Laser interferometric characterization of a vibrating speaker system A. A. Freschi, N. R. Caetano, G. A. Santarine, and R. Hessel Citation: American Journal of Physics 71, 1121 (2003); doi: 10.1119/1.1586262 View online: http://dx.doi.org/10.1119/1.1586262 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/71/11?ver=pdfcov Published by the American Association of Physics Teachers This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.3.46 On: Thu, 16 Jan 2014 13:49:09 http://scitation.aip.org/content/aapt/journal/ajp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/test.int.aip.org/adtest/L23/33263977/x01/AIP/HA_Pub2Web_ReregisterToCAlert_AAPTCovAd_1640x440_10_2013/aapt_aipToCAlerts.png/7744715775302b784f4d774142526b39?x http://scitation.aip.org/search?value1=A.+A.+Freschi&option1=author http://scitation.aip.org/search?value1=N.+R.+Caetano&option1=author http://scitation.aip.org/search?value1=G.+A.+Santarine&option1=author http://scitation.aip.org/search?value1=R.+Hessel&option1=author http://scitation.aip.org/content/aapt/journal/ajp?ver=pdfcov http://dx.doi.org/10.1119/1.1586262 http://scitation.aip.org/content/aapt/journal/ajp/71/11?ver=pdfcov http://scitation.aip.org/content/aapt?ver=pdfcov This ar Laser interferometric characterization of a vibrating speaker system A. A. Freschi,a) N. R. Caetano, G. A. Santarine, and R. Hessel Departamento de Fı´sica, IGCE, UNESP, Caixa Postal 178, 13500-970, Rio Claro, SP, Brazil ~Received 4 November 2002; accepted 2 May 2003! An experiment that combines opto-mechanical and electrical measurements for the characterization of a loudspeaker is presented. We describe a very simple laser vibrometer for evaluating the amplitude of the vibration~displacement! of the speaker cone. The setup is essentially a Michelson-type interferometer operated by an inexpensive semiconductor laser~diode laser!. It is shown that the simultaneous measurements of three amplitudes~displacement, electrical current, and applied voltage!, as functions of the frequency of vibration, allow us to characterize the speaker system. The experiment is easy to perform, and it demonstrates several useful concepts of optics, mechanics, and electricity, allowing students to gain an intuitive physical insight into the relations between mathematical models and an actual speaker system. ©2003 American Association of Physics Teachers. @DOI: 10.1119/1.1586262# e ct th re te g f r ob as s li la rc y- re ur as b d e nd e a a e tio e m ne tr in e try e is ha th nts in rgy ng s etic etic ker a t il is in- nc- e ex- ra- ssi- com- scil- I. INTRODUCTION Since the advent of the laser in the early 1960s it has b natural to study vibrations by means of the Doppler effe1 The idea is to use an optical interferometer to measure frequency shift of a coherent laser beam that is scatte from a small area of the vibrating object. The interferome mixes the scattered light with a reference beam, resultin an optical signal whose~beat! frequency is equal to that o the difference between the mixed beams. By using this p cedure both the velocity and/or the displacement of the ject can be precisely measured. Several variations of the l Doppler technique have been developed for application fields such as structural dynamic testing, acoustics, qua control, industrial plants, and medical diagnosis.1–4As a con- sequence of the wide acceptance of the technique, the re instruments~usually called Laser Doppler Vibrometers! have become important tools for those involved either in resea or applications involving experimental vibration and d namic system analysis. However, despite the fact that the extensive coverage on this subject in the technical literat there are relatively few papers devoted to the use of the l Doppler technique in undergraduate laboratories.5–8 This work presents a very simple laser vibrometer suita for characterizing electromechanic transducers such as namic loudspeakers. The aim is to give the student a gen introduction to the Doppler effect, optical interference, a vibration analysis, by illustrating an application of the las Doppler technique in the field of acoustics. Vibration me surements on a speaker system are one of the typical c where, because of the speaker cone lightness, optical t niques are much preferred over the usual instrumenta ~such as accelerometers!. We consider one of the most common speaker mod which consists of a light coil suspended on a strong per nent magnetic field fixed on the center of the moving co In Sec. II A the basic theoretical expressions relating elec cal and mechanical quantities of the vibration of the coil the magnetic field are derived. Section II B presents a gen explanation of the Doppler effect and optical interferome It is assumed that the student has a thorough knowledg the fundamental laws of mechanics and electromagnet and is familiar with calculus and complex numbers. Emp sis has been given to explaining the basic concepts toge 1121 Am. J. Phys.71 ~11!, November 2003 http://aapt.org ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 en . e d r in o- - er in ty ted h is e, er le y- ral r - ses ch- n ls, a- . i- ral . of m, - er with a description of the experimental setup and compone ~in Sec. III!. The results are presented and discussed Sec. IV. II. THEORY A. The dynamic loudspeaker A loudspeaker is a device that converts electrical ene into sound.9 The most common models consist of a movi cone firmly cemented to a small light cylindrical coil in it center, as depicted in Fig. 1. The coil is in a strong magn field of a permanent magnet. The lines of the radial magn field lie in a plane perpendicular to thex axis shown in Fig. 1. Alternating current through the coil causes the spea cone to vibrate, thus producing sound waves to the air. Consider a loudspeaker in which the coil is fed with steady sinusoidal voltage,V, of amplitudeV0 and angular frequency v. The alternating electric currenti flowing through the coil has amplitudei 0 . The forceF on the coil ~and the resulting motion! is in thex axis direction and can be written asF5Bli , with B the magnetic field strength a the coil, andl the length of wire in the coil. Because the co moves perpendicularly to the magnetic field, and the wire perpendicular to the field and to the motion, there is an duced electromotive force given byE5Blu, whereu is the coil speed.9,10 If the loudspeaker vibrates with small amplitude~low in- put power!, we may think of it as a linear system.10 Thus all oscillatory quantities can be expressed by sinusoidal fu tions with temporal angular frequencyv. Let us callx the position of the central point of the speaker cone along thx axis, and definex50 as its equilibrium position~which oc- curs in the absence of any external applied voltage!. The amplitude of the displacement isx0 . As will be shown, it is possible to characterize the speaker system by fitting the perimental data of the current normalized amplitude of vib tion, x0 / i 0 , and the impedance,Z05V0 / i 0 , as functions of v, using the corresponding theoretical expressions. A cla cal approach to obtain these expressions makes use of plex numbers to represent the physical parameters that o late. We write a little caret (̂) over the parameter to 1121/ajp © 2003 American Association of Physics Teachers bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09 in hu th th or m ro th d te d th r- ld i- f er to onal ci- et- s lex ll s total t n- e of box This ar represent that it is a complex number; the correspond physical quantity being the real part of the expression. T we may write, F̂5Blî ~1! and Ê5Blû5 j vBlx̂, ~2! where the temporal derivativedx̂/dt was replaced by j v x̂ (d/dt→ j v),10 with j 5A21. Because the productBl is real, the j in Eq. ~2! simply means thatE and u lead the displacementx by 90°. We now write two more equations for our analysis. LetR and L be the electrical resistance and the inductance of coil, respectively. The coil and the speaker cone altoge have massm, the overall suspension elastic constant isk, and the dissipation constant~due to friction plus useful sound! is b. We assume that the speaker cone and the coil move gether as a ‘‘rigid piston,’’ which is a good approximation f low frequencies, at which the sound wavelength is long co pared with the diameter of the speaker cone.9,11,12 For the electrical part of the speaker, the applied voltageV must overcome the induced electromotive force, the voltage ac the resistor and the inductor, V̂5Ê1Rî1 j vLî , ~3! with the voltage across the inductor (Ldi/dt) written as j vLî . On the mechanical side, the force accelerating massm can be expressed as F̂2kx̂2 j vbx̂52mv2x̂, ~4! where 2kx→2kx̂ is the elastic force,2bu52bdx/dt →2 j vbx̂ is the frictional plus air resistance force, an md2x/dt2→2mv2x̂ is the mass times acceleration. No that the problem expressed by Eq.~4! represents a dampe oscillator subjected to an external forceF̂, with b the damp- ing constant~which depends on the characteristics of bo the speaker and the surrounding air!. If we eliminate F̂ in Eqs.~1! and ~4! and calculatex̂/ î , we obtain x0 i 0 5U x̂ î U5 Bl/k H F12v2 m k G2 1v2 b2 k2 J 1/2 ~5! with the parametersa15Bl/k, a25m/k, anda35b/k. Fig. 1. Speaker parameters and physical cross section. 1122 Am. J. Phys., Vol. 71, No. 11, November 2003 ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 g s e er to- - ss e An important conclusion that may be drawn from Eq.~5! is the existence of a resonance atv5@(k/m) 2(b2/2m2)#1/2. In practice, the damping constant is gene ally small (b!Akm), so the resonance occurs atv 'Ak/m. Also, as we might expect, a strong magnetic fie B, a long length of the wire in the coill, and a small elastic coefficient k, result in greater motion, and the termBl/k appears naturally in the numerator of Eq.~5!. The impedanceẐ5V̂/ î can be calculated by solving s multaneously Eqs.~1!–~4!. We eliminateF̂, Ê, and x̂: Z05uẐu5 V0 i 0 55 F R1 ~Bl !2/b 11 1 v2 k2 b2 S 12v2 m k D 2G 2 1v2F L1 ~Bl !2/k S 12v2 m k D1 1 1 v2 k2 b2 S 12v2 m k D G 2 6 1/2 ~6! with three more independent parameters:a45R, a55L, and a65(Bl)2/b @the last parameter, (Bl)2/k, can be calculated from the previous ones#. It is clear that the determination o the parametersa1 to a6 represents a solution for the speak problem because the productBl5a3a6 /a1 , and therefore the parametersb, k, andm, can be easily calculated. In the analysis of loudspeakers it often is convenient replace the actual speaker system by an equivalent moti electrical system.9 By writing Ẑ5R1 j vL1ẐM , it is pos- sible to show that a parallel combination of a resistor, capa tor, and inductor can represent the termẐM , sometimes also called motional impedance. The equivalent electrical n work is depicted in Fig. 2, whereCm5m/(Bl)2 is the analog of m, Lk5(Bl)2/k is the analog ofk, andRb5(Bl)2/b the analog ofb. ~This circuit is but one of a number of circuit that can be devised from the rationalization of the comp impedance functionẐ.) From inspection of Fig. 2 we can te that the frequency of resonance is defined primarily byCm andLk ~indeed,m/k5LkCm). The termb/k5Lk /Rb which appears in Eqs.~5!, and ~6! has dimension of time, and i related to the sharpness of the resonance peak. Of the resistive componentR1Rb in this analogy, only that par given by Rb is associated with the transfer of electrical e Fig. 2. Equivalent all-electrical network to a loudspeaker. The resistanc the wire isR andL is the inductance. The impedance inside the dashed is called ‘‘motional impedance,’’ withCm5m/(Bl)2, Lk5(Bl)2/k, andRb 5(Bl)2/b. The applied voltage isV and E is the induced electromotive force ~in volts!. 1122Freschiet al. bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09 k e he - s ut ity e tr n o . l ar x- am th m . on pe ur is u s he a ng h e a av o b bine ce es tput rav- tor or- and n- ew bal- ally . In is ed- in- , olt- This ar ergy to acoustic energy~plus losses due to friction in flexing the speaker cone!, whereas the energies stored inLk andCm correspond, respectively, to the mechanical potential and netic energy of the system. The average powerW dissipated as sound ~plus frictional losses! is then given by W 5Erms 2 /Rb5burms 2 , with Erms and urms the rms ~root mean square! amplitudes of the voltageE and the velocityu, re- spectively. Note that atv5A1/LkCm, the same current flows throughR and Rb ~the currents throughLk and Cm cancel each other!, and the ratio of the radiated sound power~plus frictional losses! to the total power dissipated by the speak system is simplyRb /(R1Rb). As we might expect from the analysis of the electrical network depicted in Fig. 2, t power dissipated byRb is greatly attenuated at low frequen cies due to the shunting effect ofLk , and at high frequencie due to the shunting effect ofCm ~and also because the inp current is blocked byL!. B. Doppler effect and wave mixing The Doppler effect1 can be used to measure the veloc ~and/or the displacement! of an object that scatters light. Th idea is to detect the change in the frequency of an elec magnetic wave due to the relative motion of the object a the receiver. Consider a light beam sent out from a laser t object moving with velocityu in the direction of the beam To first order inu/c, with c the speed of light, the fractiona shift in frequency of the backreflected beam is1 2u/c. Thus, for an object moving at 1 mm/s, the frequency shift is 1 p in 1.531011. Although extremely small, this shift~usually called the Doppler shift! can be precisely measured by mi ing the reflected light from the object with a reference be from the same laser. A common problem in the laser Doppler technique is discrimination of the direction of the velocity. This proble is usually solved by employing optical devices~such as Bragg cells! to shift the frequency of the reference beam1 However, when only the amplitude of a sinusoidal vibrati is of interest, all we need is an ordinary Michelson-ty interferometer,13 which is the basis of the setup used in o experiments~Fig. 3!. In this configuration, a laser beam divided into a reference beam and a signal beam by the c beamsplitter BS. The reference beam is directed onto a tionary object, O1 , and the signal beam is directed onto t vibrating test object~loudspeaker!, O2 . The retro-reflected beams return to the beamsplitter, where part of the be coming from O1 is transmitted, and part of the beam comi from O2 is reflected toward the photodetector D. When t test object moves, the frequency of the signal beam shifted, resulting in an optical power modulation of th mixed wave due to interference between the reference signal beams. The expression for the output voltageVD from the photo- detector can be derived from the well-established w analysis of two-beam interference6,13 and may be written as VD5kP0H 11m cosF4p l ~x12x2!G J , ~7! with k the voltage responsivity of the photodetector,P0 the average optical power,m the modulation depth,l the light wavelength, andx1 andx2 the optical path lengths of the tw arms of the interferometer. Note that the path difference 1123 Am. J. Phys., Vol. 71, No. 11, November 2003 ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 i- r o- d an t e be ta- m e is nd e e- tween the signal and reference beams when they recom is 2x122x2 , and anything that changes this path differen will cause a change in the output voltageVD . Each complete cycle ~2p phase shift! on the alternating component ofVD corresponds to an object movement ofl/2 and the frequency of this modulation is the Doppler shift, given bynD 52u/l. Thus, by counting the number of complete cycl through which the movement of the object causes the ou voltage to change, we can determine the total distance t eled by the object. If the test object oscillates harmonically with amplitudex0 and angular frequencyv, we may letx25x201x0 sin(vt), so the total phase excursion in one period of vibrationT 52p/v is 16px0 /l. Therefore, ifx0@l andn is the num- ber of complete cycles described by the voltageV in a time interval T, we may write 2pn516px0 /l, or x05nl/8 ~x0@l!. ~8! III. EXPERIMENTAL REALIZATION We used a 3.5 mW optical output power semiconduc laser diode14 operating atl50.65mm for the light source of the setup illustrated in Fig. 3. The diode laser module inc porates a mounted laser diode, adjustable focusing lens, driver circuit into a compact cylindrical package~12 V dc operating voltage!. The cube beamsplitter15 transmits and re- flects approximately 50% of the incoming light and the i terferometer arms were set for equal lengths to within a f millimeters (x1'x2'15 cm). The laser coherence length13 was about 10 cm, much larger than the beam’s path im ance. The laser beam is focused onto the reference (O1) and test (O2) objects: a small metallic plate and a low-power~1 W! loudspeaker, respectively. The loudspeaker has nomin 8 V impedance, with a speaker cone 60 mm in diameter order to measure the vibration of the coil, the laser beam made normally incident~in thex-axis direction! in the center of the speaker cone. The photodetector~D! consists of an inexpensive silicon photodiode connected to a transimp ance amplifier integrated on a single monolithic chip.16 It Fig. 3. Experimental setup, illustrating the interferometer and electrical struments. BS: beam splitter, D: photodetector, O1 : static reference plate O2 : loudspeaker, A: milliammeter, G: function generator. The applied v age to the speaker isV, the current isi, the detector output isVD , andx1,2 are the arm lengths of the interferometer. 1123Freschiet al. bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09 a lt- an ( p- er a op of gl o ag tle llo ol y ig te a on- is not ay ing the e- eri- fit- the s. ow to on n This ar was enclosed in a small aluminum box, which has a sm hole for collecting the incoming light beam. The output vo age, VD , is the product of the photodiode current and external feedback resistor,RF , and is proportional to the radiant power falling on the photodiode’s active areaA 55.2 mm2). At the wavelength of 0.65mm and RF 510 MV, the full photodetector voltage responsivity is a proximatelyk54.5 V/mW. All components, with the exception of the loudspeak were mounted on a small aluminum breadboard. The bre board was placed on a foam sheet to reduce the noise serted by mechanical vibrations in the optical setup. A dr let of reflective ink17 was placed on the illuminated areas both objects~reference plate and loudspeaker! to increase the collected light power, and the detected voltage accordin The loudspeaker is fed with a steady sinusoidal voltage amplitudeV0 and frequencyn5v/2p from a function gen- erator ~G!.18 An ammeter19 ~A! measures the currenti rms 5 i 0 /A2 through the speaker~see Fig. 3!. The input imped- ance of the oscilloscope is set to 1 MV, much higher than the impedance of the loudspeaker. The voltageV that feeds the loudspeaker, and the ac component of the output volt from the photodetector@VD(ac)#, are visualized on an oscilloscope20 screen~triggered usingV as reference!. IV. RESULTS AND DISCUSSION A typical measurement is illustrated in Fig. 4, where lit more than one vibrating period was acquired in the osci scope. The experimental input is the sinusoidal applied v age V ~curve 1 in Fig. 4!. In the example, the frequenc n5125 Hz and the amplitudeV0520 mV. The measured current i 052.36 mA. The amplitudex0 is determined by counting the number of peaks~complete cycles! in a half- period of theVD(ac) signal~curve 2 in Fig. 4!. This number can be underestimated by a maximum of 1 peak, so h precision demands a large number of peaks to be coun Fig. 4. Typical figure observed on the oscilloscope screen. Curve 1 sh the applied voltageV, and curve 2 shows the ac component of the detec output voltage,VD(ac). Each complete cycle~or peak! of theVD(ac) curve corresponds to a speaker cone displacement ofl/2, or 0.325mm. The am- plitude of the vibration is obtained by counting the number of peaks in ~or half! period of vibration. The arrows on the time axis show the locatio that correspond to instantaneous zero velocity,u50 ~and the positionx 56x0). 1124 Am. J. Phys., Vol. 71, No. 11, November 2003 ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 ll , d- in- - y. f e - t- h d. All our measurements involved more than 25 peaks in half-period (n/2.25), sox0 was always calculated with a precision better than 4%. Three arrows mark on the horiz tal axis of Fig. 4~time axis: 1 ms/division! the locations that correspond to instantaneous zero velocity,u50 ~and the po- sition x56x0). Note that the amplitude of theVD signal drops as the Doppler shift~and the velocityu! increases due to the finite frequency bandwidth of the photodetector. It not a problem as long as the maximum Doppler shift is far beyond bandwidth. If desirable, larger bandwidths m be achieved at the expense of lower responsivity by reduc the resistance of the feedback resistorRF of the photodetec- tor. In Fig. 4, the number of complete cycles~or peaks! in a half-period of vibration isn/2528, which results inx0 5nl/854.6mm. Thus atn5125 Hz, x0 / i 051.93 mm/A, andV0 / i 058.47V. The current normalized amplitude of vibrationx0 / i 0 was measured for frequencies ranging from 25 to 500 Hz, and impedanceZ05V0 / i 0 was measured up to 30 kHz. The r sults are illustrated in Fig. 5. The dots represent the exp mental data whereas the solid curves correspond to the tings with Eq.~5! in Fig. 5~a!, and Eq.~6! in Fig. 5~b!. The fittings were performed using a least-squares routine and uncertainties were estimated by calculating the variance21 From the first fitting @Fig. 5~a!# we get a15Bl/k5(1.15 60.04) mm/A, a25m/k5(6.2360.02)31027 s2, and a3 s r e s Fig. 5. Experimental results~dots! and fitted curves~solid lines!. ~a! Current normalized amplitude of vibration,x0 / i 0 , and~b! amplitude of the imped- ance,Z05V0 / i 0 , as functions of the vibrating frequency. 1124Freschiet al. bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09 a ily g cie n g m e h o nt se u a se th an th e o im t nd lo . I tu ou - od riz in th ge can The ry ode . The ase m- om- ker vi- ler s,’’ F. o- - a t,’’ al ect ker In- 09. ler len that iode .S., 06, ome signal is un d This ar 5b/k5(6.860.3)31025 s. Note that each variable plays different role in the behavior ofx0 / i 0 : At low frequencies x0 / i 0'Bl/k; the frequency of resonance is defined primar by m/k; andb/k acts on the sharpness~quality factor! of the resonance peak. The calculated values ofm/k5a2 andb/k5a3 were then substituted into Eq.~6! to perform the second curve fittin @Fig. 5~b!#, resulting in a45R5(7.8860.14)V, a55L 5(0.1560.02) mH, anda65(Bl)2/b5Rb5(29.160.4)V. Note that at low frequenciesZ0'R; at resonanceZ0 is domi- nated by the motional impedance and at high frequen Z0'vL. We checked that the last term of Eq.~6!, which has (Bl)2/k5Lk in the numerator, does not play a significa role in the fitting, so thatLk was better calculated by makin Lk5a3a6 and the capacitanceCm5a2 /a3a6 . From the quantities reported above we getBl5a3a6 /a1 . All the pa- rameters that characterize the speaker system are sum rized in Table I. It is important to mention that some care is required wh using a diode laser as the light source in interferometry. T main advantages of such lasers are their low cost and c pactness, which make them well suited for incorporation i undergraduate teaching labs, particularly when several la are required at one time. On the negative side, we have fo that our diode laser is more unstable than HeNe lasers. C must also be taken to avoid light returning back to the la cavity. In addition, small changes in the temperature or injection current cause a variation in the laser spectrum, sometimes, two or more longitudinal modes may exist in laser beam.22 As a result, the output voltage of the photod tector, VD , may become noisy. In practice, we have n found this to be much of a problem because of the short t required for each measurement point and the ability change laser parameters such as the injection current a the temperature. The goal is to obtain a well-defined curve on the oscil scope screen that allows the number of peaks,n, to be counted. For that we typically had to wait a few seconds desirable, an ultrastable current supply and tempera controller can be mounted to produce a frequency-stable put beam from the laser diode.23,24 However, satisfactory re sults were obtained using the simple commercial laser di module. V. CONCLUSIONS We have shown that a speaker system can be characte by measuring three quantities as functions of the vibrat frequency: the amplitude of the electrical current through speaker coil terminals, the amplitude of the applied volta and the amplitude of the vibration~displacement!. Ordinary Table I. Computed values of the speaker system~MKS units!. Definitions of the parameters are given in the text and are summarized in Figs. 1 an Bl ~T3m! b ~N3s/m! m (31023 kg) k (3103 N/m) 1.7360.16 0.1060.02 0.9460.13 1.5060.19 R ~V! L (31024 H) Cm (31026 F) Lk (31023 H) Rb ~V! 7.8860.14 1.560.2 316619 1.9860.12 29.160.4 1125 Am. J. Phys., Vol. 71, No. 11, November 2003 ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 s t a- n e m- o rs nd re r e d e - t e o /or - f re t- e ed g e , electrical devices, such as multimeters or oscilloscopes, be used to measure the current and the applied voltage. amplitude of the vibration is evaluated by using a ve simple laser vibrometer operated by an inexpensive di laser at the wavelengthl50.65mm. The vibrometer is com- pact, robust, and also easy to assemble and disassemble technique has been experimentally verified on a simple c study, a small low-power loudspeaker. The structural para eters of the speaker were determined, as well as the c ponents of an analog all-electrical network to the spea system. ACKNOWLEDGMENTS This work was supported by FAPESP—Fundac¸ão de Am- paro àPesquisa do Estado de Sa˜o Paulo, Brazil~Proc. 97/ 13231-6! and CNPq—Conselho Nacional de Desenvol mento Cientı´fico e Tecnolo´gico, Brazil ~PROFIX, Proc. 540294/01-2!. a!Electronic mail: afreschi@rc.uncsp.br 1L. E. Drain,The Laser Doppler Technique~Wiley, New York, 1980!. 2F. Durst, A. Melling, and J. H. Whitelaw,Principles and Practice of Laser- Doppler Anemometry~Academic, New York, 1981!, 2nd ed. 3B. E. Truax, F. C. Demarest, and G. E. Sommargren, ‘‘Laser Dopp velocimeter for velocity and length measurements of moving surface Appl. Opt. 23, 67–73~1984!. 4A. A. Freschi, A. K. A. Pereira, K. M. Ahmida, J. Frejlich, and J. R. Arruda, ‘‘Analyzing the total structural intensity in beams using a hom dyne laser Doppler vibrometer,’’ Shock Vib. Dig.7 ~5!, 299–308~2000!. 5T. D. Nichols, D. C. Harrison, and S. S. Alpert, ‘‘Simple laboratory dem onstration of the Doppler-shift of laser-light,’’ Am. J. Phys.53 ~7!, 657– 660 ~1985!. 6R. H. Belansky and K. H. Wanser, ‘‘Laser Doppler velocimetry using bulk optic Michelson interferometer: A student laboratory experimen Am. J. Phys.61 ~11!, 1014–1019~1993!. 7T. J. Belich, R. P. Lahm, R. W. Peterson, and C. D. Whipple, ‘‘Optic Doppler measurements,’’ Am. J. Phys.65 ~3!, 186–190~1997!. 8K. Dholakia, ‘‘An experiment to demonstrate the angular Doppler eff on laser light,’’ Am. J. Phys.66 ~11!, 1007–1010~1998!. 9L. E. Kinsler and A. R. Frey,Fundamentals of Acoustics~Wiley, New York, 1962!, 2nd ed. 10R. P. Feynman, R. B. Leighton, and M. Sands,Lectures on Physics ~Addison-Wesley, Reading, MA, 1977!, 6th ed. 11G. L. Rossi and E. P. Tomasini, ‘‘Vibration measurements of loudspea diaphragms by a laser scanning vibrometer,’’ Proceedings of the 13th ternational Modal Analysis Conference, Nashville, 1995, pp. 1205–12 12G. M. Revel and G. L. Rossi, ‘‘Sound power estimation by laser Dopp vibration measurement techniques,’’ Shock Vib. Dig.5 ~5–6!, 297–305 ~1998!. 13E. Hecht,Optics ~Addison–Wesley–Longman, 1998!, 3rd ed. 14Part ALA12-3.5G-650, Creative Technology Lasers, 5057 Heather G Lane, Concord, CA 94521-3092,^www.laser66.com&. The price per laser is approximately US$40. We emphasize that regardless of the fact satisfactory results were obtained using this laser, other types of d lasers may work as well, and our choice might not be the best one. 15Part D32,505, Edmund Scientific~Industrial Optics Division!, 101 East Gloucester Pike, Barrington, NJ 08007-1380, U ^www.edmundoptics.com&. 16Part OPT210P, Burr-Brown, 6730 S. Tucson Blvd., Tucson, AZ 857 U.S., ^www.burr-brown.com&. 17CODIT—Silver ~Code: 7210!, 3M Worldwide, ^www.3m.com&. This ink has immersed on it a large amount of spheres, with diameters of s tenths of micrometers. The surface treatment increases the detected due to the high reflectivity of this ink and because the incident light backreflected over a small angle. 18Model CFG253 Function Generator, Tektronix, 14200 SW Karl Bra Drive, P.O. Box 500, Beaverton, OR 97077, U.S.,^www.tek.com&. 19Model TX3 Digital Multimeter, see Ref. 18. 20Model TDS420A Digitizing Oscilloscope, see Ref. 18. 2. 1125Freschiet al. bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09 th J he r- This ar 21A. Papoulis,Probability, Random Variables, and Stochastic Processes~IE- McGraw–Hill, UK, 1991!, 3rd ed. 22R. A. Boyd, J. L. Bliss, and K. G. Libbrecht, ‘‘Teaching physics wi 670-nm diode lasers—experiments with Fabry-Perot cavities,’’ Am. Phys.64 ~9!, 1109–1116~1996!. 1126 Am. J. Phys., Vol. 71, No. 11, November 2003 ticle is copyrighted as indicated in the article. Reuse of AAPT content is su 200.145.3.46 On: Thu, 1 . 23C. C. Bradley, J. Chen, and R. G. Hulet, ‘‘Instrumentation for t stable operation of laser diodes,’’ Rev. Sci. Instrum.61 ~8!, 2097–2101 ~1990!. 24K. G. Libbrecht and J. L. Hall, ‘‘A low-noise high-speed diode laser cu rent controller,’’ Rev. Sci. Instrum.64 ~8!, 2133–2135~1993!. e nineteen liquid s ppa h the v Free Fall Apparatus. The Wm. Gaertner Company of Chicago was usually regarded as a source for good-quality optics apparatus, but in th twenties it made a line of general laboratory equipment, such as this free-fall apparatus. The glass plate at the bottom was covered with whitehoe polish, and allowed to fall past the electrically-maintained tuning fork of known period that can be seen, tines downward, in the upper half of the aratus. A stylus attached to one tine scratched a trace down the shoe polish layer. From the trace a record of position vs. time could be obtained, from whicalue of ‘‘g’’ could be found. The apparatus is in the Greenslade Collection.~Photograph and notes by Thomas B. Greenslade, Jr., Kenyon College! 1126Freschiet al. bject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 6 Jan 2014 13:49:09