρ (k) -m(k) u(c) u (c) -n(k) ρ (k) ρ(k) (f (k) -δ)
. This corresponds to the recent twin-peaks of economic growth hypothesis raised by researchers like Quah [START_REF] Quah | Twin peaks: Growth and convergence in models of distribution dynamics[END_REF], Jones [START_REF] Charles | On the evolution of the world income distribution[END_REF] and Azariadis [START_REF] Azariadis | The economics of poverty traps part one: Complete markets[END_REF] and is depicted in Figure 1.

In order to understand where the multiplicity of equilibria comes from we re-write equation ( 6) by solving for V (k), which gives

ċ(t) = - u (c(t)) u (c(t)) f (k(t)) -δ -ρ(k(t)) - ρ (k(t)) ∞ t u(c(s))e -θ(s) ds u (c(t)) (11)
The multiplicity of steady states occurs since when an agent is very poor then increases in consumption are necessary for survival (u (0) = ∞), and the agent is so impatient that the preferences are nearly exclusively directed toward today, implying that most wealth will be directly consumed. Similarly, if the future expected utility is very low, then our agent will prefer to consume current wealth. For k > k m however, this implies that overall wealth has already been built up sufficiently in order to incorporate far-sighted goals, such that the discount rate is relatively low (or the life expectancy high enough). For example, with a low enough mortality rate agents will need to carefully plan ahead to the distant future, whereas a high mortality rate implies that agents don't expect to become old and thus do not plan ahead. If we use Stone-Geary preferences to analyze the impact of changes in the subsistence level of consumption, then we find that increases in the subsistence level decrease the last term in equation ( 11). This means that the higher the subsistence level or the closer is consumption to subsistence the less willing are agents to postpone consumption and the less important becomes the impact of wealth on the discount rate. In other words, consumption close to subsistence and a low expected future utility, both a likely phenomena in less developed countries, are likely to trap countries in the low wealth equilibrium. One can easily show that improvements in total factor productivity lead to shifts in the steady state curves as depicted in Figure 2. The intuition is that if wealth is more productive, then it is more efficient to increase the level of wealth in order to raise overall utility by reducing the discount rate. In recent years, some of the countries that were believed to belong to the group of low steady state countries (South Korea, Taiwan, etc.) were able to reach a convergence path which led them to a high steady state. It is plausible that this is due to improvements in technology or health (mortality). This should provide a new and interesting possibility for policy interventions which focus on international convergence of wealth.

The case of unique steady state has been treated in Schumacher[START_REF] Schumacher | Endogenous discounting and the domain of the felicity function[END_REF].

The author is grateful for helpful comments by Raouf Boucekkine. Financial support from the Chaire Lhoist Berghmans in Environmental Economics in CORE and from the Chair AXA Assurance et