Tuning the pitch of a wine glass by playing with the liquid inside

Transcription

1 IOP PUBLISHING Eur. J. Phys. 29 (2008) EUROPEAN JOURNAL OF PHYSICS doi: / /29/2/011 Tuning the pitch of a wine glass by playing with the liquid inside Matthieu Courtois, Boris Guirao and Emmanuel Fort Physique Expérimentale, UFR de Physique, Université Paris Diderot, BP 7021, Paris, France emmanuel.fort@espci.fr Received 21 August 2007, in final form 20 December 2007 Published 28 January 2008 Online at stacks.iop.org/ejp/29/303 Abstract It is well known that the pitch of the sound produced by an excited glass shell can be tuned by adding some liquid in it. In this paper, it will be proved that the distribution of the liquid inside the shell plays a crucial role in this frequency shift. Thus it provides another way to tune the pitch of the sound by modifying the liquid distribution inside the glass. Both adding a cylinder in the liquid or rotating it results in a pitch lowering. A simple model based on energy conservation is in good agreement with the measured experimental results. This paper assumes some basic knowledge about mechanics and hydrodynamics. It addresses instructors of physics at the undergraduate and advanced secondary school level as well as their students. (Some figures in this article are in colour only in the electronic version) 1. Introduction Musical glasses have been appreciated for centuries because of the ethereal sound they produce. These pure tones are commonly produced by rubbing a moistened finger around the rim of a wine glass. The pitch of the sound can be tuned by adding some liquid inside the glass. It can be easily checked that pouring some water into the glass lowers the pitch of the note. This technique is commonly used in musical glass instruments. For instance, Seraphim, also called glass harp, is based on the use of several glasses filled with different water levels tuned to produce harmonic notes. This phenomenon can be explained by simple energy arguments. Basically, the liquid contribution to the total energy of the glass liquid system is limited to its kinetic energy component. The vibration of the liquid increases the inertial factor. The total potential energy, associated with the elastic energy of the glass shell, remains unchanged. Consequently, the frequency of the emitted sound decreases as the liquid quantity increases [1 3] /08/ $30.00 c 2008 IOP Publishing Ltd Printed in the UK 303

2 304 M Courtois et al A more detailed analysis of the liquid contribution to the pitch lowering shows that it depends on the distribution of the liquid within the glass shell. Since the amplitude of the vibration is not constant within the liquid, two elementary volumes of liquid will contribute differently to the pitch lowering depending on their respective position within the shell. For instance, the part of the liquid in the direct vicinity of the glass rim, where the larger shell vibrations take place, contributes more significantly to the pitch lowering than the part of the liquid situated at the centre of the shell. One can get a qualitative idea of the radial amplitude profile of the liquid vibrations by observing the amplitude of the ripples on the liquid surface (see for instance [4]). In this paper, the spatial influence of the liquid contribution to the pitch of the sound will be investigated. It will also be proved that the opportunity to tune this pitch with an identical amount of liquid by changing the liquid distribution inside the glass is offered. Two alternative experiments had been performed to change the sound frequency of the glass without modifying the liquid volume. In the first experiment, the liquid distribution had been changed using a stirring rod to rotate the liquid. In the second experiment, a cylindrical rod was immersed inside the liquid to change its distribution inside the shell. In both cases, it was pointed out that models based on energy conservation can successfully explain the observed frequency shifts. The following section is devoted to this energy model. Then, details of the experimental setup are given in section 3. Finally, experimental results are discussed in section 4. The experiments presented in this paper have been performed during an undergraduate experimental course in which students were asked to build an experiment from scratch. This paper thus addresses mainly teachers of physics at the undergraduate or at an advanced secondary school level as well as their students. 2. Theoretical developments In this section, we give simple theoretical considerations meant to clarify the role of the liquid in the pitch lowering. This theory dates back to Lord Rayleigh [5] and has been applied successfully to filled wine glasses [1 3] as well as other problems involving filled or immersed tanks [6 8]. The vibration of the glass shell is treated thanks to the use of Rayleigh s classical theory of sound [5]. The glass shell is assumed to be cylindrical with a radius R, a height H and a thickness a. We take for granted that the shell remains unstretched, i.e. extension is negligible compared with bending. This hypothesis is clearly applicable in the case of crystal glass. Let us consider the displacement (δr, δθ, δz) of a point of cylindrical coordinates (r,θ,z). Using the unstretched hypothesis, we have the following equations for the radial and azimuthal displacements for the quadrupolar (2, 0) eigenmode, which is the mode excited in the musical glasses [4], { δr = Af (z) cos 2θ cos ωt Rδθ = (A/2)f (z) sin 2θ cos ωt, where A and f(z) represent, respectively, the vibration amplitude and the amplitude profile of the quadrupolar mode along the glass, ω is the angular frequency of the vibration. After integration over the entire shell, the following expression for the shell kinetic energy [2] had been obtained 1 : K shell = Tk shell ω 2 where Tk shell = 5π 8 ρ gara 2 sin 2 ωt ρ g is the density of the glass. 1 Coefficient π is missing in [2], equation (5) due probably to a typo. H 0 [f(z)] 2 dz, (1)

3 Tuning the pitch of a wine glass by playing with the liquid inside 305 The elastic energy of the glass is given by the integration over the entire shell of the energy of flexure. For the quadrupolar mode, a very good approximation of the following expression for the potential energy of the shell [2] had been found: [ U shell = 3πYa3 A 2 cos 2 ωt 1+ 4 ( ) ] R 4 H [f(z)] 2 dz, (2) 8R 3 3 H 0 where Y is the crystal Young s modulus. The expression describing the potential energy is composed of two terms which are respectively associated with radial and transversal displacements of the shell. The contribution of the liquid to the total energy stored in the glass fluid system is limited to the kinetic energy if we consider the liquid inside the shell as inviscid and incompressible. The velocity potential satisfies the Laplace equation: 2 (r, θ, z) = 0. The fluid velocity v is related to by v = grad. The Laplace equation may be solved by setting the boundary conditions. These boundary conditions represent the free surface ( = 0) and the equality of the normal velocity at the fluid/shell interface, i.e. / r r=r = d(δr)/dt. Under these assumptions, the kinetic energy dk liq (r) associated with a cylinder of liquid with r as radius, dr as thickness and h(r) as height is given by dk liq (r) = ω 2 T liq k (r) dr where T liq k (r) = 10π h 9 ρ lrg 2 (r)a 2 cos 2 ωt [f(z)] 2 dz (3) 0 where ρ l represents the liquid density. Here g(r) represents the normalized radial profile of the liquid vibration. Note, the difference in the amplitude for the kinetic expression of the fluid dk liq (r) and that of the shell K shell (see equation (1) originates in the slipping motion between the glass and the fluid due to incompressibility condition [10]). From the energy conservation of the vibrating glass liquid system, we obtain the Rayleigh quotient when normalized to the pulsation ω 0 of an empty glass [5], ω 2 0 ω liq Tk = 1+ 2 T shell k with T liq k = R 0 T liq k (r) dr. (4) The height profile of the vibration amplitude f(z) can be fitted accurately by the dimensionless function f(z) = (z/h ) 2 [1]. The normalized radius profile g(r) is well fitted by a simple linear function r/r [2]. This scaling can be obtained after some arithmetics by solving the Laplace equation with the proper boundary conditions mentioned above (see for instance [1] for details). These two functions had been used in the calculations. In the following, we consider the three types of fluid distributions associated with each experiment that is to say a liquid at rest, a rotating liquid and a liquid with an immersed cylinder Liquid at rest For the liquid at rest (see figure 1(a)), equation (4) can be integrated readily over the entire liquid volume of height h, we obtain [1] ω0 2 ( ) h 3 ω = 1+ξ with ξ = 4ρ lr 2 H 9ρ g a. (5) The pitch decreases as the liquid quantity increases. The term h/h to the power 3 is related to the larger contribution of the liquid as it gets closer to the rim of the glass. In other words, the closer the liquid gets to the rim, the more it moves, and consequently the more it

4 306 M Courtois et al (a) (b) (c) (d) (e) Figure 1. Experimental configurations: (a) sketch of the cylindrical glass partially filled with liquid at rest; (b) sketch of the glass filled with liquid and immersed stirring rod apparatus; (c) sketch of the glass with liquid and immersed cylinder. (d) Sketch of the shell deformations associated with the quadrupolar (2, 0) mode. The arrows show the directions of the bending (red colour inwards, blue colour outwards); (e) power spectrum of the impulse response of the glass. The resonances can be attributed to the flexural (m, 0) modes with m = 2, 3 and 4. contributes to the pitch lowering. It is noteworthy that the power of the term h/h is directly related to the shape of the shell [1] Liquid in rotation The liquid rotation is induced by a stirring rod immersed in the liquid (see figure 1(b)). The profile of the liquid free surface can be calculated by assuming that the vortex vector ω is nonzero along the vertical axis and that it decreases to zero at the glass surface. Solving the Laplace equation with these boundary conditions gives the following profile for the vortex vector: { ω = ω0 for 0 <r<r 0 ( ω = ω 0 1 ln(r/r 0 ) ) (6) ln(r/r 0 for R ) 0 <r<r, where R 0 is a small parameter compared with the glass radius R. Although the following results merely depend on the value of R 0, the best results are obtained with the smallest values. The arbitrary value: R 0 = 1 mm may be used. Using the Stokes theorem, we determine the tangential velocity and apply the Euler equation for circular trajectories. An equation for the profile of the liquid surface is obtained.

5 Tuning the pitch of a wine glass by playing with the liquid inside 307 For a given amount of liquid in the glass, this radial height profile h(r) depends on the single parameter ω 0 when the volume conservation is applied to the liquid. Experimentally it is quite simple to measure the height reached by the rotating liquid along the glass h max, thus we have expressed the liquid height profile h(r) using the parameter h = h max h 0 where the height of the liquid at rest is h 0. The pitch lowering induced by the rotating liquid profile, may be easily obtained if equation (4) is solved by introducing the expression of the surface profile h(r, h). Itgives the following polynomial function: ω 2 0 ω 2 1+ξ ( h0 H ) 3 ( h h ( ) h 2 ( ) ) h (7) In order to simplify, the numerical value R = 2.6 cm associated with our glass had been chosen Immersed cylinder The liquid distribution had been changed by immersing a cylinder of radius r c into it (see figure 1(c)). The height of the liquid is thus defined by { h(r) = hmin for r<r c (8) h(r) = h max for r>r c. Two different kind of experiments were performed using immersed cylinders. In a series of experiments, the cylinder is immersed into a given volume of liquid. In that case, the two ( heights h min and h max are related through the volume conservation of the liquid in the glass V = πr 2 c h min + π ( ) ) R 2 rc 2 hmax. This type of experiment really gives the influence of the liquid distribution inside the glass but the range of the liquid level is limited due to the fixed water volume. We consequently performed another series of experiments in which the liquid is added around a fixed cylinder centred in the glass and maintained at a close distance to the shell bottom. In that case h min is fixed at a value of about 1 mm while h max can be changed independently up to the maximum value H. The liquid profile is introduced in equation (4) to obtain the pitch lowering ω 2 0 ω 2 = 1+ξ ( hmax H ) 3 ( 1+ r4 c h 0 ( ) h 3 min hmax) 3. (9) R 4 h 3 max h 0 3. Experimental setup The experimental setup consists of a cylindrical glass with a stem and constant thickness of the glass shell that have been made by the glass-maker of the university. The glass dimensions are radius R = 2.6 cm, height H = 14.7 cm and thickness a = 0.1 cm. Commercial glasses have also been used and the results are very similar to those obtained with our cylindrical glass. However some slight discrepancies appear with the model due to the more complex shape of a real glass and to the presence of a non-constant thickness profile of the commercial glasses [3]. We connected a microphone to a computer to save the acoustic signals and to record the power spectra. The impulse response of the glass liquid system is obtained by hitting the glass gently with a metallic stick. The spectral resonance of the quasi-degenerated (2, 0) quadrupole modes (see figure 1(d)) can also be obtained by a simple finger rubbing excitation. However, in that case, the sound

6 308 M Courtois et al Figure 2. model). Pitch lowering versus water level in the glass (square: experimental data, full line: amplitude is modulated in front of the microphone leading to additional spectral peaks since the vibrational structure of the glass follows the exciting finger. Besides, as the liquid level increases it becomes more and more difficult to excite the glass with this method. Figure 1(e) shows a typical spectrum obtained by gently hitting the glass with a stick for a partially filled glass. Only a few resonance frequencies are visible in the displayed spectrum. They can be attributed to the flexural (m, 0) modes with m = 2 4 [9]. The higher modes (n 3) are difficult to excite when the glass is empty or filled with small amounts of liquid. This is due to the thickness of the glass. Like for the (2, 0) mode, the higher modes shift towards lower frequencies when liquid is added into the glass [3]. For the experiments with a rotating liquid, the rotation is ensured by a standard stirring rod apparatus. We proceed as follows: first, the glass is partially filled up to a desired height h max. Then, the stirring rod is centred and immersed to the bottom of the glass (without contact). For each measurement, we select a rotation frequency of the rod and wait a few seconds for the dynamic equilibrium to be reached. We measure the pitch level by gently hitting the glass and the associated new higher and lower liquid levels (resp. h max and h min, see figure 1(b)). For the experiments with immersed cylinders, various cylinders with different radii made of aluminium or rubber were used to change the water distribution in the glass. We proceed as follows: the chosen cylinder is centred and positioned to the bottom of the glass with no contact. The glass is then filled with water up to a given upper water level h max (see figure 1(c)) for which we measure the associated pitch level. 4. Results and analysis Figure 2 shows the measured evolution resonance frequency for the (2, 0) ovalling quadrupolar mode of vibration versus the water height in the glass. Small amounts of liquid do not significantly change the pitch level. Thus, the pitch is nearly constant at about 1660 Hz as the glass is filled up to the quarter of its total height, i.e. about 4 cm. As more and more water is poured into the glass, its effect on the pitch lowering is more important. It decreases regularly

7 Tuning the pitch of a wine glass by playing with the liquid inside 309 and significantly down to 940 Hz as more liquid is added into the glass shell. The full line in figure 2 represents the theoretical curve given by equation (5) without any fitting parameter. The model is in very good agreement with the experimental data. Basically, the lowering effect of the liquid is simply due to the added vibrating mass in the glass liquid system as mentioned earlier. In order to understand the increasing effect of water on the pitch level as more and more liquid is added, it is necessary to take into account the vibrating profile of the shell. Since the bottom of the shell cylinder is clamped, its vibration is very small in the lower part of the glass inducing negligible oscillations in the liquid. As the water reaches higher heights, it comes into contact with parts of the shell which vibrate with larger amplitudes. Therefore, the upper part of the liquid contributes more significantly to the total kinetic energy of the liquid glass system. Besides, it is interesting to note that the very good fit of the experimental data shows that the quadratic height profile used for the vibration amplitude of the vibrating mode in the model, i.e. f(z) = (z/h ) 2, is quite accurate. Note that the base of the glass (together with the stem) is assumed to be rigid in agreement with experimental observations of the oscillations performed by holography in [3]. Figure 1(d) shows a graphical visualization of the oscillations. Indeed, the frequency shift versus the liquid level in the glass shell is very sensitive to the vibrating mode profile. Figure 3(a) shows the pitch lowering when rotating the liquid in the glass with the use of a standard stirring rod. The frequency of the sound is plotted versus the upper water level reached for various initial amounts of water in the glass. In figure 3, each symbol (square, triangle,...) is associated with a particular experiment with a given amount of water, i.e. a specific initial water level at rest. The maximum rotating speed is limited by the fact that the stirring rod is not immersed any more in the liquid or alternatively the water reaches the rim of the glass. From any initial resting amount of water, the rotation of the liquid clearly lowers the pitch level by a significant amount. For instance, a glass filled with an initial level of h = 10 cm has a pitch level of 1225 Hz. Rotating the liquid to reach an upper level of about 12 cm lowers the pitch to 1130 Hz. This value should be compared with the frequency of 1080 Hz associated with a resting glass filled to this level. The full lines in figure 3(a) show the energy model for each initial condition of water volume using equation (7). The model agrees remarkably with the experimental data considering the fact that there is no fitting parameter. This experiment gives evidence of the spatial inhomogeneity of the liquid contribution to the pitch lowering. Hence, it confirms the increasing contribution of the liquid as higher parts of the shell are reached in agreement with the results of the experiment performed with liquid at rest (see figure 2). Furthermore, it gives access to the relative importance of the radial and height contributions since the results depend on both vibration profiles. The good agreement with the model confirms the validity of a linear radial amplitude profile for the vibration amplitude. These results prove that the part of the liquid near the central axis has only a weak contribution in the pitch lowering. The small discrepancy between the model and the experimental data probably originates from the fact that the radial profile of the water surface given by the model differs from the experimental one. Figure 3(b) shows the experimental data (squares) and the calculated values (full line) for the minimum height reached by the liquid level h min versus the upper liquid level h max for various initial water quantities. The calculated values are always higher than the experimental ones for the rotating liquids. The model overestimates the measured values by a few millimetres. From volume conservation, this means that the calculated radial profile underestimates the redistribution of the liquid from the centre to a position near the upper part of the shell where the vibration is more important. The model does not take into account the viscosity of the liquid which slows it down at the liquid glass interface which contributes to

8 310 M Courtois et al (a) (b) Figure 3. (a) Pitch lowering versus upper water level in the glass for various initial water levels: experimental (symbols) and theoretical values (full lines); (b) lower water level versus upper water level for the same various initial water levels: experimental (symbols) and theoretical values (full lines). pack the liquid towards the shell. Direct experimental observations of the lower level water show that the liquid exhibits a nearly horizontal tangential plane. Another experiment, which consists of immersing a cylinder into the liquid, had been performed to evaluate the water distribution inside the glass. Figure 4 shows the lowering of the pitch for a glass filled with various amounts of water for the same glass with a rubber (full circle) or an aluminium cylinder (full triangle) placed at the centre along the symmetry axis. The presence of the cylinder at the centre of the glass appears to lower the pitch in the same manner as if this volume was filled without water. At first sight, the reference curve without any cylinder is very similar to the one obtained with the rubber or the aluminium cylinder. The simple model developed in section 2.3, which consists of removing the influence of the liquid in the cylinder volume in the pitch lowering, is in good agreement with the measured

9 Tuning the pitch of a wine glass by playing with the liquid inside 311 Figure 4. Pitch lowering versus upper water level in the glass for various immersed cylinders. experimental data (full line in figure 4). These results clearly confirm the weak contribution of the central part of the liquid because of the decaying radial amplitude of the vibration. Although our simple model permits us to give a good description of the observed influence of the cylinder on the pitch lowering, it cannot explain the small differences measured for each cylinder. Indeed, our model does not take into account the cylinder material. The lowering obtained by using an aluminium cylinder is slightly more pronounced than the one induced by the rubber cylinder. The explanation probably originates from the difference between the reflection coefficient of the acoustic wave of two materials. The acoustic impedance between the rubber and the water being about the same, no reflection is induced. This is not the case between aluminium and water for which the main part of the acoustic energy is reflected (rigid boundary condition). A reflection coefficient of approximately 80% is obtained from the typical values found in the literature, inducing a reflection of the acoustic wave at the interface. Our simple model does not take this aspect into account assuming that the interface with the cylinder is similar to a surface with an adapted acoustic impedance (i.e. no reflection of the acoustic wave). This hypothesis, which has the advantage of simplicity, slightly underestimates the kinetic energy associated with the liquid and, consequently, the pitch lowering (see figure 4). Taking into account proper boundary conditions would require a more complicated approach to obtain the exact fluid dynamic expression (as compared with the solution given in equation (4)). Recently, Chen et al have addressed this issue [10, 11]. They explained that a reduction of the capacity due to the immersion of a solid cylinder is accompanied by an increase in the lateral liquid flow since the fluid is incompressible. This lateral flow enhances the pressure on the glass wall which prevents the glass vibration and lowers the pitch. In [10], the authors have calculated the influence of this additional pressure on the pitch lowering in the case of two concentric infinite cylinders, the outer one of radius R, which is made of glass, and the inner one of radius r c. They showed that the coefficient ξ of our equation (9) should be corrected by a factor (1+(r c /R) 4 )/(1 (r c /R) 4 ) for the quadrupolar mode. If we assume that this correction factor can be applied to our case with a partially filled finite cylinder, this correction factor equals The associated curve can be seen in figure 4 (dashed line). The two curves (full and dashed line) are associated with the acoustic impedance matching of the

10 312 M Courtois et al immersed cylinder with water (perfect and poor matching respectively). The experimental data are framed by the two theoretical limiting curves. 5. Conclusion We have shown that the spatial distribution of the liquid inside a vibrating glass plays a crucial role in the pitch lowering. Thus, the frequency of a vibrating glass can be tuned not only by pouring some liquid into a glass but also by changing its distribution inside the glass shell with constant liquid volume. Two different experiments have been performed to modify the liquid distribution: rotating the filled glass or immersing a cylinder into the liquid. These experiments gave evidence of the crucial role of the liquid vibration profile both in the axial and in the radial position. A simple model based on energy conservation of the liquid glass system gives good agreement with the observed pitch changes. However, in the case of immersed cylinders, the acoustic impedance must be taken into account for a more accurate analysis. The additional pressure on the glass shell induced by the lateral liquid flow is not negligible any more. Acknowledgment We would like to acknowledge Fanny Fourier for her participation into the initiation of this project. References [1] Banerji S 1919 On the vibrations of elastic shells partly filled with liquid Phys. Rev [2] French A P 1983 In vino veritas: a study of wineglass acoustics Am. J. Phys [3] Jundt G, Radu A, Fort E, Duda J, Vach H and Fletcher N 2006 Vibrational modes of partly filled wine glasses J. Acoust. Soc. Am [4] Apfel R E 1985 Whispering waves in a wineglass Am.J.Phys [5] Rayleigh L 1894 Theory of Sound vol I (London: Macmillan) [6] Bentley P G and Firth D 1971 Acoustically excited vibrations in a liquid-filled cylindrical tank J. Sound Vib [7] Amabili M J 1999 Vibrations of circular tubes and shells filled and partially immersed in dense fluids Sound Vib [8] Koopmann G H and Belegundu A D 2001 Tuning a wine glass via material tailoring an application of a method for optimal acoustic design J. Sound Vib [9] Rossing T D 1994 Acoustics of the glass harmonica J. Acoust. Soc. Am [10] Chen K W, Wang C K, Lu C L and Chen Y Y 2005 Variations on a theme by a singing wineglass Europhys. Lett [11] Chen Y Y 2005 Why does water change the pitch of a singing wineglass the way it does? Am.J.Phys