Recent Advances in Fluid Mechanics and Heat & Mass Transfer

A Mathematical Model for a Willow

TABITA BOBINSKA ANDRIS BUIKIS Faculty of Physics and Mathematics Institute of Mathematics and Computer Science University of Latvia University of Latvia Zellu Street 8, LV1002 Riga Raina Blvd 29, LV1459 Riga LATVIA LATVIA [email protected] [email protected] http://ww3.lza.lv/scientists/buikis.htm

Abstract: - In the article we examine a mathematical model for a Norwegian flute. We find the possible frequencies of the sound produced by the flute, analyse how the pitches can be altered when changing the parameter h in the boundary condition at x = l, and determine the energy distribution in the sound.

Key-Words: - willow flute, harmonics, natural scale, wave equation, and energy distribution

1 Introduction end of the tube closed produces another harmonic The willow flute (Norwegian: seljefløyte, Finnish: scale. pitkähuilu or pajupilli, Swedish: sälgflöjt or sälgpipa) is a Scandinavian of the recorder family existing in two forms: an end-blown 2 Problem Formulation flute, often called a whistling flute, and a side-blown Playing the flute causes periodic oscillations of the flute. air pressure inside the instrument. These pressure disturbances are governed by the wave equation.

Fig.2: Definition of geometrical parameters for the flute Assuming for simplicity that the pressure over the cross-section of the tube is constant, the willow flute can be modelled by  2 p  2 p B Fig.1: A willow flute  c 2  xF )( , c 2  , (1) t 2 x 2  This paper focuses on the side-blown flute (see where txp ),( denotes the acoustic pressure, Fig.1, from [6]) which is between 40 and 80 centimetres in length. It consists of a tube with a function xF )( is some external source,  density transverse mouthpiece that is constructed by of air (  2041.1 mkg 3 ). Let’s suppose that there putting a wood plug with a groove in one end of the is no appreciable conduction of heat, therefore the tube, and cutting a sound hole (edged opening) a behaviour of sound waves is adiabatic. So, B is the short distance away from the plug (see Fig.2, from adiabatic bulk modulus of air, which is [7]). The air flow is directed through a passageway approximately given by  pB . Here  is the across the edge creating a sound. a As the flute has no finger holes, different pitches ratio of the specific heats of air at constant pressure are produced by overblowing and by using a finger and at constant volume (  4.1 ), and pa is the to cover, half-cover or uncover the hole at the far ambient pressure (  ,101 325 Pa). end of the tube. The seljefløyte plays tones in the Let’s establish boundary and initial conditions. harmonic series called the natural scale. When the At x  0 we have 3rd type boundary condition end of the tube is left open, the flute produces one   tptp  0),0(),0( . (2) fundamental and its , playing it with the x 1

ISBN: 978-1-61804-026-8 188 Recent Advances in Fluid Mechanics and Heat & Mass Transfer

The boundary condition at the far end of the tube To determine the functions n tT )( , we represent the depends upon whether the end is open, closed or source function and initial conditions as Fourier partially open. For an open end of a tube, the total series. So tT )( will satisfy the simple initial value pressure at the end and atmospheric pressure must n be approximately equal. In other words, the acoustic problem 22 pressure p is zero: n   nn )()(  FtTctT n , . (3) tlp  0),( Tn )0(  n , At a closed end there is an antinode of pressure, as T  )0(  . most of the sound is reflected: n n Hence, the solution of problem (1) is tlp  0),( . (4) x txp ),(  In many research papers and works where wind l  2 l instruments have been considered (see, e.g., [1] -    ),,()(),,()( dtxGFdtxG  [3]), only these two types of boundary conditions  2  0 t 0 are used. But one should bear in mind that the l  l player can also close the end of the tube only     )0,,()(),,()( dxGFdtxG  (7) halfway. In that case the Robin’s condition 0 t 0 x  2 tlptlp  0),(),( (5) with Green’s function  txG ),,(  must be applied (see [4]). The value of  2 depends on how much the end is closed.  cos  tc  XxX  )()( n nn . Conditions (3)-(5) can be written in the following  22 2 n1 c  X form n n

 x tlhptlph  0),(),()1( , h  10 . (6) But the initial conditions are 2.1 Geometrical parameters and functions   xxp )()0,( , In the paper we use the linear wave equation

t  xxp )()0,( . referring to the acoustic pressure. Actually the Nonhomogeneous equation (1) can be solved by process is much more complex, as there are many finding a solution expressed in the form things that should be taken into account. First of all, nonlinear equation could be used when dealing with   nn tTxXtxp )()(),( , n mass-transfer. Secondly, there is a question about hydrodynamic behaviour of the jet of air. Both where xX )( , n  ,3,2,1 are the eigenfunctions n  experimental and theoretical results suggest (see, of the associated homogeneous problem, having the e.g., [5]) that the airstream tends to form vortices following expressions: when impinging the edged opening. In the case of  one dimensional plane wave, we use a source sin)(  xxX   , tan  n , n nn n function to describe this process. Thirdly, the mass 1 flow in the passageway can be approximated by 3rd l  sin l  X 1 n l  2cos   . type boundary condition (2). n  n n  All results in the next two sections are obtained 2   nl  with the following parameter values: The corresponding eigenvalues  are roots of n R these transcendental equations: l  656.0 , l1  01.0 , H  , end open 5 H 2  n   p , p  324,101 , tan l  ; 11 2 1 n  R  1 and the initial conditions: end closed  x  0)( ,  n cot nl  ;  0 0  lx 1 1   lx 1  ll 1 end partially open  x)( 1 1 lxl 1   ,   ,   d 1    0 1   lxl  21  .  cot nl    n     21   n  where d is a positive constant, and the source function:

ISBN: 978-1-61804-026-8 189 Recent Advances in Fluid Mechanics and Heat & Mass Transfer

  xl defined as 440 Hz. Many people should be capable xF )( 1  . 100 l of detecting the difference if it is as little as 2 Hz. Willow Note Piano flute 3 Pitch and Frequency C4 261.63 261.63 One of the fundamental properties of sound is its C5 523.25 523.27 pitch. Pitch is a subjective attribute of sound related G5 783.99 784.90 to the frequency of a sound wave. Increasing the C6 1046.50 1046.53 frequency causes a rise in pitch. But when E6 1318.51 1308.16 decreasing the frequency, the pitch of the note G6 1567.98 1569.80 diminishes. B♭6 1864.66 1831.43 The tone of the lowest frequency is known as the C7 2093.00 2093.06 fundamental or first harmonic. All other possible Table 1: Frequenc ies (Hz) of wi llow flute’s open-end C pitches whose frequencies are integer multiples of scale compared with frequencies of 12-TET scale the fundamental frequency are called the upper th harmonics or overtones. The frequency of the m key in 12-TET can be Using relation (7), we obtain the frequencies of found from the equation m49 the vibration mode of the willow pipe: 12 f m  2440 . a n f n  , n  ,3,2,1  2

If the parameter 1 is sufficiently large, the possible 3.2 Closed-end scale frequencies for the flute are close to harmonics: Closing the end drops the fundamental an octave an below the pitch of the pipe open at the end. The next f n  , n  ,3,2,1  (end open) 2l possible note G4 has approximately three times the an frequency of the fundamental C3, the next one, E5, f n  , n  ,5,3,1  (end closed) five times, and so on. This means that only the odd 4l harmonics are present. As  n is dependent on l , we can change the pitch by ch anging the length of the tube. If the pipe has finger holes, one can also change the effective length of the instrument by opening different holes. By altering the pressure of the air blown into the Fig.4: Notes of closed-end C scale instrument, we change the value of n , that is, we Table 2 shows that the willow flute’s pitches do not jump between solutions , resulting in discrete pn quite conform to those produced by the piano. differences in pitch. The pitch can also be changed Willow by modifying h in the boundary condition at  lx . Note Piano flute C3 130.81 130.82 3.1 Open-end scale G4 392.00 392.45 For the given parameters (see Section 2) the E5 659.26 654.08 fundamental for the open flute is Middle C or C4. B♭5 932.33 915.72 The second harmonic is then an octave above the D6 1174.66 1177.35 fundamental; the third harmonic is G5, and so on. F♯6 1479.98 1438.98 G♯6 1661.22 1700.61 B6 1975.53 1962.25 Table 2: Frequenc ies (Hz) of wil low flute’s c losed-end C scale compared with frequencies of 12-TET scale Fig.3: Notes of open-end C scale One can notice (see Table 1) that some of these 3.3 Intermediate scale frequencies differ from the frequencies of the notes We can get further effects, when covering the end on a piano in twelve-tone equal temperament (12- th hole only halfway (this implies that we should TET) with the 49 key, the fifth A (called A4), change the parameter h in the boundary condition

ISBN: 978-1-61804-026-8 190 Recent Advances in Fluid Mechanics and Heat & Mass Transfer

(6)). In this case the flute produces an intermediate set of pitches that fall in between those produced with the end closed and those when the end is open.

Fig.5: Approximate location of the willow flute’s pitches on a piano. C scale Pitches produced with end closed Pitches produced with end open Pitches produced with end partially open In Fig.6 you can see how the parameter h affects location of pitches played by willow flute when leaving the end partially open.

Fig.6: Approximate location of pitches when h  5.0 ; h  7.0 ; h  9.0

4 Energy of Tones As the sound of the willow flute playing is relatively harmonic, the energy of the sound is concentrated at certain frequencies of vibration. The more dominant the certain pitch, the greater the concentration of energy at that frequency. A person playing this particular instrument can get different pitches through manipulation of the supplied air, which is, changing initial conditions. If the initial conditions are defined as in Section 2, you can modify the distribution of energy between the fundamental and its overtones by changing  . Fig.7: Energy distribution (J) across the frequencies You must blow as softly as possible to produce when   l ;   l 3 ;   l 6 the fundamental, as it is hard to get. Blow a little harder and you get the first and so on. The harder you blow the higher harmonics you get to be 5 Conclusion dominant (see Fig.7 for open-end flute). We have examined how different boundary Formula for calculating the energy distribution is conditions affect the frequencies of pitches 1 l tl produced by the willow flute. And we have also E( )   2 ),( dxtxpt   Bpd 2  ),(),( dxxpx . 2  t  x t determined the distribution of energy of tones. 0 00 Although linear differential equation (1) could be considered as an adequate

ISBN: 978-1-61804-026-8 191 Recent Advances in Fluid Mechanics and Heat & Mass Transfer

approximation when describing acoustic [2] R. W. Hall and K. Josić, The Mathematics of properties of the willow pipe, it is necessary to take Musical Instruments, American Mathematical into account many other things, e.g., nonlinear Monthly, Vol.108, No.4, 2001, pp. 347-357. effects. But that is a subject for future research. Available from Internet: http://www.sju.edu/~rhall/newton/article.pdf [3] T. D. Rossing (Ed.), Springer Handbook of Acknowledgments: Acoustics, Springer Science+Business Media, LLC New York, 2007. [4] L. Simanoviča, Boundary Conditions Effect on Musical Instruments Timbre, Faculty of Physics and Mathematics, University of Latvia, This work is partially supported by the project Bachelor’s Thesis, 2010 (in Latvian). 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008 and [5] P. A. Skordos, Modeling flue pipes: subsonic the project “Support for Doctoral Studies at flow, lattice Boltzmann, and parallel University of Latvia" of the European Social Fund, distributed computers, Department of Electrical and by the grant 09.1572 of the Latvian Council of Engineering and Computer Science, MIT, PhD. Science. Dissertation, January 1995. Available from Internet: http://hdl.handle.net/1721.1/36534 [6] http://farm1.static.flickr.com/132/322465679_c References: d82a91398.jpg [1] N. H. Fletcher, T. D. Rossing, The Physics of [7] http://www.hf.ntnu.no/rff/organisasjonar/skrifte Musical Instruments, Second Edition, Springer r/vol1/html/olakailedang.html Science+Business Media, Inc., 2010.

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