# A Physical Model of Recorder-Like Instruments

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Page 37 ï~~A PHYSICAL MODEL OF RECORDER-LIKE INSTRUMENTS M.P. Verget R. Causset A. Hirschberg* IRCAMt, 1 Place Igor Stravinsky, 75004, Paris, France. Eindhoven University of Technology*, W & S, Postbus 513, 5600 MB, Eindhoven, The Netherlands. verge@ircam.fr, causse@ircam.fr ABSTRACT: In this paper, a simple one-dimensional representation of recorder-like instruments for sound synthesis by physical modeling is presented. The jet oscillation model used is a modification of the semi-empirical model by Fletcher (1976b). The steady-state drive of the acoustical oscillations in the pipe by the jet motion is represented by a pressure jump in the mouth of the instrument. Vortex shedding at the edge of the labium during steady-state operation is taken into account by the use of a free jet model. The combined effects of this non-linearity and the jet-drive model enable one to predict correctly the steady-state amplitude of the fundamental. The transient response of the model is triggered by the initial volume injection into the mouth of the instrument and is dependent on the steepness of the driving pressure rise. Introduction The objective of this paper is to present a simple model of recorder-like instruments that could be used for sound synthesis by physical modeling. The model is based on a description of the flow in the mouth of the instrument. It was developed from extensive sets of measurements and flow visualizations performed on a small experimental flue organ pipe (Verge et al., 1994a; Verge et al., 1994b). Our main concern in this work was to develop a model that could reproduce the main features of measurements with as little arbitrary parameter adjustment as possible. The basic model The simple one-dimensional model shown in Figure 1 is used to simulate sound production by recorder-like instruments. This simple low-frequency approximation representation was obtained by "unfolding" the idealized two-dimensional geometry of a recorder-like instrument shown in Figure 2. The resonator is represented by a larger tube of cross section Sp = H2 and length Lv + 6p where 6b is an end correction associated with sound radiation. Because the mouth region is small compared to the wavelengths considered, the flow in this region can be assumed to be incompressible. The complicated two-dimensional geometry of the mouth of a recorder can therefore be represented by an equivalent end correction of length 6m and cross section Sm,= WH. In the one-dimensional representation of the instrument the flue exit is located a distance 6ou from the "outside world" and a distance 6i, from the entrance of the resonator (cm = 6in + bout ). The details of the determination of these parameters have been discussed by Verge et al. (1994b). In the same paper, the flue exit was shown to be a volume source that acts as a sound source when the jet velocity Uj fluctuates. This effect, combined with the initial interaction of the jet with the labium, provides the signal which triggers the attack transient. During steady-state operation, the jet is submitted, at the flue exit, to the transverse acoustic ICMC PROCEEDINGS 1995 3 37

Page 38 ï~~dm Lp + 6p --------'-: Ap h 1* Qi W Q 1Huly $ x=O x Figure 1: Simple one-dimensional model used to Figure 2: Two-dimensional representation of simulate sound production by recorder-like instru- recorder-like instruments. ments. field Qmn which controls the oscillating motion of the jet in the mouth of the instrument. At the labium, this movement results in a complex flow which can be lumped, as suggested by Powell (1961), Elder (1973) and Coltman (1976),into two complementary flow sources Q1 and Q2 such as shown in Figure 2. As discussed by Verge et al. (1994a), these two sources induce a pressure difference across the mouth of the instrument which constitutes the jet-drive mechanism for the acoustic oscillations in the pipe. In a one-dimensional representation, the complementary sources can be represented by a dipole placed in the end correction Sm. The distance between the sources, determining the strength of the dipole, can be calculated from the two-dimensional geometry of the instrument, in the potential flow approximation, by using conformal mapping techniques (Verge et al., 1994a). Such a dipole induces a fluctuating pressure jump Ap across the pipe segment representing the mouth of the instrument such as shown in Figure 1. The model is driven by a pressure signal pf that represents the pressure in the foot of a flue organ pipe or in the mouth of a musician. A relationship between the pressure drop pf- Pm across the flue canal and the resulting jet velocity Uj at the flue exit is given by the Bernoulli equation (Paterson, 1983) dUj 12 POlC d-y+.2PoU -p1 pm, (1) where Po is the air density and l is the length of the flue canal. At the flue exit, the mass conservation law can be applied Qj + Qout = Qin. (2) At the flue exit (x = -Sin), the pressure pn is determined by the value of the radiation impedance Zout. In a low frequency approximation (with the sign convention of Figure 1) (Rayleigh, 1894) Pm -Poco 1w +m Â~+ i b,(3) _UT n C (3) Qo Sm 4 \CoJ where w is the angular frequency, rm the radius of a circle having the same cross section as that of the mouth (irr2m - Sm) and co is the speed of sound. By inverse Laplace transform, a time-domain relationship between the pressure Pm at the flue exit and the flow Qot is obtained POCO( 1 r d2Q out bout dQO it(4) Pm-Sm 4\j d12 -Co dt " The pressure Pp at the entrance of the resonator can be related to the pressure Pm at the flue exit by applying the Bernoulli equation Pp~~~m- PO i dQi +Li P, 5 pp p - Sm di C 38 8ICMC P ROCE E D I N GS 1995

Page 39 ï~~where continuity of pressure at the entrance of the resonator (x = 0) was assumed and where the jet-drive mechanism, associated with the motion of the jet at the labium, was represented by a pressure jump tpjd across the end correction bin. At the interface between the mouth of the instrument and the pipe, flow continuity is also assumed Qp(O)= Qi,, (6) where Qp is the flow in the resonator. The amplitude of the pressure jump L pjd associated with the jet-drive is determined by the time derivative of the amplitude of the flow source Q1 corresponding to the portion of the jet flow entering the pipe at the labium (Verge et al., 1994a). Po6d dQ1 APd- m dt ' (7) where 6d is the "distance" between the sources Q1 and Q2 in the one-dimensional representation of the instrument. For our experimental pipe, this distance 6d was estimated, in a potential flow approximation, to be of the order of 3.5 mm (Verge et al., 1994a). The flow Q1 is calculated, as suggested by Cremer & Ising (Cremer and Ising, 68), from the estimated jet position q(W, t) at the labium Q1 = H Uj(y)dy, (8) where H is the jet width, yo the labium position with respect to the flue exit axis. By assuming, as proposed by Fletcher & Thwaites (1979), the jet to have a Bickley velocity profile (U,(y) = Uosech2(y/b)) the following expression is obtained Qi =bHUÂ~ (1+tanh (-YO)b)b '(9) where U0 is the jet central velocity and b the jet width parameter which determines the jet velocity profile and the stability properties of the jet. Equations 1, 2, 4, 5, 6, 7, and 8 constitute our basic recorder model. The details of the implementation of the different elements of this model are now discussed. The jet oscillation model The problem of the jet oscillation is very complex. There is, at the present time, no available physical model that enables an accurate description of the jet motion and the determination of the transverse position 77(W, t) of the jet at the labium. Verge et al. (1994a) have proposed to use a modification of the semi-empirical formula by Fletcher & Thwaites (1979) Ww_1 '2 Qp 0.38Q \r1- cosh We -i ) i X(4sW)exp,(10) W- - Sm 1 (iW) where W is the distance between the flue exit and the edge of the labium, p(Stb) is an amplification coefficient, u(Sib) the propagation velocity of the hydrodynamic perturbation on the jet, Stb = wb/U, is the Strouhal number and where the fluctuating flow Qi is defined as follows = Qi - l bHu0. (11) The first term in the parenthesis of equation 10 represents the transverse acoustic velocity v$ at the flue exit. This velocity is the sum of a contribution due to the acoustic field from the pipe and a direct hydrodynamic feedback from the edge of the labium which may account for an I C M C P ROC EE D I N G S 1995 3 39

Page 40 ï~~edgetone behavior of the instrument (Verge et al., 1994a). The hyperbolic cosine in the second term of equation 10 represents the amplification of the perturbation by the jet and the complex exponential the phase delay due to its convection towards the labium. The effect of these two terms depend strongly on the value of the parameters,(Stb) and u(Stb) and we propose to use the data, based on a spatial stability analysis, calculated by Mattingly and Criminale (1971). Since the value of these parameters used in the simulations depend on the Strouhal number Stb based on the jet width parameter b, the estimation of this latter parameter is crucial. Calculations based on the conservation of the jet momentum flux, show that this parameter should be equal to 0.4h for a Poiseuille flow and to 0.75h for a uniform velocity profile (Verge et al., 1994a) at the flue exit. Since the jet velocity profile depends on the convergence of the flue canal, this parameter should be characteristic of a given instrument. Note that equation 10 displays an unrealistic behavior at zero frequency because of the factor 1/iw. This model can be simplified by modifying equation 10 as follows 1(2(Q-.38Q1 _____ 7(x, w,U)-i- S S )[1-cosh(IW)] exp UW, (12) which appears to be a band-pass filter (when removing the singularity at zero frequency) having a frequency response that depends on the jet velocity. This function enables to treat the amplification and the delay effects separately and fits the same boundary condition as equation 10. In a first approximation, the propagation velocity u of the perturbation on the jet can be assumed to depend on the jet velocity only which allows to represent this effect by a simple delay line. An order three Butterworth band-pass filter enables one to obtain a reasonable fit of the frequency response of equation 12. The total phase shift introduced by such a filter and a delay line of length 0.6U1 /W appears to give a good fit of the phase response of equation 12 and of experimental data by Coltman (Verge et al., 1994a; Verge, 1995) for Strouhal numbers Stw = fW/UU > 0.15. The coefficients of the filter must be chosen so that the frequency of the maximum of the response coincides with that of the jet model response for a given jet velocity. The transfer function of the filter can then be multiplied by a constant in order to match the amplitude. In order to obtain a filter model that can be used for different jet velocities, the steps just described must be repeated for the whole range of jet velocities that will be used during the simulations. Different sets of data must also be calculated for different flue exit to labium distance W and width parameter b. In the simulations which will be presented, filter coefficients were calculated for a set of jet velocities and stored in a look-up table. The resonator The well-known D'Alembert's solution of the wave equation enables one to represent the resulting pressure signal pp as the sum of two traveling-waves po and pi p= pv(x - cot) + pj(x + cot). (13) For an observer placed at the entrance of the pipe, the incoming wave pi is equal to the outgoing wave po delayed by the time of a round-trip in the resonator and filtered by the effects of the visco-thermal losses and sound radiation at the end of the resonator. Using Euler's equation and this expression, the flow Q, in the pipe can be expressed as =p _ p(o-P) (14) Poco An efficient implementation method of the wave-equation consists in the use of so-called digital waveguides (Smith, 1992). This representation is based on the traveling-wave solution (13) of the 40 0IC MC PROCEEDINGS 1995

Page 41 ï~~wave equation and can be implemented with the use of delay lines. The effects of visco-thermal and radiation losses can be efficiently implemented by filtering the pressure signals at the end of the delay lines. Vortex shedding at the labium Fabre et al. (1995) have shown that during steady-state operation of the instrument, the major non-linear amplitude limiting mechanism is due to vortex shedding at the labium. This effect can be taken into account by assuming flow separation of the acoustic flow Qp at the labium and the formation of a jet in the mouth of the instrument. In this quasi-stationary free jet model, proposed by Ingard and Ising (Ingard and Ising, 1967), the effects of vortices are represented by a fluctuating pressure jump/Apa across the mouth of the instrument -1 / 2_ Apa=---Po y Q)sign(Qp), (15) where av is the vena-contracta factor of the jet which depending on the geometry can vary within the range 0.6 < at < 1. We used later a value a, = 0.6. This pressure drop opposes the acoustical flow in the mouth and corresponds to the dissipation of the kinetic energy of the free jet by turbulence. This crude model should constitute an upper limit approximation of the effects of this vortex shedding. Energy losses Ea induced by this model scale with the third power of the acoustic flow (Fabre et al., 1995). The initial interaction of the jet with the labium plays a crucial role during the attack transient (Verge et al., 1994b). When the jet hits the labium, a vortex which is shed which appears to be important in the triggering of the attack transient and the generation of high frequencies during the transient. In order to determine the jet position during its formation, we propose to represent the flue exit as a point source Qy(t) flowing into a semi-infinite space. The forming jet therefore appears as a growing half-circle. The vertical position of this half-circle is determined by the ratio of the flow Qou and Qin in the mouth. When the half-circle hits the labium, a pressure pulse/Apt, having an amplitude proportional to the square of the jet front velocity Up. is sent in the pipe. The proportionality constant has been determined by fitting experimental data. Time domain implementation of the model Equations 1, 2, 4, 5, 6, 7, 9, 12, 13 and 14 constitute the complete recorder model. The driving pressure signal p! is considered to be a known parameter as well as the incoming pressure wave pi (0) at the entrance of the resonator since it can be determined from past values of the outgoing signal p,,(0). We therefore have a set of ten equations with ten unknowns which must be solved simultaneously. By using equations 2, 4, 5 and 6, a relationship between the pressure pp and the flow Qp at the entrance of the resonator can be obtained. Then by using equations 13 and 14, an expression with a single unknown is obtained r SP a2(po - pi) _porm d2Q _ 6,mSp 'a(po- p) PO6O d(16Q) po 4C2Sm 8t2 4coS, dt2 cos t S dt -p. (16) The outgoing wave po is determined by the effects of inertia in the mouth of the pipe, feedback from the resonator, the pressure source in the mouth of the instrument and the jet velocity fluctuations. During steady-state operation, the pressure source/Ap is the main driving mechanism, while during the attack transient, the sound production is dominated by the effects of the jet velocity fluctuations. This equation can be solved numerically by using finite differences schemes. Note that the different sound producing mechanisms at the edge of the labium have been included in IC M C PRO C EE D I N G S 1995 4 41

Page 42 ï~~the pressure jump Ap /p = Apjd +/APa +/Apt,. (17) In addition, a turbulence noise source Apt, scaling with the square of the jet velocity, can also be included in this term (Verge, 1995). Simulation results The recorder model presented in the previous section has been programmed in C language. An optimized version of the code has been implemented on an SGI Indigo workstation for real-time sound synthesis. The numerical version of the model was validated by checking that the program could reproduce results obtained in the linear analysis of smooth transients presented by Verge et al. (1994a). Figure 3 shows a comparison between a transient response of the experimental flue pipe described by Verge et al. (1994a) and a simulated signal. The measured driving pressure signal is presented in Figure 3a and was used as the input signal of the simulation program. The response of the model is triggered by the initial volume injection into the mouth of the instrument resulting into the pressure pulse appearing at approximately 5 ms. The model is sensitive to the steepness of the driving pressure rise since the amplitude of this pulse depends on the time derivative of the jet flow Qj. This enables one to simulate the differences between a fast and a slow attack (Verge et al., 1994b). In this simulation, the half width parameter b of the jet was set to a value of 0.35h. Transients obtained with a smoother driving pressure rise could also be matched but with a value of the parameter b of 0.39h. The steady-state amplitude of the pressure signal measured under the labium in a recorder is close to the value of the steady-state driving pressure. This feature is observed in the simulated signal presented in Figure 4a. It is interesting to evaluate the influence of the distance 6d and the non-linearity associated with the flow separation at the labium on the playing amplitude since they are the main features that distinguish our model from the widely accepted model by Fletcher (1976a; 1976b). In Fletcher's model, the flow separation at the labium is ignored. Furthermore, only the flow source Qi at the labium is considered. This is equivalent to considering that the acoustic oscillations in the pipe are driven by a monopole rather than by a dipole. In the onedimensional representation of Figure 1 this can be obtained by using a distance 3d equal to the total end correction 6m. This-brings the source Q2 "outside" the instrument and prevents it from performing acoustic work. This modification of the original model increases, such as can be observed in Figure 4b, the playing amplitude by more than a factor four. Taking into account the vortex shedding decreases, such as shown in Figure 4c, the amplitude by a factor two. Conclusion The simple model presented in this paper appears to grasp some basic features of recorder-like instruments. In particular, the model predicts the correct playing amplitude which is, in view of the existing literature, a significant step forwards. This appears to be the result of the combination of the use of a "dipolar" jet-drive mechanism and of a simple quasi-stationary model of the separation of the acoustic flow at the edge of the labium. Furthermore, the response of the model is triggered by its reaction to the driving pressure signal. This may seem straightforward, but models, such as that proposed by McIntyre et al. (1983), need to be triggered by an arbitrary additional impulse. The main weakness of this recorder model is its sensitivity to the value of the width parameter 6 of the jet. This parameter determines the velocity profile of the jet and hence its stability properties. In our simulations, this parameter was used for "fine-tuning". 42 2IC MC PROCEEDINGS 1995

Page 43 ï~~a) Pf (Pa) b) Pp (Pa) c) Pp (Pa) time (s) -100 -so-150 -200 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 time (s) ISO so -50O 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 time (s) a) Pp (Pa) b) Pp (Pa) C) Pp (Pa) 300 200 l00 0 -100 -200 -300 0 0.01 0.02 0.03 0.04 0.05 0.06 time (s) so -1500 0 0.01 0.02 0.03 0.04 005 0.06 time (s) 0.02 0.03 0.04 time (s) Figure 3: a) Driving pressure rise measured in the foot of the experimental flue pipe during the attack transient; b) corresponding response of the instrument measured under the labium; c) simulated response at the entrance of the pipe using the driving pressure rise shown in a). In the measurements, the distance W was set to 4 mm and the transverse position of yo of the labium was adjusted in the middle of the flue exit. In the simulations, the distance 5d = 3.5 mm and the width parameter b = 0.35h. Figure 4: Simulated signal obtained by using a steady-state driving pressure pi = 270 Pa. a) response obtained with the model given by equation 16; b) response obtained when neglecting the vortex shedding at the labium and using a distance 6d = 6m and c) response obtained when taking into account vortex shedding at the labium and using a distance 6d = ICMC PROCEEDINGS 1995 4 43

Page 44 ï~~References Coltman, J. W. (1976). Jet drive mechanism in edge tones and organ pipes. J. Acoust. Soc. Am., 60(3):725-733. Cremer, L. and Ising, H. (1967-68). Die selbsterregten Schwingungen von Orgelpfeifen. Acustica, 19:143-153. Elder, S. A. (1973). On the mechanism of sound production in organ pipes. J. Acoust Soc. Am., 54:1554-1564. Fabre, B., Hirschberg, A., and Wijnands, A. P. J. (1995). Vortex shedding in steady oscillations of a flue organ pipe. submitted for publication in Acta Acustica. Fletcher, N. H. (1976a). Jet-drive mechanism in organ pipes. J. Acoust. Soc. Am., 60(2):481-483. Fletcher, N. H. (1976b). Sound production by organ flue pipes. J. Acoust. Soc. Am., 60(4):926-936. Fletcher, N. H. and Thwaites, S. (1979). Wave propagation on a perturbed jet. Acustica, 42:323 -334. Ingard, U. and Ising, H. (1967). Acoustic nonlinearity of an orifice. J. Acoust. Soc. Am., 42(1):6-17. Mattingly, G. E. and Criminale, W. O. (1971). Disturbance characteristics in a plane jet. The Physics of Fluids, 14(11):2258-2264. McIntyre, M. E., Schumacher, R. T., and Woodhouse, J. (1983). On the oscillations of musical instruments. J. Acoust. Soc. Am., 74(5):1325-1345. Paterson, A. R. (1983). A First Course in Fluid Dynamics. Cambridge University Press, Cambridge, UK. Powell, A. (1961). On the edgetone. J. Acoust. Soc. Am., 33(4):395-409. Rayleigh, L. (1894). The Theory of Sound. Dover, reprint (1945), New-York. Smith, J. (1992). Physical modeling using digital waveguides. Computer Music J., 16(4):75-87. Verge, M. (1995). Aeroacoustics of Confined jets, with Applications to the Physical Modeling of Recorder-Like Instruments. PhD thesis, Eindhoven University of Technology, The Netherlands. Verge, M. P., Causs6, R., Fabre, B., Hirschberg, A., Wijnands, A. P. J., and van Steenbergen, A. (1994a). Jet oscillations and jet drive in recorder-like instruments. Acta Acustica, 2:403-419. Verge, M. P., Fabre, B., Mahu, W. E. A., Hirschberg, A., van Hassel, R., Wijnands, A. P. J., de Vries, J. J., and Hogendoorn, C. J. (1994b). Jet formation and jet velocity fluctuations in a flue organ pipe. J. Acoust. Soc. Am., 95(2):1119-1132. 44 I4 CMC PROCEEDINGS 1995