Jean-Loïc Le Carrou
email: jean-loic.lecarrou@univ-lemans.fr

Francois Gautier

Nicolas Dauchez

Joël Gilbert

Modelling of Sympathetic String Vibrations

Keywords: numbers: 43, 40, Cw, 43, 75, Gh

String instruments are usually composed of several strings connected to a vibrating body allowing efficient sound radiation. For some special string tunings, sympathetic vibrations can occur: if one string is excited, some others are also excited via the body.

In order to investigate this phenomenon, an analytical model of a simplified generic string instrument has been developed. The body of the instrument is represented by a beam clamped at both ends, to which several strings are attached. The state vector formalism and the transfer matrix method are used to describe the propagation of bending and extensional waves in each sub-structure (strings and beam). Coupling conditions between sub-structures take into account the angle formed by the beam and the strings. This leads to a linear system from which the normal modes of the assembly are computed. Numerical computations are carried out in the case of the beam-2 strings assembly and mode shapes are classified thanks to a criterion, the kinetic energy ratio (KER). Four kinds of modes may be identified: beam modes, string modes, beam-string modes and string-string modes. The latter are responsible for sympathetic response. An experimental modal analysis carried out on a clamped beam equiped with two strings confirms our theoretical results.

I. INTRODUCTION

String instruments are usually composed of several strings connected to a vibrating body allowing efficient sound radiation. The plane part of this vibrating body is called soundboard and is generally a stiffened and tapered plate of complex shape, made from different materials. When no dampers are applied to the strings, important coupling between them can occur via the soundboard: if one string is excited, other strings can vibrate and contribute significantly to the musical sound. This phenomenon is known as sympathetic vibration and is defined in the dictionary of acoustics [START_REF] Morfey | Dictionary of acoustics[END_REF] as "resonant or near-resonant response of a mechanical or acoustical system excited by energy from an adjoining system in steady-state vibration".

For some instruments such as the viola d'amore, the baryton or non-european instruments like the sitar or the sarangi, sympathetic string vibrations result in a characteristic sound quality: these instruments have additional strings designed to be sympathetically excited by the main strings [START_REF] Greilsamer | Le baryton du prince esterhazy[END_REF]. In the case of the violin, sympathetic vibration is highlighted in [START_REF] Hutchins | Sympathetic vibration and coupling of resonances[END_REF]: "if one plays the D on the violin G string which matches the pitch of the open D string and alternately damps and undamps the open D with a finger, the reading on the meter can be observed to increase about 1-2 dB as the open string is damped. The effect is more noticeable on a cello or bass".

Numerous strings of the piano are organised in pairs or triplets (bichord or trichord) in which the strings are almost tuned in unison. The coupling of these strings has been investigated in [START_REF] Gough | The theory of string resonances on musical instruments[END_REF][START_REF] Weinreich | Coupled piano strings[END_REF][START_REF] Capleton | False beats in coupled piano string unisons[END_REF]: a model for the normal modes of strings pairs has been developed taking into account mistuning from unison and the finite admittance of the bridge at the connection point of the strings. The role of string's polarization is also pointed out: resistive and reactive parts of the admittance may differ for 'horizontal' and 'vertical' vibration. With this 'anisotropic boundary conditions', interactions in the time domain between normal modes lead to beats and double decay rates in the aftersound. Since the hammer excites simultaneously the group of two (or three) strings, the tone does not involve as such a sympathetic vibration. This sympathetic vibration exists when the una corda pedal is used. In this case, the hammer is mechanically shifted and only one string out of two (or two out of three) is struck. Coupling between the initially excited string and the other string leads then to sympathetic vibration [START_REF] Weinreich | Coupled piano strings[END_REF].

Note that the sympathetic vibration effect has been included in time domain simulation of piano tones to get realistic synthesis sound. For such simulations, a phenomenological model of the coupling between strings is introduced and no physical model of the coupling are described [START_REF] Valimaki | Towards high-quality sound synthesis of the guitar and string instruments[END_REF].

The harp is the main instrument to which this study can be applied. For this instrument, the sympathetic vibration effect appears as soon as a string is excited because all the strings are undamped. This effect remains even if the excited string is damped. Although this effect is a fundamental characteristic of the instrument, the instrument maker has to design the harp in such a manner that sympathetic vibration remains reasonable. Note that the case of the harp has some particularities: each string is tuned differently from the others and no pair of strings exists. Different strings can interact even if they are connected to remote points of the soundboard. In this case, the local admittance of the soundboard is not a sufficient parameter and structure-borne sound inside the soundboard has to be taken into account. The whole soundboard and the strings have to be studied as a coupled system.

The aim of this paper is to present a modelling of sympathetic vibration on a simplified string instrument. The main application is related to the case of the harp but the formulation can be adapted to the piano's configuration. This model allows us to determine the normal modes of a structure coupled to a great number of strings and to identify those which are responsible for the sympathetic phenomenon.

The paper is structured as follows: first, the state vector formalism and the transfer matrix method are presented and applied to obtain the normal modes of a beam-N strings assembly. The case of a beam-2 strings assembly is then highlighted. Numerical results for modes shapes of the assembly are given and classification of these modes is proposed using an appropriated criterion. Experimental modal analysis is also performed on the studied beam-2 strings assembly and confirms the theoretical results. Finally, the influence of inharmonicity on the modes, responsible for sympathetic vibration, is studied.

II. MODEL OF A BEAM-N STRINGS ASSEMBLY

In order to investigate couplings between strings via the instrument's soundboard, we consider a simplified configuration: the soundboard is represented by a simple beam, on which several strings are connected. Such a choice allows to describe the structure-borne coupling in an analytical way. The aim of this section is to present a method for calculating modes of a beam-N string assembly: the state vector formalism and the transfer matrix method are applied in this context in order to obtain the normal modes (eigenfrequencies and mode shapes) of the assembly.

A. Description of the assembly

The beam-N strings assembly is described in figure 1. This plane assembly is constituted of a uniform, prismatic beam on which N different strings are connected. The beam is supposed to be clamped at points A and B. The extremity O i of the i-string (i=1...N ) is the string connection point to the beam. The other extremity C i of the i-string is supposed to be simply supported. The angle α i between the axis (A, x b ) of the beam and the axis (A, x si ) of the i-string induces coupling between longitudinal and bending vibrations in the beam and in the i-string. In the case of the harp, this angle is about 40 • .

The chosen configuration is well adapted to the case of the harp: for this instrument, the strings are connected on the symmetry line of the soundboard, and thus its vibrations can be modelled using an equivalent beam. In harmonic regime (e jwt ), the vibratory state of each sub-structure (string or beam) of the assembly is described at any point of its neutral line by a state vector denoted by X (x) whose components are kinematic and force variables [START_REF] Pestel | Matrix method in elastomechanics[END_REF][START_REF] Wang | Waves in cylindrical shells with circumferential submembers: a matrix approach[END_REF][START_REF] Moulet | Les jonctions en mécanique vibratoire: représentation par matrice de diffusion et caractérisation expérimentale pour des poutres assemblées[END_REF]. The same generic name x is used to denote the space coordinate in the beam and in the strings. It permits us to present in the same manner the motion equations related to beam and to strings. In figure 1, the coordinate x correspond to x b for the beam and to x si for the i-string. For the i-string, the state vector is the 4-by-1 vector,

X s (x) =      u s (x) w s (x) N s (x) Q s (x)      , (1)
where u s , w s are the longitudinal and transversal displacements and N s , Q s are the longitudinal and transversal forces. In the beam, the state vector is the 6-by-1 vector:

X b (x) =           u b (x) w b (x) θ b (x) N b (x) Q b (x) M b (x)           , (2)
where u b (x), w b (x) and θ b are respectively the longitudinal, transverse displacements and the slope of the beam cross section (figure 2) and where N b (x), Q b (x) and M b (x) are re-spectively the longitudinal transverse forces and bending moment. Note that longitudinal displacements are taken into account to ensure kinematic continuities. The beam and strings bending motions are respectively described in the framework of the elastic string's theory [START_REF] Valette | Mécanique de la corde vibrante[END_REF] and of the Euler-Bernoulli beam model [START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF]. Extensionnal motions are described in the framework of classical rod theory [START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF]. No dissipative phenomena are included. For all these kinematic models, the equations of motion of a sub-structure (beam or string) can be written as a first order differential matrix equation, called state equation,

dX (x) dx = H(ω)X (x). (3)
Matrix H(ω) is the characteristic matrix of the sub-structure. Since each sub-structure is supposed to be uniform, the matrix H(ω) is a space invariant matrix depending only on the angular frequency ω. The solution of (3) can be written [START_REF] Pestel | Matrix method in elastomechanics[END_REF][START_REF] Wang | Waves in cylindrical shells with circumferential submembers: a matrix approach[END_REF] as

X (x) = T(x, x 0 )X (x 0 ), (4)
where T(x, x 0 ) is the transfer matrix between the state vector at point x 0 and the state vector at point x. This transfer matrix can be straightforwardly determined from matrix H. It is given by T

(x, x 0 ) = EQ(x, x 0 )E -1 , (5)
where E is the matrix of eigenvectors of matrix N, defined by H = N. Matrix E obeys the fundamental relationship, NE = EΛ, [START_REF] Capleton | False beats in coupled piano string unisons[END_REF] where Λ is the diagonal matrix whose terms are the eigenvalues ((λ k ) k=1...N ) of the matrix N, Λ = diag(λ k ).

Each eigenvalue λ k corresponds to a wavenumber. Since the beam and the strings have an axial symmetry, these eigenvalues appear by pairs (λ k , -λ k ) corresponding to wave propagation in two opposite directions. In relation [START_REF] Weinreich | Coupled piano strings[END_REF], the matrix Q(x, x 0 ) is the propagator matrix given by

Q(x, x 0 ) = diag e λ k (x-x 0 ) . (8)
For beam and string's segments, the expression of the characteristic matrix H(ω) and the transfer matrix T(x, x 0 ) are given in appendices A and B.

C. Coupling equations at beam-string connection point

The connection between the beam and the i-string is shown in figure 2. Points O - i and O + i are limit points of the connection volume belonging to the beam's parts. Point O i is the limit point belonging to the i-string. Since the connection volume is supposed to be very small, kinematic continuity relations are assumed:

u b (O - i ) = u b (O + i ) (9)
w b (O - i ) = w b (O + i ) ( 10
)
θ b (O - i ) = θ b (O + i ) (11)
u s (O i ) = u s (O + i ) cos(α i ) + w s (O + i ) sin(α i ) (12)
w s (O i ) = -u s (O + i ) sin(α i ) + w s (O + i ) cos(α i ). (13)
Mass and rotational inertia of the connection volume are ignored, it implies that the sum of the force and of the moment continuity applied to the connection equal zero. Projecting such relations on the local axis (x b , y b , z) leads to

N b (O + i ) -N b (O - i ) + N s (O i ) cos(α i ) -Q s (O i ) sin(α i ) = 0 (14) Q b (O + i ) -Q b (O - i ) + N s (O i ) sin(α i ) + Q s (O i ) cos(α i ) = 0 (15) M b (O + i ) -M b (O - i ) = 0. ( 16
)
Using the state vector formalism introduced in ( 1) and ( 2), coupling equations ( 9) to ( 16) can be written on matrix forms as:

X b (O + i ) = X b (O - i ) + K i X s (O i ) (17) and LX s (O i ) + M i X b (O + i ) = 0, (18)
where

L = 1 0 0 0 0 1 0 0 , M i = cos(α i ) sin(α i ) 0 0 0 0 -sin(α i ) cos(α i ) 0 0 0 0 . and K i =           0 0 0 0 0 0 0 0 0 0 0 0 0 0 -cos(α i ) sin(α i ) 0 0 -sin(α i ) -cos(α i ) 0 0 0 0          
.

The matrices L and M i are called continuity matrices. The matrix K i is a coupling matrix, depending on the connection angle α i , which is responsible for the coupling between longitudinal and transverse motions in each string and in the beam.

D. Modes of the assembly

The state vector at one point of the assembly depends on the state vector on its boundary points: A, B, C 1 , ..., C N . For these particular points, the number of state vector components is 12+4N : 12 for the beam state vectors (X b (A) and X b (B)) and 4N for the N string state vectors (X s (C 1 ), ..., X s (C i )). Half of these 12+4N state vector components are imposed by boundary conditions. In the case of the clamped beam and of the fixed strings ends that are considered here, every kinematic variables, indicated by index k (displacements and slopes), are set to zero. Forces variables, indicated by index f (forces and moments) are unknown and do not equal zero. Equations of the assembly impose 6+2N transfer relations between the 6+2N components of the boundary state vectors.

A first set of 2N equations is given by the following relation, proved in appendix C, for i = 1...N ,

LT s (O i , C i )X s (C i ) + M i T b (O + i , A)X b (A) + i j=1 M i T b (O - i , O + j )K j T s (O j , C j )X s (C j ) = 0.
(19) A second set of 6 equations is obtained when the transfer relation between state vectors X b (O + N ) and X b (B), given by

X b (O + N ) = T(x O N , x B )X b (B), (20)
is associated to the recurrence relation (C1), leading to:

X b (A) -T b (A, B)X b (B) + N j=1 T b (A, O + j )K j T s (O j , C j )X s (C j ) = 0. ( 21
)
The set of 6+2N independent scalar relations given above may be written in matrix form

Le Carrou et al.: Modelling of sympathetic string vibrations as:

R         X b (A) X b (B) X s (C 1 ) . . . X s (C N )         = R 1 R 2 R 3 R 4         X b (A) X b (B) X s (C 1 ) . . . X s (C N )         =          0 . . . . . . . . . 0          (22)
where

R 1 = I 6 -T b (A, B) , (23)
R 2 = T b (A, O + 1 )K 1 T s (O 1 , C 1 ) • • • T b (A, O + N )K N T s (O N , C N ) , (24)
R 3 =    M 1 T b (O + 1 , A) 0 . . . . . . M N T b (O + N , A) 0    , (25)
R 4 =       LT s (O 1 , C 1 ) 0 • • • 0 M 2 T b (O - 2 , O + 1 )K 1 T s (O 1 , C 1 ) LT s (O 2 , C 2 ) 0 • • • . . . . . . 0 M N T b (O - N , O + 1 )K 1 T s (O 1 , C 1 ) • • • M N T b (O - N , O + N -1 )K N -1 T s (O N -1 , C N -1 ) LT s (O N , C N )       . (26)
Thus, the 12+4N-by-1 vector of state vectors at the ends, explicited in the relation ( 22), can be written on the following form taken into account the components set to zero because of boundary conditions:

        X b (A) X b (B) X s (C 1 )
. . .

X s (C N )         =                  X k b (A) X f b (A) X k b (B) X f b (B) X k s (C 1 ) X f s (C 1 ) . . . X k s (C N ) X f s (C N )                  =                  0 X f b (A) 0 X f b (B) 0 X f s (C 1 ) . . . 0 X f s (C N )                  =                     0 3 • • • • • • • • • 0 3 I 3 • • • • • • • • • . . . 0 3 0 3 • • • • • • . . . 0 3 I 3 0 3 • • • . . . . . . 0 2 0 2 • • • . . . . . . . . . I 2 0 2 . . . . . . . . . 0 2 . . . . . . . . . . . . . . . . . . 0 2 0 2 • • • • • • 0 2 I 2                             X f b (A) X f b (B) X f s (C 1 ) . . . X f s (C N )         = ZX f , (27)
where X f denotes the force variables of all state vectors at the boundaries. Therefore, 6+2N relations remain along with the 6+2N unknown variables of state vectors and may be written as

(R • Z) X f = R R X f = 0. ( 28
)
In the case of free vibrations, vector X f is not equal to zero and the matrix R R is not an inversible matrix, implying:

det (R R (ω)) = 0. ( 29
)
The angular eigenfrequencies ω i of the system are obtained from equation (29). For each ω i , relation (28) provides the components of X f (with an unknown multiplicative factor) from which we can perform all state vector components at any point on the assembly by equations (C1) and (20) and thus obtain the associated mode shape.

III. THE BEAM-2 STRINGS ASSEMBLY CASE

Using the beam-N strings model presented in section II, a numerical application is carried out in the case of the beam-2 strings assembly and validated by an experimental modal analysis.

A. Description

The structure used in experimental and theoretical studies is presented in figure 3. It is composed of 2 strings tuned on octave (the frequency of one almost equals the double of the other one's) at E 2 =82.4 Hz and E 3 =164.8 Hz on the equal temperament scale based on A 4 =440 Hz. Note that such values correspond to fundamental frequencies of uncoupled strings, allowing to determine the string's tension if we assume that the beam behaves like a fixed end. In order to avoid static deformation during the experiment, we choose to fix the strings not only on either side of the beam but also close to each other, figure 3. The strings and beam characteristics used in the numerical applications are presented in table I.

B. Numerical results

Eigenfrequencies

Eigenfrequencies of the structure are computed from equation (29). The logarithm of matrix R R determinant is shown in figure 4 as a function of frequency. Eigenfrequencies are given by each drop of the curve. We thus obtain 14 eigenfrequencies in the frequency range 0-500 Hz, shown in table II. For high frequencies, above 750 Hz, a numerical divergence occurs in the calculation of this determinant. As the frequency increases, the matrix R R becomes ill-conditionned because of exponential terms with real argument present in the propagator matrix Q, defined by [START_REF] Pestel | Matrix method in elastomechanics[END_REF]. This limitation of the transfer matrix method is well known and discussed, for example, in [START_REF] Pestel | Matrix method in elastomechanics[END_REF].

Modes shapes and orthogonality check

For each eigenfrequency, we perform mode shapes of the structure. Table II shows the eigenfrequencies of modes and in figure 7-A are presented the corresponding mode shapes. Note that because of the transfer matrix method's frequency limitation, only transversal modes are obtained (the first longitudinal mode is about 2000 Hz for a clamped beam with the same characteristics as for the coupled one). To verify the validity of our calculation, we perform an orthogonality test of the modal basis using the dot product matrix of modes. This i-by-j Dot Product Matrix can be explicited for each term,

DP M (i, j) = < Φ i , Φ j > √ < Φ i , Φ i > < Φ j , Φ j > , (30)
where vector Φ i is the mode shape i,

Φ i = u w i ( 31
)
whose components are longitudinial and transversal displacements, and where the dot product is defined [START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF] by

< Φ i , Φ j >= S ρ S Φ T i (x) Φ j (x) dx. (32)
The dot product between modes i and j for a conservative structure (S) equals one when i = j and otherwise equals zero. This is illustrated in figure 5, which shows the orthogonality of modes.

Definition of the criterion for modes classification

In order to classify modes of the assembly, we define an indicator, called kinetic energy ratio (KER), allowing us to determine in which sub-structure the modal shape displacement is the most important. Each sub-structure is indexed by k: k=0 for the beam, k=1 for the E 2 string and k=2 for the E 3 string. For each mode Φ j , we define

KER j (k) = k ρ k ω 2 j Φ T j (x)Φ j (x) dx 3 r=1 r ρ r ω 2 j Φ T j (x)Φ j (x) dx = k ρ k Φ T j (x)Φ j (x) dx 3 r=1 r ρ r Φ T j (x)Φ j (x) dx (33)
where ρ k is the mass per unit length of the sub-structure and x is the generic space variable defined in the section II.

Modes classification

Modes of the studied structure are classified in four families depending on the relative importance of KER for each sub-structure. This classification is qualitative. Note that an arbitrary threshold on criterion KER could be defined to fix this classification in a quantitative manner.

• Modes 1, 3, 6, 11, 13 are beam modes: beam KER is one order higher than the string KER therefore the beam is the predominant sub-structure. If we compare beam modes (table II left column) to uncoupled beam modes (middle column), we notice that the beam-string coupling increases the beam mode's eigenfrequencies: each string acts as a stiffener [START_REF] Oniszczuk | Free transverse vibrations of an elastically connected complex beam-string system[END_REF]. Moreover, the first eigenfrequencies of an uncoupled beam constrained by no displacement at the positions of string supports (table II right column) have the same order of magnitude than those of the beam modes (left column). It indicates that strings approximatively behave as rigid supports. However, the discrepancies between the two sets of eigenfrequencies are all the more important as the frequency is high, and show the importance of the beam-string coupling.

• Modes 2, 7, 10, 12 are string modes: E 2 or E 3 string KER is one order higher than the two other KER therefore one string is the predominant sub-structure. In table II, string mode frequencies are slightly affected by the beam. This shows that the beam almost behaves like a fixed end for the strings.

• Modes 4, 5, 14 are string-string modes: both string KER are of the same order and one order higher than beam KER, therefore the two strings are at the same time the predominant sub-structures. Modes 4 and 5 almost correspond to the first mode of the E 3 string and the second mode of the E 2 string. The difference is the relative phase between strings: for mode 4, the two strings are in phase at their connection point on the beam, whereas they are out of phase for mode 5 (figure 7-A). This explains the small shift of frequency and the lower participation of the beam for mode 5. Mode 14 corresponds to the third mode of the E 3 string and the sixth mode of the E 2 string. The out of phase corresponding mode cannot be found because of the frequency limitation of our model.

• Mode 9 is a beam-string mode: both beam KER and E 3 string KER are much higher than E 2 string KER therefore beam and E 3 string are the predominant sub-structures. Note that in association with mode 10, this mode should be expected to be a coupled string-string mode, corresponding to the second mode of the E 3 string and the fourth mode of the E 2 string.

FIG. 6: E 3 string initial shape w s (x, 0) obtained by plucking.

Definition of the sympathetic mode

We define the string-string mode as the Sympathetic Mode where strings KER have the same order of magnitude and are much higher than the beam KER. This qualitative definition of the sympathetic mode is consistent with the musician's definition of sympathetic vibrations: when one string is excited another string vibrates. Actually, according to the modal superposition principle [START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF], the response of the structure in the time domain can be obtained by summing the response of all modes,

q(x, t) = u(x, t) w(x, t) = j a j Φ j (x) cos(ω j t), ( 34
)
where t is the time and a j is the modal amplitude depending on the initial condition

a j =< q(x, 0), Φ j >= S ρ S u(x, 0) w(x, 0) T Φ j dx. ( 35
)
The initial velocities equalling zero, terms on the form sin(ω j t) do not exist in (34) as they would be expected to, in the general case. The initial shape w(x, 0) of a string is given by figure 6 in the case of string plucking. If this shape is such that a j does not equal zero, mode j is excited. If this mode is a sympathetic mode, both strings will be excited at the same time and the ratio of energy in each string is given by the KER. The beam-2 strings assembly presented in figure 3 is studied by modal testing in order to confirm theoretical results.

The experimental setup is shown in figure 8. The beam is clamped onto a frame filled with sand to avoid interference with assembly modes. The steel strings are fixed, on one side, by a ring stuck in the beam, and on the other side, by a guitar tuning machine fixed on the frame. A guitar tuning machine is used to tune the strings at E 2 and E 3 .

The experimental modal testing consists in measuring the frequency response functions between velocity and applied force at different points on the structure. From these data, we may extract eigenfrequencies, mode shapes and damping parameters of the studied structure using the Least Square Complex Exponential Method [START_REF] Ewins | Modal testing: theory and practice[END_REF] as implemented in LMS software.

Because it is difficult to hit a string with a hammer, the hammer is used on one point of the beam to excite our system while a laser vibrometer measures the velocity response at all points: 11 on the beam and 3 on each string.

Results

In table III, the eigenfrequencies resulting from the modal analysis are presented in frequency range 0-550 Hz. In order to compare experimental eigenfrequencies with theoretical ones, we plot experimental against theoretical values, in figure 9. Each number of experimental mode corresponds to the number of the associated theoretical mode, obtained by studying mode shapes, (figure 7). Note that, in table III, a theoretical mode is missing, mode 10. We suggest that this problem is due to E 2 string measurements. In figure 9, we show that there is a good agreement between measured and theoretical modes: the average difference is about 8% with a maximum of about 25% for modes 3 and 8. However, a slight frequency deviation exists for beam modes and this suggests that the ideal clamped end condition is not well satisfied in practice.

For each measured mode, the criterion KER, defined by equation (33), is also computed from the mode shape and then the kind of mode is determined. For the last modes, mode 12 to 14, the two strings KER are questionable because of few measurement points on strings, as shown in figure 9-B, and will not be discussed. Four kinds of mode are also obtained and particularly, string-string modes, although only one string-string mode has been extracted: mode 4.

The good agreement between experimental and theroretical results shows that the transfer matrix method is a valuable approach for the vibratory behaviour of the studied structure, and validates the proposed modes categorization.

D. Influence of inharmonicity on string-string mode

In order to investigate the sympathetic behaviour of the beam-2 strings model, we study the influence of the inharmonicy of the two strings on the KER criterion of string-string modes. Inharmonicity quantifies how out-of-tune the strings are and is defined by

= f 2 -2f 1 2f 1 , (36)
where f 1 and f 2 are the fundamental frequencies of the uncoupled E 2 and E 3 strings. For this study, we fix f 2 (f 2 = 164.8 Hz) and only f 1 is modified (from f 1 = 80.4 Hz to f 1 = 84.4 Hz).

For a set of strings pair's frequencies, modes of the assembly are performed, by using the theoretical approach given section II, and kinetic energy ratio of each sub-structure for four modes are computed, see figure 10. Moreover we show the evolution of mode shapes for two inharmonicities: = -0.6% and = 0.6% which respectively correspond to f 1 = 82.9 Hz and f 1 = 81.9 Hz. These mode shapes allow us to link the kinetic energy ratio to its implications for string-string modes when inharmonicity differs from zero. In figure 10, the case studied in section III can be seen when the two strings are tuned on octave, = 0, and the kinetic energy ratios of the two strings do not equal: 25% and 69% for E 2 string and E 3 string respectively in the case of mode 4, 43% and 37% in the case of mode 10, 0% and 89% in the case of mode 9 and 97% and 0% in the case of mode 10. To obtain equal kinetic energy ratio for the two strings (close to 50%), the inharmonicity value is approximatively 0.25% which corresponds in musical scale at 4 cents. In order to have the two strings taking an equal part in the sympathetic mode, they must be 4 cents out-of-tune. Note that this value is obtained for curves which have the same aspect, modes 4, 9 and 10, but also for mode 5's curve which is more disrupted. Actually, for this mode there are three values in the case of equipartition of the strings's kinetic energy ratios as shown in figure 10.

For modes 4, 5, 9, 10, the strings's KER have the same evolution: while one of the strings's KER goes from 0% to 100% the other one goes from 100% to 0%. When the strings's KER switch, they have the same order of magnitude and thus the string-string FIG. 10: kinetic energy ratio, KER j (k) for each sub-structure k (k = 0 for the beam, k = 1 for the E 2 string and = 2 for the E 3 strings) in relation to inharmonicity ( = f 2 -2f 1 2f 1 ) on modes j 4, 5, 9 and 10. The two mode shapes corresponding to = -0.6% (f 1 = 82.9 Hz) on the left and = 0.6% (f 1 = 81.9 Hz) on the right of each curves are shown. mode may appear. Inharmonicity range of the strings's KER switch is different for each mode: mode 4 and 5 ranges are larger than mode 9 and 10. This fact can explain why in the case studied in section III, = 0, string-string modes are not obtained for mode 9 and 10, section III B 4, for the strings are not exactly 4 cents out-of-tune.

Note that the beam can play an important part in the string-string mode. Actually, for mode 5, the coupling with the beam is simultaneous with the strings's KER switch. Thus, at the switching moment it may be obtained not only sympathetic modes but also a beam-string mode.

IV. CONCLUSION

In order to consider the sympathetic phenomenon, an analytical model of a generic string instrument, composed of a beam-N strings assembly, has been studied by means of the transfer matrix approach, then the modal basis has been derived. A simplified system, a beam-2 strings assembly, has been extensively analysed. One string being tuned on octave away from the other (E2 and E3), several specific modes called "string-string modes" among three other kinds of modes, are extracted from the modal basis. The modes have been analysed and classified through the comparison of the kinetic energy ratio of each substructure (E 2 string, E 3 string and beam). String-string modes are responsible for the sympathetic phenomenon.

Furthermore, the modal basis of the beam-2 strings assembly has been extracted by measurements. A good agreement between eigenfrequencies and mode shapes is highlighted through theoretical results and experiments. Experimental results have been interpreted in terms of kind of modes previously given, and string-string modes have been observed. Moreover we have focussed our attention on the theoretical influence of string's inharmonicity on the string-string mode. On one hand, it is noticeable that the particular case of null inharmonicity between the two strings, does not correspond to the equipartition of the string's kinetic energy ratios. On the other hand, the inharmonicity range in which the string-string mode exists, when that the two string's kinetic energy ratios have the same order of magnitude, is narrow.

The model is one-dimensional and this fact is the main restriction for direct application to a real string instrument. Other limitations are related to the conservative hypothesis, that is retained for the normal mode computation. Moreother, torsional motions of the strings are not taken into account. The limitation of the method is the numerical difficulties of the transfer matrix method which occur at high frequencies.

The method and the results that we have presented are applicable to systems with many strings comparable to real string instruments, such as the harp. Harp players are annoyed by the acoustic consequence of the sympathetic phenomenon caused by the high number of strings. A question is, how can this annoyance be reduced? A way to investigate this problem is to do time domain simulations based on the modal basis of the string instrument, and to show the influence of string-string modes on the radiated sound, for instance. A parametric study of the different characteristics of the simplified instrument can be carried out to see their influence on string-string modes. T b11 (x, x 0 ) 0 0 T b14 (x, x 0 ) 0 0 0 T b22 (x, x 0 ) T b23 (x, x 0 ) 0 T b25 (x, x 0 ) T b26 (x, x 0 ) 0 T b32 (x, x 0 ) T b22 (x, x 0 ) 0 T b35 (x, x 0 ) T b36 (x, x 0 ) T b41 (x, x 0 ) 0 0 T b11 (x, x 0 ) 0 0 0 T b52 (x, x 0 ) T b53 (x, x 0 ) 0 T b22 (x, x 0 ) T b56 (x, x 0 ) 0 T b62 (x, x 0 ) T b63 (x, x 0 ) 0 T b65 (x, x 0 ) T b22 (x, x 0 )

          . (B4)
After calculations, we find:

T b11 (x, x 0 ) = cos(k t (x -x 0 )), T b14 (x, x 0 ) = 1 k t E b A b sin(k t (x -x 0 )), T b22 (x, x 0 ) = 1 2 c 1 (k f (x -x 0 )), T b23 (x, x 0 ) = k f 3 E b I 2ρ b ω 2 s 1 (k f (x -x 0 )), T b25 (x, x 0 ) = - k f 2ρ b ω 2 s 2 (k f (x -x 0 )), T b26 (x, x 0 ) = - k 2 f 2ρ b ω 2 c 2 (k f (x -x 0 )), T b32 (x, x 0 ) = ρ b ω 2 2k f 3 E b I s 2 (k f (x -x 0 )), T b35 (x, x 0 ) = - 1 2k 2 f E b I c 2 (k f (x -x 0 )), T b36 (x, x 0 ) = 1 2k f E b I s 1 (k f (x -x 0 )), T b41 (x, x 0 ) = -A b E b k t sin(k t (x -x 0 )), T b52 (x, x 0 ) = - ρ b ω 2 2k f s 1 (k f (x -x 0 )), T b53 (x, x 0 ) = - k 2 f E b I 2 c 2 (k f (x -x 0 )), T b56 (x, x 0 ) = - k f 2 s 2 (k f (x -x 0 )), T b62 (x, x 0 ) = ρ b ω 2 2k 2 f c 2 (k f (x -x 0 )), T b53 (x, x 0 ) = k f E b I 2 s 2 (k f (x -x 0 )), T b65 (x, x 0 ) = - 1 2k f s 1 (k f (x -x 0 )).

FIG. 1 :

 1 FIG. 1: Diagram of a clamped-clamped beam connected to several fixed strings. It includes local coordinate sytems and the notation used.

FIG. 2 :

 2 FIG.2: Connection between beam and string including local coordinate sytem and notations. In grey, the connection volume, supposed to be very small.

FIG. 3 : 2 A 2

 322 FIG. 3: Diagram of beam-2 strings tuned on octave.

FIG. 4 :

 4 FIG. 4: Logarithm of the R R matrix determinant as function of frequency. Each drop corresponds to an eigenfrequency.

E 2

 2 string and k=2 for the E 3 string) and kinds of modes computed for the beam-2 strings assembly (left column). Eigenfrequencies of uncoupled sub-structures (beam * , E 2 string, E 3 string) with same characteristics as for coupled ones (middle column). Eigenfrequencies of beam constrained by nodes at the positions of string supports (denoted beam * * ), obtained by FEM (right column).

13 FIG. 7 :FIG. 8 :

 1378 FIG. 7: A-Theoretical mode shapes obtained by transfer matrix method. Mode number is associated to eigenfrequency shown in table II. B-Experimental mode shapes obtained by modal analysis on 11 measurement points on the beam and 3 on the string. Eigenfrequency of modes are shown in table III.

  waves propagate with wavenumbers k b , -k b , k b , -k b , k r , -k r , wherek b = √ ω 4 ρ b E b I and k r = ω ρ b A b E b . (B2)These six values are the eigenvalues of matrix N b = -H b . Calculation of the associated eigenvectors gives the transfer matrix from relation[START_REF] Gough | The theory of string resonances on musical instruments[END_REF]. It is convenient to introduce the Duncan's functions[START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF], defined by the relations, in which ξ is a real parameter,s 1 (ξ) = sin(ξ) + sinh(ξ) c 1 (ξ) = cos(ξ) + cosh(ξ) s 2 (ξ) = -sin(ξ) + sinh(ξ) c 2 (ξ) = -cos(ξ) + cosh(ξ),(B3)to express the transfer matrix, T b (x, x 0 ) =

  FIG.5: Dot product matrix for the 14 first theoretical modes of the assembly.

	Mode Freq.		KER j (k)		Kind	Uncoupled freq. (Hz)
	j	(Hz)	k=0 k=1 k=2	of mode	Beam * E 2 String E 3 String Beam * *
	1	60.25	100	0	0	Beam	24.85	76.98
	2	82.35	0	100	0	String	82.40
	3	114.95 100	0	0	Beam	68.50	141.18
	4	164.45	6	25	69 String-String	164.80	164.80
	5	164.80	0	63	37 String-String
	6	174.20	99	0	1	Beam	134.30	212.72
	7	247.10	0	100	0	String	247.20
	8	292.95	99	1	0	Beam	222.00	389.34
	9	329.30	11	0	89 Beam-String 331.60 329.60	329.60 418.72
	10	329.70	3	97	0	String
	11	335.95	97	0	3	Beam	331.60	418.72
	12	412.00	2	98	0	String	412.00
	13	492.10	91	9	0	Beam	463.15	690.04
	14	494.40	1	25	74 String-String	494.40	494.40

TABLE II: Mode number, eigenfrequencies, kinetic energy ratio (k=0 for the Beam, k=1 for the

TABLE III :

 III Experimental eigenfrequencies, kinetic energy ratio and kinds of modes obtained by modal analysis for 2 strings connected to the beam.

  FIG. 9: Experimental eigenfrequencies as a function of theoretical eigenfrequencies. Mode number correspond to table II theoretical results and table III experimental results.

Acknowledgments

The authors acknowledge financial support from the CNRS and the Region des Pays de la Loire for JLLC's PhD studentship. They also acknowledge the students of ENSIM: R. Arnaud, O. Doutres and S. Paillasseur for setting up the experiment and for the first computerized model.

APPENDIX A: TRANSFER MATRIX OF A STRING'S SEGMENT

Each string is considered as perfectly flexible and stretched at tension T 0 made from a material whose Young's modulus is E s and mass per unit length is ρ s . The area of the cross section is denoted A s . Strings are submitted to longitudinal and transverse displacements described by the equations of motion given by rod's theory and by string's theory [START_REF] Valette | Mécanique de la corde vibrante[END_REF]. These equations may be written in state form [START_REF] Hutchins | Sympathetic vibration and coupling of resonances[END_REF], in which the characteristic matrix, is given by

and is associated to the state vector [START_REF] Morfey | Dictionary of acoustics[END_REF]. Calculation of the eigenvalues of matrix N s = -H s provides the wavenumbers of the longitudinal and transverse waves travelling in the strings:

where

Transfer matrix between points at coordinates x and x 0 is obtained from relation (5) as

where

APPENDIX B: TRANSFER MATRIX OF A BEAM'S SEGMENT

The beam is supposed to be prismatic and made from material whose Young's modulus is E b and mass per unit length is ρ b . The area of the cross section is A b and the second moment of area is I. Transverse and longitudinal motions of the beam are respectively described using Euler-Bernoulli theory and rod's theory [START_REF] Géradin | Théorie des vibrations, application à la dynamique des structures[END_REF]. Using state vector formalism, the equations of motion can be written on the form (3), in which the characteristic matrix

In this appendix we demonstrate the relation (19). This relation can be demonstrated in two steps. The first step is the demonstration of the recurrence relation, defined by, for i = 1...N ,

• We verify the relation (C1) for i=1. In that case (C1) is written as

in accordance with (17), the relation (C2) is true.

• We assum that the relation (C1) is valid for the i-string , we show that this relation C1) is still valid for the i + 1-string . Let's consider the coupling relation (17) induced by the connection between the beam and the i + 1-string,

and taking the form of

As we have the relation (C1), the relation (C2) is equivalent to

Relation (C7) proves that (C1) is valid for i+1 string.

• Finally, we prove the relation (C1), for i = 1...N :

The second step is the insersion in the coupling relation (18) of the recurrence relation (C1). These two relations, (18) and (C1), along with the transfer relation between state vectors X s (O i ) and X s (C i ), given by

leads us to obtain the relation (19).


Jean-Pierre Dalmont

Joël Gilbert

Sébastien Ollivier

Matthieu Blondet

Non-linear characteristics of single reed instruments : quasi-static volume flow and reed opening measurements

Introduction

Sound production in reed wind instruments is the result of self-sustained oscillations.

A mechanical oscillator, the reed, acts as a valve which modulates the air flow entering into the mouthpiece. The destabilisation of the mechanical element is the result of a complex aeroelastic coupling between the reed, the air flow into the instrument driven by the mouth pressure of the musician, and the resonant acoustic field in the instrument itself. Following McIntyre et al. (1983), wind instruments can be described in terms of lumped model formed by a closed feedback loop operating as a self-sustained oscillator. In their model the loop is made up with two elements, a lumped non-linear element -the mouthpiece -and a linear passive element -the resonator, that is the instrument itself. The modelling and the measurement of the resonator has been extensively studied since [START_REF] Bouasse | Instruments à vent, vols I & II, Librairies Delagrave[END_REF] by many authors (see for example references in Nederveen, 1998, or Fletcher and[START_REF] Fletcher | The physics of musical instruments[END_REF]. On the contrary, the non-linear element has only recently been the subject of thorough studies (see for example the review given by [START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF] and the knowledge of this elements action is considerably less than that of the resonator. The non-linear element can be defined by a relationship between the pressure difference across the reed and the volume flow at the inlet of the pipe of the instrument. Assuming a quasi-static response of the reed which neglects inertia and damping, the volume flow is thus an explicit function of the pressure difference. This function is called the characteristics of the reed. Most authors converge on an elementary model of the characteristic presented in section 2.1 [START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Fletcher | Air flow and sound generation in musical wind instruments[END_REF][START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF][START_REF] Saneyoshi | Feedback oscillations in reed woodwind and brasswind instruments[END_REF][START_REF] Kergomard | Elementary considerations on red-instruments oscillations[END_REF][START_REF] Kergomard | Calculation of the spectrum of selfsustained oscillators using a variable truncation method[END_REF]Ollivier et al. 2002). It is based on a quasi-stationary model of the air flow through the mouthpiece, and on a mechanical model of the reed discussed respectively in sections 2.2 and 2.3. The aim of the present paper is to obtain experimental data for the non-linear element in order to check the validity of the elementary model and to find realistic values of the parameters useful for physical modelling synthesis. The data may also give information helpful for the understanding of the physical phenomena involved.

In the present paper, a method for measuring the characteristics is proposed, the measurements being done in quasi-stationary conditions. The dynamic aspects related to the reed are not considered in the present paper. These aspects are certainly not negligible with regard to the spectrum of the instrument (Thomson, 1979). However when the playing frequency is small compared to the reed resonance frequency, it has been shown that these aspects are not essential from the strict point of view of the auto-oscillation [START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Dalmont | Some aspects of tuning and clean intonation in reed instruments[END_REF]. For the experiments an artificial blowing machine is used (section 3.1). To perform the volume flux measurements, a constriction (diaphragm) is used as a pressure reducing element for a differential pressure flowmeter (section 3.2). The diaphragm takes the place of the resonator. The major advantage of such a device is that the diaphragm also plays the role of a non-linear absorber [START_REF] Ingard | Acoustic non-linearity of an orifice[END_REF] which thwarts a possible standing wave in the mouthpiece. Oscillations of the reed are thus in most cases impeded and avoided if the diaphragm is well chosen. This makes it possible the measurement of the complete characteristics of real clarinet mouthpieces. The choice and dimensions of the diaphragm which depends on the reed properties are discussed section 3.3. The diaphragm does not induce additional sensors for the pressure sensors used other than those commonly present in an artificial mouth. In addition to the flow measurement, the opening of the reed is measured optically. Then the measurements of the reed opening and of the pressure difference across the reed gives information on the evolution of the reed stiffness which is important for the understanding of the quasi-static reed mechanics. For obtaining relevant results, an accurate calibration of the sensors has to be done. Thus a specific experimental procedure has been developed which is discussed in section 3.5. Some typical experimental results are presented section 4.1. Section 4.2 and 4.3 are focused on two aspects of the mechanics of the reed, viscoelasticity and stiffness respectively. Section 4.4 is dedicated to the flow aspects.

Finally in section 4.5 typical values of the parameters of the model are summarized and the accuracy of their determination is discussed. [START_REF] Backus | Small-vibration theory of the clarinet[END_REF] has presented the first measurements of the characteristics of a single reed instrument under steady flow conditions. The main result of Backus is a non-linear expression relating the volume flow U through the reed and two variables : the pressure difference ∆P across the reed, and the opening H between the tip of the reed and the mouthpiece. The pressure difference ∆P across the reed is equal to the mouth pressure P m minus the pressure in the mouthpiece P in . Backus fitted the experimental data by means of an expression in which U is proportional to (∆P) 4/3 and to H 2/3 . There are good arguments to abort Backus formula [START_REF] Hirschberg | A quasi stationnary model of the air flow in the reed channel of single reed woodwind instruments[END_REF][START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF] and most flow models are now based on the stationary Bernoulli equation [START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Fletcher | Air flow and sound generation in musical wind instruments[END_REF][START_REF] Saneyoshi | Feedback oscillations in reed woodwind and brasswind instruments[END_REF][START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF] which states that the volume flux U in the reed channel is given by :

State of art

Elementary model

ρ P wH U ∆ = 2 , ( 1
)
where w is the width of the reed channel, and ρ is the air density.

The steps leading to equation (1) are summarised by [START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF]. A reasonable prediction of the flow through the reed channel is obtained by assuming flow separation at the neck of the flow channel (figure 1a). For a uniform reed channel of height H the pressure in the reed channel is equal to the pressure in the mouthpiece. Assuming a turbulent dissipation of the kinetic energy in the jet without pressure recovery and neglecting friction in the reed channel we find, from Bernoulli equation, the velocity v b :

ρ P v b ∆ = 2 . ( 2
)
Multiplying v b by the jet cross section S j provides a prediction for the volume flow entering into the mouthpiece. If there are rounded corners at the inlet of the reed channel as in figure 1b for example), it is furthermore assumed that the jet cross section S j is equal to wH, where w is independent of H. The motion of the reed determines the opening cross-section area and controls the volume flow entering in the mouthpiece. Assuming the mechanical response of the reed to be reduced to its stiffness (see section 2.3 for discussion) the variation of the reed opening

H H y - = 0
, where H is the reed opening and 0 H the reed opening in the absence of flow, is proportional to the pressure difference ∆P across the reed :

k P y ∆ = , ( 3
)
where k is a stiffness per unit area. Equation ( 3) is meaningful until H is equal zero (reed blocked on the lay). This corresponds to a limit value P M of the pressure difference ∆P given by : 0

H k P M = . ( 4
)
If the pressure difference is larger than P M , the reed closes the opening and no flow enters into the mouthpiece. Finally the volume flow U can be written as a function of the pressure difference ∆P : which is just below the threshold of oscillation [START_REF] Kergomard | Calculation of the spectrum of selfsustained oscillators using a variable truncation method[END_REF].

if 0 if 2 ) 1 ( 2 ) (      ≥ ∆ = ≤ ∆ ρ ∆ ∆ - = ρ ∆ ∆ - = M M M

FIG. 2. Theoretical characteristic (equation 5) : volume flux U as a function of the pressure

difference ∆P (arbitrary scales). P M is the value of the pressure difference corresponding to the reed blocked on the mouthpiece. [START_REF] Hirschberg | A quasi stationnary model of the air flow in the reed channel of single reed woodwind instruments[END_REF] and [START_REF] Van Zon | Flow through the reed channel of a single reed music instrument[END_REF] have studied experimentally and theoretically the volume flow control by the motion of the reed in order to explore the limits of validity of the elementary model presented above, and to provide a better understanding of the results of [START_REF] Backus | Small-vibration theory of the clarinet[END_REF].

Quasi-stationary model of air flow

Following [START_REF] Van Zon | Flow through the reed channel of a single reed music instrument[END_REF] due to the abrupt transition from the narrow reed channel of height H to the inner part of the mouthpiece of diameter D, flow separation occurs for sufficient high Reynolds numbers Re=ρU/(µw)>10 (µ is the dynamic viscosity of air). A free jet is formed in the mouthpiece. For large values of D/H>10, which is a typical value for single reed instruments, the pressure recovery upon deceleration of the flow in the mouthpiece is negligible. Hence the pressure P in in the mouthpiece is assumed uniform and equal to the pressure at the end of the reed channel. Measurements by [START_REF] Van Zon | Flow through the reed channel of a single reed music instrument[END_REF] confirm this assumption : the pressure variations within the mouthpiece are less than 3% of the dynamic pressure in the jet.

In the limit of high Reynolds numbers and a short channel (Re.H/L>1000 where L is the length of the reed channel) the volume flow can be estimated by assuming a uniform flow in the reed channel and by applying Bernoulli's equation. Ignoring the flow separation at the entrance of the channel, the volume flow is found to be given by equation 1 (elementary model). As noted in [START_REF] Hirschberg | A quasi stationnary model of the air flow in the reed channel of single reed woodwind instruments[END_REF] separation occurs when the edges at the entrance are sharp, which is the case for clarinet mouthpiece and its reed. A free jet with a section S j lower than the reed-mouthpiece opening cross-section S will be formed within in the channel.

For short reed channels (L/H<3) no reattachment of the flow occurs within the channel and the volume flow U will be given by : ρ α

P wH U ∆ = 2 where S S j = α . ( 6
)
The coefficient α is a "contraction" coefficient which is strongly dependent on the geometry of the reed channel inlet. For typical 2D mouthpiece geometry, values in the range 0.5<α<0.61 are expected. Laser Doppler flow measurements and flow visualisation experiments by [START_REF] Van Zon | Stromingsgeïnduceerde klepinstabiliteiten[END_REF] confirm the typical values of α for the geometry considered.

For low Reynolds number (Re<10) and long reed channel (L/H>10), the flow is well approximated by a fully developed Poiseuille flow. This corresponds to the case of the reed almost closed for which the volume flow is thus given by :

L P wH U ν ρ 12 3 ∆ = . ( 7
)
Both Poiseuille and Bernoulli limits were also found by [START_REF] Gilbert | Etude des instruments de musique à anche simple : extension de la méthode d'équilibrage harmonique, rôle de l'inharmonicité des résonances, mesure des grandeurs d'entrée[END_REF] and [START_REF] Maurin | Confrontation théorie-expérience des grandeurs d'entrée d'un excitateur à anche simple[END_REF].

The intermediate flows between the two extreme cases mentioned above are discussed in Van [START_REF] Van Zon | Flow through the reed channel of a single reed music instrument[END_REF] and [START_REF] Hirschberg | Non-linear behaviour of single-reed woodwind musical instruments[END_REF]. For L/H>4, the jet formed by the separation of the flow from the sharp edge of the reed at the entrance of the reed channel reattaches to the wall after a distance of about 2H. If the channel is short the friction is negligible, the volume flow U approaches the value given by equation ( 1) then the section S j of the jet is equal to the reed-mouthpiece opening cross-section S. The quasi-stationary models described above assume a fixed separation point at the inlet or at the exit of the reed channel and a uniform section of the reed channel. This hypothesis is questionable in the case of the clarinet mouthpiece. The transition between the "reed channel" and the mouthpiece can be smooth. In such a case the reed channel height is not uniform, and for L/H>4 the separation point is not easy to determine. As a consequence the coefficient α is not so easy to predict precisely and can be equal to values larger than 1. In other words, in such a case the volume flux U can be larger than the one predicted with the elementary model (equation 1). In the case of a fully separated jet flow (short reed channels situation, L/H<2), the channel geometry is not expected to be critical and the result given in equation ( 6) could remain valid. Notice that all the theoretical results described before have been successfully compared with experimental results obtained with a two-dimensional mouthpiece geometry. Another particularity of clarinet mouthpieces hasn't been yet mentioned. The reed channel consists of two parts : a)

the front slit delimited by the edge of the mouthpiece tip and the reed and b) the lateral slits between the lay and the reed. Then the effective section S j can be larger than the opening cross-section S. In such a case the contraction coefficient α defined by equation 6 can be larger than one if w is always defined as the width of the tip of the mouthpiece.

Quasi-static response of the reed

The mechanics of the reed is complex. The material is orthotropic and the dimension irregular. Prediction of the deformation of the reed is difficult because reed is an essentially inhomogeneous material [START_REF] Heinrich | Recherches sur les propriétés densitométriques du matériau cane de provence et ses similaires étrangers; relation avec laqualité musicale; étude associée d'une mesure de dureté[END_REF]. Its mechanical properties are variable and also depend strongly on the amount of water in the material [START_REF] Heinrich | Recherches sur les propriétés densitométriques du matériau cane de provence et ses similaires étrangers; relation avec laqualité musicale; étude associée d'une mesure de dureté[END_REF][START_REF] Obataya | Acoustic properties of a reed (Arundo donax L.) used for the vibrating plate of a clarinet[END_REF]. [START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF] suggest that a dry reed displays a viscoelastic behaviour whereas a wet reed has a viscoplastic behaviour. The reed rolls up (or not) on the lay of the mouthpiece whose geometry is said to be very critical : from the experience of craftsmen it seems that variations of some hundredths of millimeters on dimensions of the curvature of the lay lead to change of behaviour perceptible by the musician [START_REF] Hirschberg | Non-linear behaviour of single-reed woodwind musical instruments[END_REF]. Finally the lips of the musician are pressed on the reed. This means that the mechanical behaviour of the reed is also dependent on the coupling with the lip, a mechanical system which is also not easy to characterise.

The reed is usually considered to be a one degree of freedom oscillator, that is the reed tip displacement y is related to the pressure difference ∆P by the following equation :

µ ∆ = ω + + P y y g y r 2 & & & , ( 8
)
where r ω is the angular resonance frequency of the reed, g a viscous-damping coefficient and µ is a mass per area. The dynamic aspects related to the reed are not considered in the present paper and equation ( 8) is then reduced to equation (3) with 2 r k µω = .

(9).

The determination of the reed stiffness k is difficult and only orders of magnitude for these parameters can be found in the literature. The difficulty lies in the fact that these parameters are generally found only in an indirect way and under experimental conditions which are not always realistic. Thus, equation ( 9) suggests that the stiffness can be deduced from the reed resonance frequency. In fact this is questionable because the surface per area µ is itself badly known. In addition the validity of the parameter obtained from the resonance frequency for low frequencies is also questionable. The stiffness per area was measured by [START_REF] Nederveen | Acoustical aspects of woodwind instruments[END_REF] but not in a playing situation. Another solution for determining this stiffness k consists in measuring the impedance of reed [START_REF] Dalmont | Some aspects of tuning and clean intonation in reed instruments[END_REF][START_REF] Boutillon | Evaluation of the acoustical stiffness of saxophone reeds under playing conditions by using the reactive power approach[END_REF].

Unfortunately the stiffness per area is obtained by using an equivalent surface whose value is badly known. Moreover, contrary to what one could suppose, the resonance frequency of the reed itself is difficult to determine. Thus [START_REF] Facchinetti | Numerical and experimental modal analysis of the reed and pipe of a clarinet[END_REF] showed that the frequency of a squeak depends as much on the resonator as on the reed. A simple method to determine the stiffness consists in determining the beating pressure M P and the opening at rest 0 H . The beating pressure can be estimated with an artificial mouth by seeking the pressure for which the reed starts to oscillate after having been plated. This method, being based on the equation ( 4) is used implicitly by [START_REF] Kergomard | Elementary considerations on red-instruments oscillations[END_REF]. In the present paper another method allowing an accurate determination of the beating pressure is given.

In the elementary model presented the reed stiffness k is assumed to be constant. A priori it seems natural to think they are not. The most commonly admitted idea is that the vibrating length of the reed decreases with the opening. This would lead to an increase in the stiffness [START_REF] Nederveen | Acoustical aspects of woodwind instruments[END_REF]. To go further in the analysis various authors propose to model the reed as a bar [START_REF] Stewart | Functional model of a simplified clarinet[END_REF][START_REF] Sommerfeld | Simulation of a player-clarinet system[END_REF][START_REF] Stuifmeell | Vibration modes of a metal reed of a reed organ pipe: theory and experiment[END_REF][START_REF] Gazengel | Caractérisation objective de la qualité de justesse, de timbre et d'émission des instruments à vent à anche simple[END_REF][START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF][START_REF] Van Walstijn | Discrete-Time Modelling of Brass and Reed Woodwind Instruments with Application to Musical Sound Synthesis[END_REF]. Their results tend to show [START_REF] Gazengel | Caractérisation objective de la qualité de justesse, de timbre et d'émission des instruments à vent à anche simple[END_REF][START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF][START_REF] Van Walstijn | Discrete-Time Modelling of Brass and Reed Woodwind Instruments with Application to Musical Sound Synthesis[END_REF] that the reed rolls up on the table of the mouthpiece only under certain conditions which are satisfied for only very particular reed geometries. With the dimensions of a real reed and table [START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF] showed that the reed deforms without sticking to the table until a given point near the tip touches the table. This result is confirmed by van Walstijn ( 2002) who shows that the stiffness is nearly constant as long as the end of the reed does not touch the table and takes a larger value afterwards. These studies would justify the "simplistic" approach of the model suggested (equation 5) at least until the reed tip touches the lay. It remain to verify whether the transverse bending of the reed which is supported only on the side by the table does not modify this two dimensional behaviour.

FIG. 3. Theoretical characteristics (equation 5) in the case of a discontinuous reed stiffness

(according to [START_REF] Van Walstijn | Discrete-Time Modelling of Brass and Reed Woodwind Instruments with Application to Musical Sound Synthesis[END_REF], see text).

Experimental device and procedure

The artificial mouth

The artificial mouth consists of a Plexiglas box with metal reinforcement [START_REF] Gazengel | Caractérisation objective de la qualité de justesse, de timbre et d'émission des instruments à vent à anche simple[END_REF]. The artificial lip consists of a cylindrical latex balloon of small diameter (10mm) in which a piece of foam saturated with water is inserted. The lip is fixed on a rigid support which position can be translated vertically by means of a screw. The mouthpiece is inserted in a metal barrel whose horizontal position can be adjusted. Resonators can be fixed onto the other end of the barrel. The air is supplied by a high pressure system through a sonic valve.

The pressure P m in the mouth is measured by a static pressure sensor. A miniature differential pressure sensor mounted in the wall of the mouthpiece measures the pressure difference

in m P P P - = ∆
between the mouth cavity and the inside of the mouthpiece. The reed slit opening is measured using a LASER beam and a photoelectric diode. This is the method used by [START_REF] Backus | Small-vibration theory of the clarinet[END_REF] (see figure 4).

FIG. 4. Experimental device.

Flow measurement

To determine the static non-linear characteristics the clarinet is replaced by a diaphragm playing the role of a pressure reducing element. Using the pressure measurements, the flow through the reed is calculated by using Bernoulli's equation :

2 2 2 1 dia in S U P ρ = ( 10
)
where the atmospheric pressure is used as a reference, S dia being the section of the opening of the diaphragm.

Compared to another flowmeter placed upstream of the cavity, the diaphragm, apart from its simplicity of implementation, has several advantages. It makes it possible, if its dimensions are well chosen, to obtain a complete characteristic since it avoid oscillations.

Indeed apart from its pressure reducing effect the diaphragm plays, for acoustics, the role of a non-linear resistance thus preventing the appearance of a standing wave inside the barrel. This makes our experiment similar to the one suggested by Benade (1976, page 437).

3..3 The choice of the diaphragm

The diameter of the diaphragm is the result of a compromise. If this diameter is too large the pressure drop in P created by the diaphragm is too small to be measured. If, on the contrary, it is too narrow the reed closes suddenly above a critical threshold and part of the non-linear characteristic cannot be explored [START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF]. This phenomenon occurs when :

0 ) ( ) ( = ∂ + ∆ ∂ = ∂ ∂ U P P U P in m . ( 11
)
Considering that the pressure P ∆ and P in are given respectively by equations ( 5) and ( 10), it is thus necessary, to avoid phenomenon of sudden closure, that :

3 / 0 wH S dia > . ( 12
)
In practice the section of the opening of the diaphragm is chosen to be close to the section of the reed slit at rest position. This might avoid the sudden closing of the reed while ensuring a sufficiently large pressure drop.

The cross section area of the jet formed by the diaphragm is assumed here to be equal to the section of the opening of the diaphragm. This assumption is valid only in the absence of a vena contracta. To avoid a vena contracta the diaphragms has been chamfered (conical orifice). This chamfer suppresses the vena contracta and also extends the range in which the coefficient of discharge is constant (OMEGA, 1995). In order to check this the diaphragms have been calibrated by means of a volume gas meter. The calibration of the various diaphragms proves that the vena contracta coefficient is constant and equal to unity within 1% uncertainty for all the diaphragms except for the 3.5mm diameter diaphragm for which it is estimated to be 0.97.

FIG. 5. Volume flow U as a function of the total pressure drop m P for the clarinet mouthpiece

ended with a diaphragm (arbitrary scales).

Reed opening measurement

This measurement is likely to give information on the behaviour of the jet entering the instrument. In particular the opening can be compared to the effective section of the jet entering in the mouthpiece. This measurement also makes it possible to relate the opening H and the pressure difference P ∆ on both sides of the reed. This allows the determination of the evolution of the reed stiffness with the pressure difference.

The sensitivity of the optical system used to measure the opening H is calibrated by means of visual observation with a camera in macro mode. This device was used for checking the linearity of the optical system and to determine the opening at rest 0 H . A difficulty with the optical system is that the light beam can be fully stopped while the reed slit is not completely closed. To limit this problem, it is checked by visual inspection, before each experiment, that, when the reed is almost closed, the diode still detects a signal.

Experimental procedure

The two pressure signals and the optical signal are collected on a computer via a data acquisition card. The opening at rest having been measured as described above, the experiment starts without blowing pressure ( 0 = ∆P

). This state is maintained for a few seconds. This is used for the determination of the zeros of the pressure sensors and the value of the optical signal corresponding to the opening at rest. The pressure in the mouth cavity m P is then increased gradually until the reed closes completely the opening. This state is also maintained for a few seconds. This is done for the relative calibration of the differential pressure sensor ENTRAN, as in that case 0 = in P , the sensitivity of the static pressure sensor measuring m P being supposed to be known. This calibration is important because the pressure in P is obtained by making the difference between the signals from the two pressure sensors.

The pressure in the mouth is then gradually brought back to zero. A typical duration of such an experiment is 50 to 100s with a sampling frequency of 100Hz (see figure 6). For some embouchures, when the reed is almost closed reed, oscillations occur incidently which make the corresponding part of the non-linear characteristic not exploitable.

FIG. 6. Experimental signals

______ static pressure sensor.

_ _ _ _ differential pressure sensor.

-----optical sensor.

Experimental results

A typical experimental result

The non-linear characteristic was measured for various mouthpieces, reeds and embouchures. Before each measurement of the characteristic, the embouchure is tested with a cylindrical pipe of 30cm in order to check that the instrument produces a realistic sound of good musical quality. The phenomena observed are globally reproducible even if one observes a large variation in the numerical values of the various parameters. The goal of this section is to present a typical case in order to stress the most significant results. A more detailed analysis of the experimental results is provided in the following sections.

On the figure 7.a, a non-linear characteristic

) ( P NL U ∆ =
is shown for an opening at rest of = 0 H 0.6mm. The reed is a Plasticover (covered with plastic) reed of force 3. This type of reed has been chosen because it is less influenced by moisture than a standard reed. The clarinet mouthpiece is a C80 by Selmer. The curve shows an hysteresis, the maximum flow being larger for increasing pressure than for decreasing pressure. This hysteresis can be attributed to the viscoplastic behaviour of the reed. This point is discussed section 4.2. The curves are similar to the theoretical curves coming from equation ( 5), which are also plotted for reference in figure 7.a (dots). The parameters of these theoretical curves are chosen so that the maximum of the theoretical curves matches with those of the experiments. It appears that the closing is never total and that even when the reed can be considered as closed (beyond 60mbar) a weak flow remains. This flow decreases when the maximum pressure is maintained a few seconds. The theoretical model is thus valid until the reed is nearly closed. It does not take into account the residual flow when the reed channel is closed. As explained section 2.2 the jet cross section j S can be deduced from the volume flow using equation 2. Figure 7.c plots the effective cross section as a function of the variation of the reed opening measured by the optical device. The results show systematically that the jet cross section varies linearly with the aperture. In some cases, when the optical adjustment was optimum, a line passing trough zero within an uncertainty lower than 0.1mm was obtained.

The assumption of a jet cross section proportional to the opening thus seems sensible. This assumption is used for the determination, by extrapolation, of the opening corresponding to the zero value of the optical signal.

Viscoelasticity of the reed

As noticed in section 4.1, some curves exhibit an hysteresis due to a change of the rest position. This result could seem contradictory with the assumption of a static measurement.

Indeed the duration of the experiment and the speed of the pressure variations are such that the effects of the reed inertia are negligible during the experiment. The hysteresis can therefore only be explained as the result of a viscoelastic behaviour of the reed which only recovers its original rest position after a time delay larger than our experiments [START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF]. By analysing the opening as a function of time after the reed has been plated and then quickly slackened, it appears, for a given reed, that this one recovers its rest position in three steps. The reed slit opening reaches almost instantaneously 93% of its maximum value.

An exponential decay of the difference between the opening value and the maximum opening with a relaxation time s 8 1 ≅ τ is then observed. At the end of this second phase the reed slit opening reaches 97% of its maximum value. Rest position is finally reached at the end of a last phase for which the relaxation time is s 900 2 ≅ τ

. In this second case, taking into account the importance of the relaxation time, it is not unreasonable to speak of a quasi-plastic deformation [START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF]. Taking into account the typical duration of an experiment one can think that hysteresis observed in the experimental characteristics is due to a conjugation of these two effects. In particular the fact that when the reed is closed there remains a flow which tends to decrease if the maximum pressure is maintained a sufficiently long time. These viscoelastic effects essential for the musician [START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF][START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF], are probably not relevant when considering a physical model of the autooscillation process. The actual characteristics might be found somewhere in between the two static characteristics obtained respectively upon increasing the pressure and decreasing the pressure.

By chance the hysteresis being rather small the uncertainty on the relevant parameters of the model will be small. Typically it should not exceed few percent of 0 H . To limit this effect it is useful, before doing an experiment, to close the reed for a few seconds by applying a large pressure in the mouth volume in order to limit the quasi-plastic effect ( s 900 2 ≅ τ

). This is similar to some musicians practice which consists in pressing the reed with the thumb before playing. It is important to notice that the amplitude of the hysteresis can vary considerably with the reed. We observed that for some reeds, a priori not different from the others, the hysteresis did not appear. On the other hand our experiments have been done with a dry reed covered with plastic. A wet standard read would probably have emphasised a slightly different behaviour [START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF].

Reed stiffness

As noted section 4.1, the reed stiffness can be regarded as a constant until the reed beats. With some mouthpieces and reeds a slightly different behaviour has been observed. . We expect that beyond this pressure the reed stiffness increases as a result of a reduction of the free reed length due to contact with the lay. The remarkable result here is that the change in the reed stiffness value appears only for a pressure higher than the threshold of oscillation 3 / M P P ≈ ∆

. According to various measurements it seems that this phenomenon is general. This result is to be compared with the analysis of various authors [START_REF] Gazengel | Caractérisation objective de la qualité de justesse, de timbre et d'émission des instruments à vent à anche simple[END_REF][START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF]Van Walstjin, 2002) according to whom the curling up phenomenon is limited and has a small influence on the reed stiffness, the end of the reed touching the lay without smoothly curled up on the lay. Our measurements do not exclude this scenario but show that it should appear only for high pressures. Complementary measurements, allowing to determine directly the point of contact between the reed and the lay, are now in preparation to confirm this result.

FIG. 8. Experimental results (continuous line experiments ; doted lines theory) (a) non-linear characteristics, volume flow U versus pressure difference P ∆

(b) reed opening H versus pressure difference P ∆ .

Air flow

To evaluate the discrepancy between the model for the volume flow and the measurement, the measured volume flux U can be divided by the theoretical one U b 8).

It is interesting to compare the measured α values with the vena contracta coefficient obtained for simplified geometries of mouthpieces and in less realistic playing conditions. As mentioned in Section 2.2, [START_REF] Van Zon | Stromingsgeïnduceerde klepinstabiliteiten[END_REF] has obtained values in the range 0.50 to 0.61 for typical 2D mouthpiece. Furthermore [START_REF] Maurin | Confrontation théorie-expérience des grandeurs d'entrée d'un excitateur à anche simple[END_REF] observed larger values, in the range 0.60 to 0.85 for a clarinet mouthpiece mounted in an artificial mouth, the lateral parts of the reed channel being waxed to approach a 2D geometrical situation. The larger value of α could be explained by an effect of the confinement of the flow upstream of the inlet reed channel. This phenomenon would also occur in actual clarinet playing. [START_REF] Valkering | Characterization of a clarinet mouthpiece[END_REF]

Typical parameters range values

The model formulated in equation 5 is based on parameters of which the values have to be defined when dealing with simulations or physical modelling synthesis. Our experiments allow the determination of some of these parameters. In the present section the range of these parameters and the accuracy with which they can be determined using our setup are discussed.

Results are summarised in summarised. These values are in agreement with those given by other authors [START_REF] Nederveen | Acoustical aspects of woodwind instruments[END_REF][START_REF] Stewart | Functional model of a simplified clarinet[END_REF]. We guess that these values might be useful for physical modelling synthesis and simulations.

Conclusion and perspectives

The experiments presented in the present paper allow a better characterization of the clarinet mouthpiece behaviour. It also gives elements useful for a better comprehension of the physical phenomena involved. In the nineties a considerable effort has been carried out in parallel at the LAUM (Le Mans, France) and at the TUE (Eindhoven, Nederlands) to obtain a reliable model for the relationship between the jet section and the reed channel height characterised by the coefficient α of equation 6 [START_REF] Van Zon | Stromingsgeïnduceerde klepinstabiliteiten[END_REF][START_REF] Hirschberg | A quasi stationnary model of the air flow in the reed channel of single reed woodwind instruments[END_REF][START_REF] Hirschberg | Musical aeroacoustics of the clarinet[END_REF][START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF]. Using simplified 2D geometries, different stationary regimes have been identified experimentally and explained theoretically. We have now proposed an experimental procedure allowing to measure these characteristics with actual mouthpieces and reeds under conditions close to playing conditions. Our measurements with this new procedure seem to agree qualitatively with the earlier measurements in simplified geometries. The main problem in the interpretation of our results is the uncertainty in the geometry of the reed channel and of the lateral slits between the lay and the reed. Our result do confirm that a volume flux calculated from the Bernoulli law with a constant parameter vena contracta coefficient α a is reasonable first approximation for sufficiently large reed opening. For a small reed opening the effect of friction becomes significant and a correction for viscous effects should be introduced.

In the quasi-stationary basic model, the behaviour of the reed is reduced to a spring with a constant stiffness. Rather surprisingly, experimental results confirm that the latter hypothesis is reasonable, a stiffness value being associated for each embouchure adjustment.

Nevertheless for some of the embouchures, the hypothesis is correct up to a critical pressure threshold above which the equivalent stiffness is increasing when the reed-lay aperture is decreasing. Some authors proposed to consider a variable stiffness k(H) to take into account the curvature of the lay (see for example the time domain simulations of [START_REF] Ducasse | Modélisation d'instruments de musique pour la synthèse sonore: application aux instruments à vent[END_REF]. Our results show that this should be done with care. Indeed, a variation of the stiffness with the opening has been observed for some embouchures, but only when the reed is near closing.

Supplementary investigations have to be done with different reeds and mouthpieces in order to check the variability of the results. A comparison with the recent theoretical works of [START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF][START_REF] Van Walstijn | Discrete-Time Modelling of Brass and Reed Woodwind Instruments with Application to Musical Sound Synthesis[END_REF] should be done by using a reed and a lip of which characteristics would be measurable and controllable. Incidentally, our measurements confirm the visco-elastic behaviour of reeds reported by [START_REF] Marandas | Caractérisation mécanique des anches simples d'instruments à vent," 3ième Congrés Français d'Acoustique[END_REF]. It seems however that this behaviour is not relevant when predicting oscillations because the memory time scales involved are long compared to the oscillation period.

Finally, our study allows to conclude that the "simplistic" model described in section 2 is valid at least on the major part of the characteristics. This result confirms the practical interest of theoretical studies based on this model [START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Fletcher | Air flow and sound generation in musical wind instruments[END_REF][START_REF] Saneyoshi | Feedback oscillations in reed woodwind and brasswind instruments[END_REF][START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF][START_REF] Kergomard | Elementary considerations on red-instruments oscillations[END_REF][START_REF] Kergomard | Calculation of the spectrum of selfsustained oscillators using a variable truncation method[END_REF]Ollivier et al., 2002). Similar studies could be done for other instruments. For instance, a saxophone alto mouthpiece has been tested too : the same behaviour as for a clarinet mouthpiece has been observed both from the point of view of the air flow and the reed mechanics in stationary regime. This suggest that, except the size, there is no major difference between a clarinet and a saxophone mouthpiece. The case of the double reed of an oboe or a bassoon could also be investigated in the same way. This could allow to check if the hypothesis by [START_REF] Barjau | Calculation of the starting transients of a double-reed conical woodwind[END_REF] of a cross section proportional to the opening to the power two is sensible. The conjecture that there is a significant pressure recovery inside the narrow pipes on which double reeds are mounted [START_REF] Hirschberg | Aero-acoustics of wind instruments[END_REF] could also be checked.

Figure and table legends

FIG. 1 .

 1 FIG. 1. (a) Flow control by the clarinet reed involving free jet formation and turbulent

Figure 7 .

 7 Figure 7.b plots the reed opening as a function of the pressure drop. This curve shows that

FIG. 7 .

 7 FIG. 7. Typical experimental results (continuous line experiments ; doted lines theory) (a) Non-linear characteristics, volume flow U versus pressure difference P ∆ (b) reed opening H versus pressure difference P ∆ ,

Figure

  Figure8shows a result with a clarinet mouthpiece B40 by Vandoren, a Plasticover reed n°3 and a reed opening at rest mm H 6 . 0 0 = . In figure8.a a plot of the non-linear characteristics is

  measured volume flows through a reed channel formed by a stiff flat metal reed with sharp edges placed on an actual clarinet mouthpiece. It is shown that the flow threw lateral sides can increase the flux within 50% which lead to values for α compatible with our results.

FIG. 1 .

 1 FIG. 1. (a) Flow control by the clarinet reed involving free jet formation and turbulent

table 1

 1 From our measurements it appears that the range in which these parameter varies is rather small. In all the measurement realised it did not vary more than 50%. This parameter being deduced from 0 H , its accuracy is in the same range as 0 H , that is 10%. On the other hand the product of the two reed parameters 0 H and k is the beating pressure

								. The reed parameters, that is the opening at rest 0 H and the
	reed stiffness per area k, are determined using the optical device (see section 3.4 and 4.3). The
	reed opening is approximately	H	0 =	mm 1	when the reed is free, that is when the lips do not
	press the reed. When the lip presses the reed this value decreases but probably not much under
	H	0 =	0	.	4	mm	which correspond to a rather tight embouchure. Because of the small value of
	the reed opening and of the geometrical irregularities of the reed this parameter 0 H can not
	be determined with a great accuracy. The accuracy on this parameter 0 H is estimated to be
	only 10%. The reed stiffness which is deduced from	k	=	H	0	/	∆	P	is found to be in the range
	of 100mbar/mm.

deduced. In table 1 realistic values of the parameters of the model for the clarinet are

Table 1 :

 1 Summary of the parameters values of the parameters for the clarinet.

Acknowledgments

Authors are indebted to A. Hirschberg and C.J. Nederveen for helpful discussions.