Vibration of flutes studied by electronicspeckle pattern interferometry

Ole J. L0kberg and Ola K. Ledang

This paper describes how electronic speckle pattern interferometry can be used to observe in real time thewall vibrations of simple flutes excited in realistic playing conditions. Dynamic interferograms were createdby driving the phase-modulating mirror by means of the amplified sound signal picked up by a microphone.Some preliminary results are given.

1. Introduction

Sound radiating from vibrations in the wall of an in-strument contributes in varying degrees to the totalinstrumental sound. This contribution is importantfor our understanding of sound quality of a particularinstrument design. Knowledge of an instrument'sresonance vibrations is thus important both for basicunderstanding and for design purposes.

Traditionally, observation of the Chladni pattern hasbeen used for studies of instrument vibrations, see, forexample, Ref. 1. However, Chladni patterns do notprovide amplitude and phase information, only theposition of the nodal lines. In addition, the methoddoes not work on curved surfaces.

Hologram interferometry represents a major im-provement of the Chladni method as it possesses noneof the previous disadvantages and in addition is moresensitive over the whole frequency range. Holograminterferometry has been used mainly for studies ofmusical string instruments, most notably the violin.2 3

The stability requirement and the low throughput of theholographic process do limit, however, effective use ofthe technique to study instruments in realistic condi-tions. The wind instruments would be especially pro-blematic as they are excited by an airflow which easilyintroduces unwanted movements. Artificial excitationby a transducer may not necessarily represent the realbehavior of the instrument.

Ole Lkberg is with Norwegian Institute of Technology, PhysicsDepartment, N-7034 Trondheim-NTH, Norway, and Ola Ledang iswith University of Trondheim, Musicology Department, N-7055Trondheim-Dragvoll, Norway.

Received 31 October 1983.0003-6935/84/183052-05$02.00/0.© 1984 Optical Society of America.

Electronic speckle pattern interferometry (ESPI)may be described in brief introductory terms as imageholography, where video registration and processinghave replaced the recording and reconstruction pro-cesses of ordinary holography.4 The technique'scombination of short exposures and high sampling rateenables us to record vibrations of unstable objectswithout resorting to pulsed lasers.5 6 To use ESPI forstudies of the vibrations in wind instruments in naturalexcitation represents a natural extension of the tech-nique's application area. It may also lead to measure-ments of vibrations in general pipe structures.

11. Flutes and Their Excitation

The flute used in this investigation is a long willowflute which is a traditional Norwegian folk instrument.It was chosen for its simplicity and because its vibra-tions have been a matter of musicological dispute. Itis made from long straight offshoots of willow during ashort time in late spring, in Norway usually from themiddle of May, when the sap rises in the tree. Duringthis period the bark can be quite easily loosened fromthe wood core to produce a hollow pipe. A crescentshaped hole (the mouth) is cut in the wider part of thebark tube, and a small part of the wood core is formedand reinserted into the tube to act as playing section,similar to the wood block of the recorder (flute). Figure1 is a drawing of a willow flute with typical dimensionsindicated. The artificial willow-type flutes, made frommetal or plastic, were constructed in a similar way.

In a willow flute, the sound is produced by the inter-action between the airflow across the mouth, the upperlip, and the air column within the tube. The long willowflute is played by overblowing: by varying the blowingpressure, different resonances of the air column areexcited separately, thus different pitches are produced.The total frequency range which can be produced in thisway differs from specimen to specimen of the instru-

3052 APPLIED OPTICS / Vol. 23, No. 18 / 15 September 1984

Fig. 1. Willow flute.

ment, covering approximately three octaves from 400to 3000 Hz.

To excite the flute, the wood part of the flute wasmodified to make it possible to pull a flexible tube (abicycle inner tube) over part of the playing section. Thetube was connected to a stiff plastic hose which was heldin a rig away from the table. The flute could now beplayed by blowing air through the hose.

The flute was (loosely) fastened at the wood part,while the other end was laying freely in a V-type cradle.Care was taken to ensure a free path for the outcomingairflow.

The artificial flutes could be played by pressurizedair, and the various frequencies were excited with ease.The willow flute became rapidly unplayable in the drypressurized air because of dehydration of the barkmaterial. We had to play the flutes ourselves, wherebythe experimental time became limited by the lung ca-pacity of the flutist. This caused real problems only atthe highest frequencies because of the increased airpressure needed. Figure 2 shows one flutist in action,where a composite exposure has been used to indicatethe position of the hose.

111. Fringe Analysis

Electronic speckle pattern interferometry has thesame functional interferometric properties as holograminterferometry. In this case where (almost) pure har-monic vibrations were to be investigated by time-av-erage recording, we can use the results directly from Ref.7 to write the intensity distribution I(x ,y) of the videomonitor image:

I(x,y) = Io(Xy) JO (sinO1 + sinG2) ao(xY)] I (1)

where Io(x,y) is the image intensity of the objectimage at rest,

Jo is a Bessel function of first kind andzero order,

01,02 are the angles between theillumination, observation directions,and the movement vector,respectively, and

ao(x,y) is the object's amplitudedistribution.

Fig. 2. Excitation of flute.

15 September 1984 / Vol. 23, No. 18 / APPLIED OPTICS 3053

cm

a a-a

Fig. 3. ESPI layout.

The fringe function as given by Eq. (1) has two maindisadvantages for our vibration measurements. First,its measuring range in ESPI is too limited. In experi-ments with simple flutes and pipes the excitation levelat each resonance has to be held at a fixed constantmagnitude. This is different from ordinary vibrationanalysis when the excitation can be varied freely toobtain a suitable number of fringes. For ESPI workson a rough surface like bark, realistic values would befrom a lower measuring limit of -0.2X (first dark fringe)to an upward limit of only -0.8X (third-order fringe).Second, the argument of the J function contains onlythe amplitude information, and any information re-garding phase variations /0(x,y) across the object islost.

As discussed in previous papers, these drawbacks canbe remedied by the use of sinusoidal phase modula-tion8-10 usually effected by introducing a vibratingmirror into the ESPI setup. If we suppose the phase-modulating mirror to vibrate at the same frequency asthe object but with a spatially constant amplitude aRand phase R, the resulting intensity distributionImod(X,y) will be8

I..d(XY) = Io(xy) J (sinG1 + sin 2) lao(xy)

+ a - 2ao(x,y) aR * cos[o(x,y) - IR]I2) * (2)

Equation (2) shows that the fringe function remainsunchanged, but its argument is now dependent on thevectorial difference between the movements of theobjects and the phase-modulating mirror. We rely onthe fact that the bright zero-order fringe J20(0) is ob-served whenever a(x,y) = aR and 0o(x,y) = R tomeasure large amplitudes9 and to trace the phase dis-tribution.10

For our experiment we also had to use the samephase-modulation principle to detect and measure smallvibrations.1" The amplitude of the reference mirror isnow adjusted to the steepest part of the J2(0) fringewhich is, at '-/10. The reference phase R is varied

continuously from to 360°, which is equivalent tovibrating the mirror at a frequency different from theobject's frequency. As the phase relation between theobject and the reference vibration varies, the intensityof the monitor image will vary. The magnitude andtiming of the intensity variation are directly propor-tional to the object's amplitude and phase.

In these experiments we were not overly concernedabout measuring the low amplitudes with great accuracyas only the general behavior of the instruments wassought. Therefore, the photoelectric measurements ofintensity variations as described in Ref. 11 were not usedat low amplitudes as this procedure is rather time-consuming. Instead, a simpler way to get a rough in-terpolation on the J2o fringe function was used. Thismethod was based on detection of the first fringe min-imum which occurs at an amplitude a = 0.19X or'100 nm at laser wavelength X = 514.6 nm. Fromprevious calibrations we knew that our modulatingmirror had to be excited at UM1 (volts) to reach this firstfringe minimum. During the measurements the objectwas excited at constant frequency and excitation levelwith a resulting amplitude distribution a(x,y). Themodulation mirror was then driven at a slow beat fre-quency (-0.5 Hz), and its excitation level was increasedfrom zero value until the observed part of the objectbecame completely dark when the two movementspassed through the antiphase part of the beat cycle. Ifthis occultation occurs at excitation level UM2 of themirror, the object amplitude is:

ao(x,y) = 0.19X- (UM2/UM1). (3)

With care the amplitude could be determined with anaccuracy of 10-15% to amplitude levels of 5 nm. Thisvalue is not too impressive compared with the photo-electric detection level of 0.1-0.01 nm. However, thismethod gave the necessary fast indication of the vi-bration magnitude.

IV. Setup

The setup with attached electronics is shown in Fig.3. The light from the Ar laser (150 mW, single mode)

3054 APPLIED OPTICS / Vol. 23, No. 18 / 15 September 1984

was divided into two paths by beam splitter BS. Thereflected light was expanded by a short focus lens L1 andtransformed by a cylindrical lens CL into a fan to opti-mize the illumination of the long slender flute F. A lenssystem Limage stopped at f:30 imaged the object onto thetarget of the video camera after a reflection from mirrorM. The beam transmitted through the beam splitterconstituted the reference path in the setup. The beamwas first reflected from mirror Mmod before being fo-cused by a short focus lens L2 through a pinholemounted in mirror M. The expanded reference wavehit the target in line with the .object image wave, and thetwo waves interfered to create an image interferogram.The video camera transformed this interferogram intoa proportional video signal. The video signal waselectronically processed by high-pass filtering andrectification to simulate the reconstruction step in ho-logram interferometry.4 The processed signal wasthereafter fed into the video monitor to display theimage interferograms. The resulting interferogramsand the sound were also recorded by a video recorder forlater analysis.

As the dimensions of the flute were very differentfrom the format of the TV image, it was impossible toview the entire flute simultaneously with sufficientresolution. During most of the measurement we ob-served only a small part, -5 cm, of the upper playingsection of the flute which represented the area ofmusicological dispute. To get information about thewall vibrations at large, the flutes were placed ,70° tothe illumination and observation directions. In thisway almost half of the flute (-25 cm) could be observedsimultaneously.

To increase the interferometric stability the laserlight could be chopped, synchronized to the TV-framingrate.5 Exposures in the 10-msec range gave excellentstability.

To obtain phase modulation mirror Mmod was vi-brated by a piezoelectric crystal. Initially the modu-lating frequency was adjusted to the object's frequencyby a sinewave generator. This proved to be difficultespecially as the object's frequency would vary slightlybecause of the variation in the air pressure. Instead,a microphone picked up the sound from the flute. Thissignal was amplified and used to vibrate the mirror. Afrequency translator was used to shift the referencefrequency to display dynamical interferograms.10 Thefrequency translator was replaced by a phase shifterwhenever static interferograms were needed for phaseand high amplitude measurements.

V. Experimental Results

The primary purpose of these experiments was todemonstrate the ability of ESPI to detect and measurethe wall vibration of a wind instrument in almost real-istic playing conditions. Clear and stable vibrationpatterns could be observed on the flute walls even atnormal TV exposures. In fact, we could play tunes andwatch the waves traveling down the flute wall. (Ad-mittingly, slow tunes with a certain tristesse gave themost entertaining displays.)

In this paper we will not give any acoustical inter-pretation of the experiments. The different vibrationpatterns with and without phase modulations have beenrecorded on videotape for further analysis coupled toacoustical measurements. We will, however, point outthe main differences between the flutes made fromnatural and artificial materials.

The wall vibrations of the metal flute were small anddifficult to measure. Even at the most powerful reso-nances, the maximum amplitudes were <15 nm.

The plastic flute vibrated at larger amplitudes, butthe increase was not sufficiently large to give inter-pretable fringe patterns without resorting to phasemodulation. The maximum amplitudes ranged from30 to 60 nm. Mainly classical patterns were observedin the sense that phase-antiphase relations existed.Only at the second harmonics, -900 Hz, could a wavebe observed traveling from the mouth area at 450 to thelong axis.

The willow flute made from bark gave the most variedresults, which might be expected because of the pliableand irregular structure of the bark. When freshly cut,complex patterns with large amplitudes and continuousphase variations could be observed. In general, theamplitudes were five to ten times larger than the plasticflutes, which might explain the richer sound from anatural willow flute. On the bigger flutes the fringeswere too closely packed to be resolved by ordinarytime-average ESPI. Using phase modulation with ourpiezoelectric mirror at its maximum excitation level, weextrapolated these amplitudes to -3 ,um when the skewillumination and observation had been corrected. Apronounced aging effect was also observed in thesenatural flutes. The amplitudes decreased with the timeeven if the flutes were stored submerged in water andfrequently moistened during the actual experiment.The vibration patterns also transformed into moreregular patterns with phase-antiphase relations.

Figure 4 shows some time-average recordings of avibrating natural willow flute excited by a player.Figure 4(a) represents the upper part of a freshly cutflute. The crescent shaped opening or the mouth isclearly seen. Its lower lip is vibrating in a nonsym-metrical way and at a much higher amplitude than forthe artificial flutes. This supports the theory that thelip in the natural flute modulates the airflow and thuscolors the sound. Figures 4(b) and (c) show the vibra-tion patterns at various frequencies along half of thelength of the flute. In these recordings the flute was 2days older, and its vibrations have transformed intofairly regular patterns at moderate amplitudes.

VI. Concluding Remarks

We have shown that ESPI with a cw laser can be usedto observe in real time the wall vibrations of simpleflutes excited in realistic playing conditions. Theprocedure used during these experiments should alsobe applicable to studies of similar wind instrumentsprovided their vibration amplitudes are large enough.We stress again that the wall vibrations probably giveonly a minor contribution to the total sound picture

15 September 1984 / Vol. 23, No. 18 / APPLIED OPTICS 3055

Fig. 4. Vibration patterns of willow flute. (a) Strong, irregular vibrations of mouth region in a freshly cut willow flute; frequency f = 1304Hz. (b) Upper part (20 cm) vibrating at f = 566 Hz; flute 2 days old. (c) As in (b), but at f = 850 Hz. (d) As in (c), but lower part (15 cm)

of flute.

from a wind instrument. This contribution may,however, color the sound and be decisive in distin-guishing a great instrument from an ordinary one.

A more negative conclusion may be drawn from therapid stiffening of the natural willow flute resulting indrastic changes of the vibration patterns. It is sur-prising that a rough biological structure like bark, storedand used in optimal conditions, deteriorates so rapidly.This put a question mark on any extrapolation to thedynamical behavior in vivo of more delicate biologicalpreparations like, for example, the human eardrum.

Thanks are due to G. A. Slettemoen and 0. M. Holjefor a most valuable assistance and inspiring commentsduring the experiments, especially as the lung capacityof the flutists approached the limits.

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