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A Mathematical Model for a Willow Flute

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A Mathematical Model for a Willow Flute Recent Advances in Fluid Mechanics and Heat & Mass Transfer A Mathematical Model for a Willow Flute 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 folk instrument 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 F() x , c 2 , (1) t 2 x 2 This paper focuses on the side-blown flute (see where p(,) x t 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 F() x is some external source, density transverse fipple mouthpiece that is constructed by of air ( 1.2041 kg m 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 B p . 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 p(0, t ) p (0, t ) 0 . (2) fundamental and its overtones, 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 Tn () t , 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 T() t will satisfy the simple initial value pressure at the end and atmospheric pressure must n be approximately equal. In other words, the acoustic problem 2 2 pressure p is zero: Tn()() t c n T n t Fn , . (3) p( l , t ) 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 p( l , t ) 0 . (4) x p(,) x t In many research papers and works where wind l 2 l instruments have been considered (see, e.g., [1] - ()(,,)()(,,) G x t d F G x t d [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 () G (,,) x t d F ()(,,0) G x d (7) halfway. In that case the Robin’s condition 0 t 0 px ( l , t ) 2 p ( l , t ) 0 (5) with Green’s function G(,,) x t must be applied (see [4]). The value of 2 depends on how much the end is closed. cosc n t Xn()() x X n . Conditions (3)-(5) can be written in the following 2 2 2 n1 c X form n n (1 h ) px ( l , t ) hp ( l , t ) 0 , 0h 1. (6) But the initial conditions are 2.1 Geometrical parameters and functions p( x ,0) ( x ) , In the paper we use the linear wave equation pt ( x ,0) ( x ) . 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 p(,)()() x t Xn x T n t , n mass-transfer. Secondly, there is a question about hydrodynamic behaviour of the jet of air. Both where X() x , n 1,2,3, 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 X( x ) sin x , tan n , n n n 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 cosl 2 . 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 0.656 , l1 0.01, H , end open 5 H 2 n p , p 101,324 , tan l ; 1 1 2 1 n R 1 and the initial conditions: end closed (x ) 0 , n cot nl ; 0 0 x l1 1 x l1 l l1 end partially open ()x 1 l1 x l1 , , d 1 0 l1 x l 1 2 . cot nl n 1 2 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 willow 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 m49 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.
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