Musical Wind Instrument Analysis

Transcription

1 Musical Win Instrument Analysis Darren, Murray Campbell Department of Acoustics an Flui Dynamics, University of Einburgh, EH9 3JZ, Einburgh, Scotlan The acoustic impeance is a valuable parameter in etermining the acoustical behaviour of musical win instruments. It can also be use to give a reconstruction of the internal structure of the instrument. An extremely accurate reconstruction, however, is only possible if the impeance is viewe over a large range of frequencies, typically the orer of khz own to aroun 5 Hz. The high frequency components are require for resolving any small etails of the internal structure whereas low frequency content is important since its absence can lea to uner-preictions in the reconstructe imensions. The metho employe to measure acoustic impeance uses a two microphone system that measures the air pressure at two separate points along a cylinrical measurement uct. Instea of using the more common pulse excitation signal, a staning wave is set up in the uct. From the ratio of the two signals etecte by the microphones the impeance of the instrument can be erive. Previous work in this fiel has investigate the upper range of frequencies an goo agreement has been establishe between an. New work has been carrie out at frequencies from 1 khz own to 5 Hz, again revealing goo results. 1 Overview The general metho use in this paper is referre to as the Two-Microphone-Four-Calibration or TMFC metho an has been stuie in etail by van Walstijn [1]. The object we wish to stuy is couple to a cylinrical tube in which the microphones sit, referre to as the measurement uct. The plane at which the measurement uct an object meet is referre to as the reference plane. It is at this site that the input impeance of the object is erive from the ratio, y, of the microphone signals. In this paper, the object use is a close cylinrical tube of length L = 128 mm an raius equal to that of the measurement uct. Wave propagation in the system is linear an it is assume only the planar moe component is present. Various precautions are taken to ensure that this, to the closest approximation, is inee the case. Once the input impeance is obtaine, the closely relate reflection function can be calculate. An inverse Fourier transform can be performe on the ata to give its time omain representation, referre to as the input impulse response (IIR) of the object. A reconstruction algorithm can then be applie to give the bore profile of the tube. 2 Introuction to the TMFC metho The basic layout of the TMFC system is epicte in Figure 1. The river generates a plane wave which traverses the measurement uct. The wave propagates past the reference plane an into the tube to be stuie. The tube is blocke at the en by a har wall which presents an infinite impeance to the wave causing it to be fully Amplifier Driver s2 Computer DAQ measurement uct x2 x1 s1 Calibration object Reference plane En cap Figure 1: Sketch of the stanar TMFC system. Microphone one an two receive the signal an respectively an are separate by a istance, -. reflecte. This allows a staning wave to buil up in the system. The microphones in the uct respon to the fluctuations in air pressure an generate the signals enote an. From the ratio of these two signals the impeance corresponing to that particular frequency can be obtaine. In this case, the system has been use to look at frequencies from 5 Hz up to 2 khz giving the impeance curve for the close 128mm tube. 3 Theory 3.1 Higher moes As mentione, wave propagation in the uct is assume to be of planar form. Higher moes, however, can be excite for various reasons. Suen changes in cross section of the object uner stuy, for example, finger holes in win instruments or any non cylinrical portion of the ob- 671

2 Forum Acusticum 25 Buapest (a) Figure 2: The antisymmetric (1,) moe of excitation represente by (a) pressure pattern (b) flow pattern. The positive area in (a) represents the region where the pressure is 18 egrees out of phase with the negative region max s2 s1 s2 s2 max s1 max s1 min (b) y unefine y=1 (s1=s2) y= 1 (s1= s2) Figure 3: Sketch of possible staning waves that can be set up insie the measurement uct. ject will generate higher moes. Other sources of higher moes inclue the small holes in the measurement uct where the microphones sit an structural imperfections, for example, where the two tubes are joine together. There is an intimate relationship between the raius of the uct, r, an what is referre to as the cut-on frequency, cut on. In general, the expression for the cut on frequency of a cylinrical tube is given by cut on (1) At frequencies below the cut on frequency the only moe of propagation that occurs is the planar moe. All higher moes are evanescent i.e. their wave numbers are imaginary hence they are attenuate rapily an their propagation can be neglecte insie the uct. At frequencies above the cut on, higher moes can propagate. As an example, the first higher moe encountere at frequencies above the cut on is known as the antisymmetric (1,) moe as illustrate in Figure 2. As we can see, cut on is inversely relate to the measurement uct raius so the thinner we make the measurement uct the larger the range of frequencies we can consier, without having to eal with higher moes of propagation. For the system use in this paper r = 4.9 mm which correspons to a cut on frequency of 2.5 khz. A branch of acoustics known as multimoal has been evise to cope with higher moes but this topic will not be ealt with in etail in this paper. There are, however, important points to take from the. To avoi higher moes of propagation the length of the object we wish to stuy must be at least twice its own iameter, it must be cylinrical an the length of the measurement uct must be at least twice its own iameter. 3.2 Singularities Consier the first case in Figure 3. In general, the metho involves setting up a staning wave insie the measurement uct. One situation that can occur is that when the istance between the microphones is equal to half the wavelength, two of the noes of the staning wave can sit irectly uner both microphones. In this case the microphone signals ten to zero an the signal ratio becomes ill efine. In the secon an thir case the microphone istance is again equal to half the wavelength an the signals an have the same magnitue, but in the secon case they have equal phase an in the thir case they have opposite phase. In these situations the magnitue of the signal ratio is 1 irrespective of the impeance of the object uner stuy. The frequencies at which this effect occurs are known as singular frequencies or singularities. They occur when the wavelength is an integer multiple of twice the microphone separation istance (2) These frequencies must be avoie when carrying out the s. It has been shown that the most suitable region of the frequency spectrum to work in is between.3 an.8 times the first singular frequency #" [1]. For example, a microphone separation istance of 7 mm correspons to a first singular frequency of about 25 khz. This in turn gives a suitable banwith of 7 to 2 khz. Observing lower frequencies requires a larger microphone separation. 4 Theory versus Current work has involve using two slightly ifferent versions of the TMFC metho. One of these is the full TMFC metho which completely ecouples from 672

3 e $ $ Forum Acusticum 25 Buapest. This has been use in the high frequency range by van Walstijn [1] an the results obtaine via this metho have been reprouce for this paper. However, unfeasibly large calibration tubes woul be require to use the metho at the low en of the frequency spectrum. This is where the partial TMFC metho is use. For this approach, some of the information require in the equations is fille in via well ocumente plane wave. 4.1 Full TMFC metho basic measurement uct L L reference plane Figure 4: The four calibration measurements carrie out using TMFC: 1. Negative length measurement 2. Terminate at the reference plane 3. Positive length measurement 4. Double length measurement. In its simplest form the acoustic impeance can be efine as 2L $ &% ' (3) where p an U are the pressure an the volume velocity at the plane of interest respectively. It can be shown [1] that the expression for the ally etermine form of the normalise input impeance is given by ( $ *) ),+.-/ (4) where $;: is the characteristic impeance of the tube an y is the microphone signal ratio for the object uner stuy. A, B an C are complex coefficients. They can be preicte by an take the form - ==>@?BA CDFEHG #I JLK?BDMEHG IN (5) O QP >?RA CDFESG I?RA CDFESG #I N (6) T P > JUK?RDMESG 4I JUK?RDMESG IN (7) where an are the istance of microphones from the reference plane (see Figure 1) an G is the ratio of the transucing characteristics of microphone one to microphone two. A, B an C can also be obtaine ally via four calibration measurements. Consier Figure 5. We efine the reference plane to be a istance L from the en of the measurement uct. If we block the uct off, as shown in 1, in a mathematical sense we have a calibration tube of length negative L. The measure signal ratio for this calibration tube is VW XZY. In 2 the uct is terminate at the reference plane. The signal ratio is V[. In 3 an 4 the calibration tubes are of length L an 2L respectively. The signal ratio is V\U] an V respectively. Consier the measurement uct close off at the reference plane: the impeance at this plane must go to infinity. From observation of equation (4) the enominator must go to zero. If we enote the signal ratio obtaine by V [ we fin T V [ (8) Determination of the B coefficient requires a little more thought. The theoretical expression for the input impeance of a tube of length L is given by where G is the propagation constant. ( $_^S` JUKa DbESGFc I (9) Note also that the hyperbolic cosine function is o JUK?RDFE Gbc I = JUK?RD6EHGFc I (1) ( If we have a positive length tube +L of impeance ( an a negative length tube -L of impeance W XY then it follows that ( $ _e ( $ W XZY Qf (11) But what oes a negative length mean? It means we efine the reference plane to be a istance L from the en of the measurement uct (see Figure 4). It follows that -hg > O V \U] i T V4 N > V W XY V W XY O T N@j.f (12) 673

4 y ( ( E E j V Forum Acusticum 25 Buapest Substituting expression (8) into the above an rearranging gives O 1RklmZ1on#prqL361osut1RklmwvF1n phqx t1 klm vm1 n phq x3 1 s (13) Finally the A coefficient. The reflection coefficient of a close tube of length L is zu{, So for a tube of length 2L this is an so y QzL{, y y Gbc I (14) GFc I (15) (16) The reflection coefficient can also be expresse in terms of the input impeance y $ $ e Substituting (17) into (16) an rearranging gives $ ( $ ( $ ( \U] _e f (17) (18) We can use equation (4) to write the above in terms of A, B, C an y. After some rearranging this gives -. g / 1 kolrm 3}5 1 klm 3}7~8 / 1o klm 3}5 1 klm =/ 1 klm klm B (19) For best results the length L shoul be chosen as half the microphone separation [1]. The TMFC metho is alreay known to work well in the frequency range from 1 khz up to 2 khz as shown by van Walstijn [1]. However, to obtain a goo reconstruction it has been shown by Li [2] that the frequency range must be taken own to at least 5 Hz. Investigating lower an lower frequencies becomes increasingly ifficult since singular frequencies are brought closer together hence the banwith suitable for analysis is reuce. Driver s2 s1 measurement uct x2 x1 connector reference plane absorbing foam anechoic tube Figure 5: Schematic of measurement uct connecte to an anechoic termination at the reference plane. a foam wege place at the en. In this proceure there are no reflections to consier an so the wave propagation is simple an the microphone signal ratio takes the form.p zu{, ESG I (2) To account for the iniviual characteristics of the microphones the anechoic measurement is repeate with the microphones switche in position. If V an V are the signal ratios of the first an repeate measurement respectively, then we have V.PƒzL{, V P zl{, Diviing (21) by (22) an rearranging gives EHG EHG I (21) I (22) P= 1L 1 (23) This expression for G can then be plugge into (6) an (7), an are known lengths an the theoretical propagation constant is use for G. We now have expressions for the A, B an C coefficients reay for substituting into (4). 4.2 Partial TMFC metho For the partial TMFC metho an anechoic measurement is mae to obtain an expression for G. This is one by attaching a very long (effectively infinite) tube to the uct. A large fraction of the waves energy has been attenuate in traversing the tube. Any resiual energy is absorbe by 674

5 Forum Acusticum 25 Buapest 5 Results 3 Z/Z c (B) Z/Z c (B) Z (ras/π) (Z/Z c ) / π Figure 6: Example ata for the magnitue an phase of the input impeance for the close tube using the full calibration metho. Figure 8: Entire ata collecte for the magnitue an phase of the input impeance of the close tube over a frequency range of 5 to 2 Hz with 5 Hz intervals between ata points Z/Z c (B) frequency (Hz) Amplitue (Z/Z c ) / π time (ms) frequency (Hz) Figure 9: Input impulse response for the close tube..7 Figure 7: Example ata for the magnitue an phase of the input impeance for the close tube using the partial calibration metho Amplitue time (ms) Figure 1: Close up of the peaks in Figure

6 Forum Acusticum 25 Buapest 6 actual profile 6 Conclusions raius (mm) axial istance (mm) A metho for obtaining the impeance curve of a small cylinrical tube over a wie frequency range has been emonstrate. Two slightly ifferent approaches have been use to obtain the full set of ata. Goo agreement has been foun between an for both approaches. The input impeance has been constructe over a large banwith an has yiele goo results with the exception of the phase very close to the cut on frequency. An accurate representation of the input impulse response an bore reconstruction for the system has been obtaine. Figure 11: Reconstruction of the close tube. 5.1 Discussion of results Figure 8 shows the impeance magnitue an phase from 5 Hz up to 2 khz. The 5 Hz to 6 khz range was erive using the partial TMFC technique. The full metho was use from above 6 khz to 2 khz. The magnitue agrees well with the theoretical preiction over the entire range. The phase appears to behave well until it approaches 2 khz. The hypothesise reason for this eviation is that the cut on frequency, as mentione previously is close to 2 khz. 5.2 Input impeance to IIR conversion The IIR has been obtaine by calculating the inverse Fourier transform of the frequency omain expression (17) for the reflection coefficient an then taking the real part.the tube is close so we expect a large positive reflection to occur at a time c " giving a value of approximately.8 ms. This is inee the case as shown in Figure 9 an more closely in Figure Bore reconstruction The bore profile of the tube is obtaine by applying a lossy layer-peeling algorithm to the IIR. The result is shown in Figure 11. Zero on the x axis represents the position of the reference plane. The reconstruction remains fairly constant at the correct raius up to approximately 128 mm. There is then a small increase in the reconstructe raius just before the suen large ecrease corresponing to the close en. Gibb s phenomena is believe to be responsible for this. In epth reviews of bore reconstruction have been carrie out by Sharp [3] an Kemp [4]. 7 Future work Potential work to be carrie out in the near future inclues generating a reconstruction for more complex shape objects incluing open an close steppe tubes. From there it is hope the metho can be applie to obtaining the impeance curve an reconstruction of various win instruments. The possibility then opens up for comparison of the metho with the more well known evices for instrument analysis such as APR an the Brass Instrument Analysis System (BIAS). Acknowlegement Thanks to Calum Gray an Michael Newton for enlightening iscussions on programming relate issues. A special thank you to Davi Sharp for his assistance with the bore reconstruction algorithm. This work was fune by E.P.S.R.C. References [1] Maarten van Walstijn, Murray Campbell, Jonathan Kemp, Davi Sharp: Wieban Measurement of the Acoustic Impeance of Tubular Objects. Acustica. 91. (25) [2] Aijun Li, Davi B. Sharp: The problem of offset in measurements mae using acoustic pulse reflectometry. [3] Davi B. Sharp: Acoustic pulse reflectometry for the measurement of musical win instruments. Dissertation. Department of Physics an Astronomy, University of Einburgh, [4] Jonathan Kemp: Theoretical an al stuy of wave propagation in brass musical instruments. Dissertation. University of Einburgh,