Experimental demonstration of Doppler spectral broadening using the PC sound card

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1 APPARATUS AND DEMONSTRATION NOTES Jeffrey S. Dunham, Editor Department of Physis, Middlebury College, Middlebury, Vermont This department welomes brief ommuniations reporting new demonstrations, laboratory equipment, tehniques, or materials of interest to teahers of physis. Notes on new appliations of older apparatus, measurements supplementing data supplied by manufaturers, information whih, while not new, is not generally known, prourement information, and news about apparatus under development may be suitable for publiation in this setion. Neither the Amerian Journal of Physis nor the Editors assume responsibility for the orretness of the information presented. Submit materials to Jeffrey S. Dunham, Editor. Experimental demonstration of Doppler spetral broadening using the PC sound ard A. Azooz Department of Physis, College of Siene, Mosul University, Mosul, Iraq Reeived 28 November 2005; aepted 29 September 2006 The Doppler broadening of audio spetral lines is demonstrated using a simple experimental setup. The sound from a vibrating loudspeaker is reeived by a mirophone that is onneted to a omputer sound ard. A data-aquisition program is used to aquire the sound data. Fourier analysis is done to obtain the spetral power density of the sound reeived by the mirophone. The results demonstrate spetral broadening similar to that whih takes plae in atomi and moleular spetrosopy. The experiment demonstrates both the Doppler effet and spetral broadening in the undergraduate physis lab Amerian Assoiation of Physis Teahers. DOI: / I. INTRODUCTION The Doppler effet assoiated with sound propagation is well known. Consider a soure of sound S with frequeny f that moves with veloity v s relative to a laboratory. The frequeny f heard by an observer O who moves with veloity v 0 relative to the laboratory is given by f = v O v S f, where is the speed of sound. The sign onvention is to onsider v 0 and v s as positive if their diretions are parallel to the diretion of wave propagation and negative if they move opposite to the diretion of wave propagation. If the observer is stationary, Eq. 1 redues to f = f. 2 v S An experiment for demonstrating this effet using a PCsound ard as a data aquisition tool has been desribed by Bensky and Frey. 1 The experiment involves the reording of sound from a moving loudspeaker that is mounted on an air-trak art. A stationary mirophone reeives the sound from the moving loudspeaker. The sound is reorded and later analyzed by a omputer to determine the frequeny hanges due to the Doppler effet. An interesting onsequene of the Doppler effet is the Doppler broadening of spetral lines. If a gaseous soure is 1 at high temperatures, a spread in the frequeny of a given transition is produed by the atoms randomly oriented veloities with respet to a spetrograph. 2 For simpliity, we assume that a stationary soure and a spetrograph are separated by a distane D. Let us further assume that the soure is osillating with a frequeny f 0 and onstant amplitude A 0. The instantaneous value of the wave amplitude A t reeived by the spetrograph an be written as A t = A 0 sin 2 f 0 t. If the soure vibrates mehanially along the line between S and O with frequeny and amplitude X about its original stationary position x=0, the soure position x t at any time is given by x t = X sin 2 t, and its veloity v S is v S = dx =2 X os 2 t. dt 5 For this situation, Eq. 3 beomes, using Eq. 2, A t = A 0 sin 2 f X os 2 t. This result indiates that a single frequeny f 0 emitted by the soure S when deteted by the observer or the spetrograph will not be a sharp line at frequeny f 0. Instead, the observed Am. J. Phys. 75 2, February Amerian Assoiation of Physis Teahers 184

2 spetrum will extend over the range between the upper and lower limits f 0 2 X / 2 f 0 f X / 2 In Eq. 7 the root-mean-square value of the soure veloity is substituted for the atual instantaneous veloity in Eq. 6. It is lear that the broadening effet is related to both the soure osillation amplitude and the soure osillation frequeny. One further effet that needs to be onsidered is that in pratie, no real measurement instrument an reprodue a sharp, delta-funtion-like spetrum. A finite width is always assoiated with a measured peak in a frequeny spetrum. Real spetral analysis instruments produe peaks with finite values of the full-width-at-half-maximum FWHM. For a soure at a single well-defined frequeny, the value of the FWHM of the deteted peak in the frequeny spetrum is a measure of the instrument resolution. It is ommon to desribe the spetral power-density of the peaks obtained by a spetrograph in a wide range of physis appliations by Gaussian shapes of the form P f = Ke f f 0 2 / 2, 8 where P f is the power density at a partiular frequeny f, K is a normalization onstant, f 0 is the frequeny at the peak, and defines the width of the peak. If we use Eq. 8, we find the full-width-at-half-maximum f to be f = 1 2 ln 2. 9 Equation 9 defines the width of a spetral peak for a stationary soure. The parameter is determined by the resolution of the instrument. An approximate expression for the FWHM of a measured peak due to the ombined effets of instrumental resolution and Doppler broadening an be obtained by adding another term to Eq. 9. This term is equal to the differene between two mean frequenies of the vibrating soure as deteted by the spetrograph. The first is the mean frequeny deteted during the first half of the vibration yle when the soure is approahing the spetrograph, f 0 / 1 Xv/ 2. The seond is the mean frequeny deteted during the seond half of the vibration yle when the soure is moving away from the spetrograph, f 0 / 1+ av/ 2. To a first approximation, the new FWHM an be written as f D = 1 2 ln 2 + f 0/ 1 X / 2 f 0 / 1+ X / 2, whih an be simplified to yield f D = 1 2 ln 2 + 2f 0 X / 2 1 X / Fig. 1. General view of the experimental setup showing the loudspeaker LS and the mirophone MIC. about 3250 Hz. This frequeny produes optimum mathing between the osillator and the loudspeaker. The loudspeaker is rigidly fixed to one end of a vertial metal ruler-bar. The other end of the bar is rigidly fixed to the laboratory benh. This setup allows the speaker to vibrate along the diretion of the mirophone, whih is positioned about 30 m away from the stati position of the speaker. The mirophone is onneted diretly to the omputer sound ard in the usual manner. The Matlab data aquisition toolbox is used to aquire data from the sound ard. The sound ard operates as an analog-to-digital onverter. A Matlab program was written to aquire the sound data and perform the fast Fourier transformation. The program listing is given in Appendix A. III. EXPERIMENTAL PROCEDURE The purpose of the experiment is to demonstrate the effet of the speaker s speed on the frequeny spetrum of the sound reeived by the mirophone Doppler broadening. The first step is to hek the instruments reliability by turning on the speaker, running the program, and analyzing the signal reeived by the mirophone over the entire frequeny range. The frequeny spetrum is shown in Fig. 2. A single peak appears at 3250 Hz. Beause our wave generator an produe a saw tooth waveform in addition to a sinusoidal II. EXPERIMENTAL SETUP AND SOFTWARE The following experiment demonstrates this broadening effet using an ordinary general-purpose omputer soundard and a small loudspeaker attahed to the end of a vibrating metal rod. The experimental setup is shown in Fig. 1. A 0.5 W loudspeaker is onneted to a Metrix BF 817 wave generator. The frequeny of the wave generator is set to Fig. 2. Plot of the signal power density vs frequeny for a 3250 Hz sinusoidal signal applied to the line-in port of the sound ard. 185 Am. J. Phys., Vol. 75, No. 2, February 2007 Apparatus and Demonstration Notes 185

3 Fig. 3. Plot of the signal amplitude vs frequeny for a 3250 Hz saw tooth signal applied to the line-in port of the sound ard. This signal ontains a seond harmoni omponent and a third harmoni omponent at 6500 and 9750 Hz, respetively. one, a further hek is performed. This hek involves applying the saw tooth waveform to the speaker. The spetrum of this signal is shown in Fig. 3 where we have plotted the wave amplitude instead of the wave power density. The amplitude is proportional to the square root of the power density. The relative heights of the 3250, 6500, and 9750 Hz harmoni peaks are 1, 0.5, and 0.33, respetively, and are equal to the Fourier harmoni oeffiients of the saw tooth wave form. 3 One the reliability of the system is established using a stationary speaker, we an do the vibrating speaker experiment. We first selet the proper frequeny window of interest to study the peak in more detail. For this purpose, the software is set to plot the data in the frequeny range Hz instead of the entire audio frequeny range. The software is modified to run for five suessive times by putting the whole program inside a loop see Appendix B. The rod holding the speaker is displaed a ertain distane and allowed to vibrate and the program is run. Plots of the suessive peaks will start to appear on the sreen. The first peak will be the one with the greatest width. Suessive peaks will show smaller widths beause they represent lower root-mean-square speeds of the loudspeaker due to damping of the speaker s motion. The average time interval between the time of suessive peaks is approximately equal to the time the omputer needs to aquire the sound data. This time is equal to the number of data points aquired during eah run divided by the sampling rate and is typially of the order of 0.1 s negleting CPU time. The results of two experiments done with 500 and 1000 data points are shown in Figs. 4 and 5. The effet of soure vibration is lear in both figures. By omparing the two ases we see that the redution in the peak broadening rate is smaller when the sampling rate is 500 samples per seond. The time interval between any two suessive peaks is approximately s. For the ase shown in Fig. 5, this time is s. This latter ase orresponds to a greater degree of damping and thus has greater differenes in the root-mean-square of the soure veloity, resulting in greater differenes between suessive peak widths. To study the damping of the vibrations, the logarithm of the FWHM of the power density is plotted against time, as shown in Fig. 6. The plot an be fitted to a straight line with Fig. 4. Suessive peaks of a 3250 Hz sinusoidal signal. The outermost peak represents zero time and the other peaks are separated by s. The innermost peak represents a time of s. The redution in peak width is due to the vibrational damping of the loudspeaker. slope = 1.8/s. This plot shows that the vibration damping is exponential in time as expeted for ordinary fritional damping. 4 The slope orresponds to a redution of the vibration amplitude of 99% to almost a standstill over a duration of 2.6 s. Although we have no experimental means to onfirm this value aurately, visual observations suggest that suh a redution is taking plae over a period of a few seonds. The interept of the fitted line has no physial importane beause it is a measure of the initial displaement of the rod X. A further illustration of Doppler broadening an be seen by aquiring a data set over a time that is so short that damping may be negleted. The speaker is displaed by a measured distane from its equilibrium position and then released at the same moment when the program starts to aquire data. The root-mean-square veloity in this ase is proportional to the initial displaement. This proedure will produe a greater amount of broadening for large displaements. The results for initial displaements of 1, 2, 3, 4, and Fig. 5. Results of the same 3250 Hz peak analysis of Fig. 4 using time intervals of s. The innermost peak represents a time of 0.5 s. 186 Am. J. Phys., Vol. 75, No. 2, February 2007 Apparatus and Demonstration Notes 186

4 Fig. 6. Plot of the natural logarithm of the full-width-at-half-maximum for the peaks in Fig. 4 as a funtion of time. The solid straight line represents a linear fit to the data. The fitted line has a slope of 1.8/s, whih represents the damping fator. Fig. 8. Plot of the full-width-at-half-maximum for the peaks in Fig. 7 vs the orresponding initial displaements. The solid line represents a fit of the data of Fig. 7 to Eq m are plotted in Fig. 7. This type of experimentation an be triky when performed by one student. Two students an obtain better results if they synhronize the two events. The FWHM of the peaks for the five different initial displaements are plotted against the rod displaement in Fig. 8. The initial rod displaement is proportional to the vibration amplitude X. If we keep the rod vibration frequeny onstant, Eq. 11 an be approximated for / 2 1 as f D = 1 ln 2 + 2f 0BX 2 1 BX ln 2 + BX + B3 X 3, 12 B 4f The data in Fig. 8 are fitted to Eq. 12 and good agreement between the fitted urve and experimental results is observed. The fit value of 1 2 ln 2=32.5 Hz orresponds to the peak width for a stationary soure, whih is a measure of the instrument resolution the sound ard in this ase. We an estimate the rod vibration frequeny using the value of B= in Eq. 14. We substitute 331 m/s for the speed of sound and f 0 =3250 Hz for the osillation frequeny and find that the estimated value of the rod vibration frequeny is 0.9 Hz. This value is onsistent with visual observation of the vibrating rod. However, no independent measurement of this frequeny was done beause of the need for instrumentation that is not available to us. IV. EXPERIMENTAL DETAILS A few experimental hints are worth mentioning here. Better results are obtained using a loudspeaker with a smaller diameter so that the point soure beomes a better approximation. The experiment does not have to be done as we have desribed, where both data aquisition and analysis are arried out at the same time. Instead, the sound an be reorded on a hard disk using ordinary reording software and replayed at a later time for analysis. The analysis does not have to be done using Matlab and other data aquisition software an be used. Students are enouraged to write their own software using the language of their hoie. APPENDIX A: MATLAB M-FILE FOR AUDIO SPECTRAL ANALYSIS Fig. 7. Suessive peaks of a 3250 Hz sinusoidal signal with different initial displaements of the loudspeaker. Peaks orrespond to displaements of 5, 4, 3, 2, and 1 m starting from the outermost peak to the innermost peak. Greater initial displaements result in larger spetral broadening. The following software performs the following operations. 1 Aquire a preset number of data points from the omputer sound ard at a sampling rate of 8000 samples per seond statements 1 8. The number of data points is set by the samples per trigger statement. The time taken by the proess is this number divided by the sampling rate. 2 Perform a fast Fourier transformation on the data and alulate the spetral power density over 512 hannels statements Assign frequenies to these hannels. The overall frequeny range seleted is 0 12 khz statement Selet the frequeny range to be plotted statements The software is programmed to normalize all peak heights to a maximum of unity to make omparisons easier. 187 Am. J. Phys., Vol. 75, No. 2, February 2007 Apparatus and Demonstration Notes 187

5 APPENDIX B: MATLAB M-FILE FOR PERFORMING THE DOPPLER BROADENING EXPERIMENT This software is a modifiation of the audio spetral analysis program in Appendix A. A loop is added so that five suessive measurements are made and the time between suessive runs is determined by the variable spt to determine the number of data aquired in eah run. The third modifiation selets the frequeny window to be plotted. Here we use a window in the neighborhood of the 3250 Hz peak. 1 T. J. Benskya and S. E. Frey, Computer sound ard assisted measurements of the aousti Doppler effet for aelerated and unaelerated sound soures, Am. J. Phys. 69, D. Halliday and R. Resnik, Physis I&II Wiley, NY, 1966, p R. E. Sott, Linear Ciruits Addison-Wesley, Reading, MA, 1960, p Grant R. Fowles, Analytial Mehanis Holt, Rinehart and Winston, 1962, p Am. J. Phys., Vol. 75, No. 2, February 2007 Apparatus and Demonstration Notes 188