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    08 a 11 de Outubro de 2019 Juiz de Fora - MG  Anais do XXII ENMC – Encontro Nacional de Modelagem Computacional e X ECTM – Encontro de Ciências e Tecnologia de Materiais.  Juiz de Fora, MG – 08 a 11 Outubro 2019   IMPEDANCE BORE LIKE DIDGERIDOO CALCULATION TOWARDS DESIGN AND TUNING Diomar Cesar Lobão 1    –   1 Universidade Federal Fluminense, EEIMVR – Volta Redonda, RJ, Brazil    Abstract.  In the present work is shown how it is possible to make design and tuning of a  simple wind instrument like a didgeridoo (Australian absrcinal wind instrument) using the impedance concept via numerical simulations of Webster equation. Such equation is used in order to model and calculate the key notes of a given geometry for a didgeridoo. The equation is then solved via finite difference approximation to achieve the impedance distribution through the bore like conic shape. The oscillations that represent a scale keys are well determined leading to a tuning processes as need to achieve a specific key. Such approach  show the close match involving music, instrument, mathematics and also numerical  simulation.  Keywords:  Webster equation, Didgeridoo, Finite difference, Impedance, Bore wind instrument. 1. INTRODUCTION  The main reference contribution used in this paper is the pioneer article written by Arthur Gordon Webster (1919) in exact hundred years ago which has motivate so many people and also developments in the building art of wind musical instruments. In his work he called attention to a more practical use of the term 'impedance' introduced by Mr. Oliver Heaviside (1850) which has been productive of very great convenience in the theory of alternating currents of electricity. Unfortunately, as he says, engineers have not seemed to notice that the idea may be made as useful in mechanics and acoustics as in electricity. Many others contributions can be added up as for example Fletcher et al. (1991), Wolfe (2012), and research thesis by Braden (2006). There is also an inspiration paper written by van den Doel and Ascher (2008) where they apply finite difference on non-uniform grid solving Webster equation. The didgeridoo is a lip-driven (Wood, PVC or other material) instrument. The player has the mouthpiece, which requires the use of a lip-reed mechanism in order to produce notes or keys where are sounded by interaction between the player lips vibrations and the resonances of the air column in the conic instrument bore. Inside the bore the air column may resonate at   XXII ENMC e X ECTM 08 a 11 de Outubro de 2019  Juiz de Fora - MG  Anais do XXII ENMC – Encontro Nacional de Modelagem Computacional e X ECTM – Encontro de Ciências e Tecnologia de Materiais.  Juiz de Fora, MG – 08 a 11 Outubro 2019   several different frequencies for any geometric configuration. The vibrations produced goes to resonances corresponding to the natural modes of vibration of the air column contained in the tube as discussed by Fletcher et al. (1991). The player, in order to produce a specific key, he sets his lips to vibrate at a choosen frequency and injects an oscillatory air flow into the bore, which will induces vibrations into the air column. Such vibrations will build up, causing the lips to react to the increasing pressure fuctuations and stabilize in a stable vibration into the air column, producing flowing resonantes waves. Once the resonances in the air column are achieved by the player the harmonic series can  be used to define a stable regime of oscillation. As can be observed the more harmonically generated peaks that support a fundamental note, the more stable and easy to play the sounded note will be. The player also can tight his lips to produce overtones and change the flowing ressonantes waves. For a given bore like didgeridoo geometry, length and area discretized sections, is  possible numerically integrate the Webster Impedance equation and find the keys distributions through the bore length. The following sections of the present work will deal with: Impedance governing equation, Finite difference approximation, Boundary conditions,  Numerical simulation and results. 2.   IMPEDANCE GOVERNING EQUATION  To calculate the impedance for a given bore geometry it must be defined. The input impedance, is usually denoted as Z, and also is a frequency dependent value defined as: vS  p Z      (1)   where  p = Acoustic Pressure, Pa v  = Velocity, m/s S = Surface area, m² The denominator vS u   is known as volume velocity  or acoustic volume flow . The unit of Z is in ./. 3 m s Pa  It is known as an Acoustic Ohm. The readings in general are taken at a point close to the mouthpiece see Fletcher et al. (1991). Acoustic impedance is important because measures the pressure level generated by some air vibration at a particular frequency. Its knowledge allows extract information about the behaviour via impedance plots. As will be observed in the following sections, the impedance peaks on plots correspond to resonances, the larger the magnitude naturally implying a stronger response. Generated large harmonically related peaks will sustain a stable flow regime of oscillation and result in also a stable tone  being produced. In the bore conic like didgeridoo geometry, the length in one coordinate scale is larger than the transversal coordinate, this is used to stablish the dynamics of the model to be reduced to one dimension. Once assumed this no transverse modes exist. The equation to be   XXII ENMC e X ECTM 08 a 11 de Outubro de 2019  Juiz de Fora - MG  Anais do XXII ENMC – Encontro Nacional de Modelagem Computacional e X ECTM – Encontro de Ciências e Tecnologia de Materiais.  Juiz de Fora, MG – 08 a 11 Outubro 2019   used is called Webster's equation as derived by Webster (1919) and also see Bilbao (2009). The derivation yields the following equation:   t S  xS  xS  Lat        25.02222       (2)  20025.022 2       t t S t    (3)   Where   =Velocity potential )(  xS  =bore cross section area lengthbore sound  speed  La  ,of ,      pressure p t        velocity volume  x S u        viscousthermal losses term 25.00 2      S a       ,  0 S  mouthpiece area,     density of air 55.0 0      damping coefficient,  Hz  100 0      fundamental frequency At this point is important to mention that assumptions used in the Webster equation derivation implies that the instrument bore do not present any curvature, the cross section must be circular and also the wave propagation is strictly planar. The viscous and thermal  boundary layer losses are well discussed by Keefe (1984) and also by Bilbao (2009). 3.   FINITE DIFFERENCE APPROXIMATION The continum second order derivative in time and space are approximated by central finite differences and the first order continum derivative are approximated by forward finite differences as well discussed by Bilbao (2009) and Bilbao (2011). The geral second order approximation and first order (for  ,   ) are written as:        21122 2 t t  ninini         (4)        21122 2  x x ninini         (5) t t  nini        1   (6) The discretised eq(2) and (3) are solved by Leap-frog explicit nodal scheme. With the  pressure calculated in time and space the fourier transform is taken in order to get its distribution in the frequency domain. The fft(  x ) routine computes the discrete Fourier transform (DFT) of  x  using a fast Fourier transform (FFT) algorithm.   XXII ENMC e X ECTM 08 a 11 de Outubro de 2019  Juiz de Fora - MG  Anais do XXII ENMC – Encontro Nacional de Modelagem Computacional e X ECTM – Encontro de Ciências e Tecnologia de Materiais.  Juiz de Fora, MG – 08 a 11 Outubro 2019   4.   BOUNDARY CONDITIONS The bore physical domain is measured from 0 to L, the space normalization scaling is applied scaling it from 0 to 1. The discret domain is indexed from i=1  upto i=N+1  nodes. The internal nodes are from i=2  upto i=N   where the calculation are actually done. The geometric domain stablished assure that the left side (moutpiece section) of the bore is treated as being closed and here is applied an excitation 6.0 1     i . On the right side of the bore is stablished the radiating sound. A typical Neumann boundary condition accounting for zero velocity is written as 0),0(   t   x   . At the right end the boundary condition used is written in the following form ),1(),1(),( t  N t  N t  N  t t t         . The      andterms are defined by Bilbao (2011) and Silva (2009). However, the scheme will need information for nodes i=1  and i=N+1 , this is acomplished by using reflection boundary condition as for example the viscousthermal losses term: 21    as well as  N  N      1 . 5.   NUMERICAL SIMULATION AND RESULTS The Courant number is found to be of critical importance for numerical stability. The relation known as the CFL criterion: (a/L.∆t)/∆x ≤ 1  after Courant, Friedichs and Lewy (1928), who first published the result. It implies that, if we refine the space grid, that is, decrease  ∆x , we must also shorten the time step  ∆t  . The Courant-Friedrichs-Lewy (CFL) condition says that the discrete solution must not be independent of data that determine the solution of the associated partial differential equation. The numerical simulation is done using a sample rate of 44100Hz which is usual for such experiment, Braden (2006).   The Table 1 show the bore geometry used in the present simulation. In the Figure 1 is shown the bore profile for the given geometry. Table 1- Bore Geometry Radius 1(m) left Radius 2(m) right Length (m) 0.030 0.030 0.020 0.030 0.025 0.020 0.025 0.020 0.160 0.020 0.025 0.20 0.025 0.035 0.20 0.035 0.045 0.20 0.045 0.040 0.100 0.040 0.050 0.100 0.050 0.050 0.200 0.050 0.050 0.180 0.050 0.050 0.020   XXII ENMC e X ECTM 08 a 11 de Outubro de 2019  Juiz de Fora - MG  Anais do XXII ENMC – Encontro Nacional de Modelagem Computacional e X ECTM – Encontro de Ciências e Tecnologia de Materiais.  Juiz de Fora, MG – 08 a 11 Outubro 2019   Figure 1- Bore profile showing the geometry. The following data is used in the present numerical simulation. Tuning Frequency A 4 = 440.000000 Hz Density at Volta Redonda, RJ, Brazil (Kg/m³) = 1.201000 Temperature = 22.000000 (Celcius), Sound Velocity = 344.695261 (m/s) Air Density = 1.15667e-003 (g/m³) Total Number of required time samples NF= 44100 Time step k= 0.000023 CFL number = 0.055830 The following two graphics in Figure 2 shows two impedance results, the first one is logaritmic the second is linear. Figure 2- Impedance. The first plot in Figure 2 shows the impedance harmonics distribution along the bore in dB (Decibel). This impedance distribution is expect for the bore closed on the left and open
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