Refinements to the Model of a Single Woodwind Instrument Tonehole

Transcription

1 Proceeding of 20th International Sympoium on Muic Acoutic (Aociated Meeting of the International Congre on Acoutic) Augut 2010, Sydney and Katoomba, Autralia Refinement to the Model of a Single Woodwind Intrument Tonehole PACS: Ef,43.20.Mv ABSTRACT Antoine Lefebvre and Gary P. Scavone Computational Acoutic Modeling Laboratory (CAML) Centre for Interdiciplinary Reearch in Muic Media and Technology (CIRMMT) Schulich School of Muic of McGill Univerity 555 Sherbrooke Street Wet, Montreal, QC H3A 1E3, Canada Uing the Finite Element Method (FEM), a ingle unflanged tonehole wa imulated for a wide range of height and diameter in order to improve the accuracy of tranmiion-matrix calculation for intrument with tonehole of large diameter and hort height, a found on axophone and concert flute. Thee calculation confirm the validity of exiting model for tonehole of maller diameter and longer height, a found on clarinet. Revied one-dimenional tranmiion-matrix model of open and cloed tonehole are preented to extend the validity of the model baed on the FEM reult. Further, thee tonehole model are verified to be valid for ue with both cylindrical and conical (flare angle up to 6 degree) air column. For open and cloed tonehole, new formula for the low frequency value of the hunt and erie length correction are developed a a function of t/b and. Dicrepancie with current theorie are particularly apparent in the erie length correction term. At higher frequencie, the open hunt equivalent length increae fater than previouly predicted, corroborating recent experimental data (Dalmont et al., 2002). Thi effect i more important for hort tonehole. Thee reult do not take into account any poible internal or external interaction between the tonehole on an intrument, which may have an important effect for large-diameter tonehole. INTRODUCTION The deign of a woodwind intrument uing computer model require precie calculation of the reonance frequencie of an air column with open and cloed tonehole. Although there have been many theoretical, numerical, and experimental reearch tudie on the ingle woodwind tonehole (Keefe 1982b; Keefe 1982a; Nederveen et al. 1998; Dubo et al. 1999a; Dalmont et al. 2002), it i known that current theorie are not valid if the tonehole height t i horter than the radiu b (ee Fig. 1) becaue in that cae the radiation field and the inner field are coupled (Dalmont et al. 2002). Furthermore, the impact of the conicity of the main bore on the tonehole parameter ha never been tudied. Thi paper decribe improvement to the accuracy of tranmiion-matrix (TM) model of intrument with tonehole of large diameter and hort height, a found on axophone and concert flute, obtained uing the Finite Element Method (FEM). Current theoretical reult from the literature are reviewed and the methodology with which we obtain the TM parameter from FEM imulation i preented. The FEM reult are firt validated with TM method and with available experimental data. New reult are preented that extend the validity of TM parameter for tonehole of dimenion ued in mot wind intrument. The main goal of thi reearch i to obtain an accurate low frequency characterization of unflanged open and cloed tonehole (up to 1 to 2 khz). THEORETICAL RESULTS The TM repreenting a tonehole i defined a: A B T hole =, (1) C D which, when inerted between two egment of cylindrical duct, relate the input and output quantitie: pin pout = T Z 0 U cyl T hole T cyl, (2) in Z 0 U out where Z 0 = ρc/s i the characteritic impedance of the waveguide, ρ i the denity of air, c i the peed of ound in air and S i the cro-ectional area of the waveguide. The tranmiion matrix of a cylindrical duct of length L i: cokl j inkl T cyl =, (3) j inkl cokl where k = 2π f /c i the wavenumber and f i the frequency. Baed on the aumption that Z a /Z 1 (Keefe 1982b, p. 677) and that the tonehole i ymmetric, it TM may be approximated a a ymmetric T ection depending on two parameter, the hunt impedance Z = Z /Z 0 and the erie impedance Z a = Z a /Z 0 (Keefe 1981), which become: 1 Za / Za /2 T hole = 0 1 1/Z = 1 + Z a Z 2Z a (1 + Z a (4) ) 4Z. 1/Z 1 + Z a 2Z ISMA

2 25-31 Augut 2010, Sydney and Katoomba, Autralia Proceeding of ISMA 2010 ymmetry plane A ymmetry plane B = b/a b t a L Figure 1: Diagram repreenting a tonehole on a cylindrical tube. Thi equation wa further implified by Keefe, which replace all occurrence of Z a /Z by zero, an approximation that introduce mall but non-negligible error in the calculation of the reonance frequencie. The impedance Z and Z a mut be evaluated for the open (o) and cloed (c) tate of the tonehole a a function of geometry and frequency. Mathematical expreion for thee impedance are available in the literature and are reviewed below. Open Tonehole Shunt Impedance The open tonehole hunt impedance may be expreed a (Keefe 1982b): Z (o) = jkt (o) + ξ, (5) where ξ i the open tonehole hunt reitance and t (o) the tonehole equivalent length. The hunt reitance doe not influence the calculated playing frequencie of a woodwind intrument and thu, mot reearch effort are concentrated on the determination of the hunt length correction. In the mot recent literature (Dalmont et al. 2002), t (o) i written: k = kt i + tank(t +t m +t r ) (6) where t i the height of the tonehole a defined in Fig. 1, t m i the matching volume equivalent length, t r i the radiation length correction and t i the inner length correction. Nederveen et al. (1998) obtained an accurate approximation for t m : t m = b ( ), (7) 8 where = b/a i the ratio of the radiu of the tonehole to the radiu of the main bore. The term t i and t r are generally difficult to calculate analytically and, in the cae where t i hort, the coupling between the inner and outer length correction prevent their eparate analyi (Dalmont et al. 2002, ec. 2.7). The radiation length correction t r depend on the external geometry; in the low frequency approximation, it may be that of a flanged pipe (0.8216b), an unflanged pipe (0.6133b) or another intermediary value for more complicated ituation. The expreion provided in the literature for the inner length correction t i are ummarized in Table 2. Thee expreion are only valid for tonehole of large height (t > b). Note that there i an error in Eq. (5) of Dalmont et al. (2002), which refer to Eq. (55b) of Dubo et al. (1999a) the correct verion of thi equation i reported here a Eq. 21. In the limiting cae where t 0 and b 0, the low-frequency characteritic of the tonehole are thoe of a hole in an infinitely thin wall (Pierce 1989, Eq ) and the total equivalent length of the hole become: t e = t + (π/2)b. (8) If the tonehole height i large but the radiu b 0, the tonehole equivalent length become: t e = t b b = t b, (9) that i, the length of the tonehole with an unflanged length correction at the radiating end and a flanged radiation length correction inide the intrument. The reitive term ξ influence the reonance magnitude of an intrument but not their frequencie. In thi paper, we focu on tuning conideration and thu do not conider thi term further. Open Tonehole Serie Impedance The erie impedance of the open tonehole i a mall negative inertance: Z (o) a = jkt a (o). (10) No ignificant reitive term wa detected experimentally (Dalmont et al. 2002). Table 1 ummarize the equation found in the literature. In ome publication, the exponent of i 2 wherea the reult from our imulation a well a theoretical calculation by Keefe (1982b) how that the erie length correction depend on 4. Thee equation were corrected in thi paper. Cloed Tonehole Shunt Impedance The hunt impedance of a cloed tonehole behave mainly a a compliance (Nederveen 1998). Thi can be written: Z (c) = j 1 kt (c). (11) The implet expreion for the hunt length correction i that of a cloed cylinder of equivalent volume: kt (c) = tank(t +t m ). (12) An inner length correction may be conidered a well for the cloed tonehole but it influence i mall relative to the cotangent term and become ignificant only at high frequencie (Keefe 1990). A recent expreion including the inner length correction i (Nederveen et al. 1998, Eq. 7): [ ] Z (c) = j kt i cotk(t +t m ), (13) where t i i the ame a for the open tonehole a defined in Eq ISMA 2010, aociated meeting of ICA 2010

3 Proceeding of ISMA Augut 2010, Sydney and Katoomba, Autralia Keefe 1982b, Eq. (68b) Nederveen et al. 1998, Fig. 11 Dubo et al. 1999a, Eq. (74) Dubo et al. 1999a, not numbered t a (o) 0.47b = 4 tanh(1.84t/b) (14) t a (o) = 0.28b 4 (15) t a (o) b = tanh(1.84t/b) (16) t a (o) = ( )b 4 (17) Table 1: Comparion of the expreion for the erie length correction a Nederveen 1998, Eq. (38.3) Keefe 1982b, Eq. (67a) Nederveen et al. 1998, Eq. (40) Dubo et al. 1999a, Eq. (73) i = ( )b (18) i = ( )b (19) i = ( )b (20) i = t (o) t a (o) /4, (21) t (o) = ( )b Table 2: Comparion of the expreion for the inner length correction i Cloed Tonehole Serie Impedance The cloed tonehole erie impedance behave a a mall negative inertance, a for the open tonehole cae. Thi can be expreed a: Z (c) a = jkt a (c), (22) where t (c) a i the erie length correction. Keefe (1981, Eq. 54) propoed: t a (c) 0.47b 4 = coth(1.84t/b) , (23) wherea Dubo et al. (1999b, Eq. 74) calculated the length correction in the ame ituation a: t a (c) b 4 = 1.78coth(1.84t/b) , (24) where we corrected the error in the exponent of (4 intead of 2). INVESTIGATION METHOD In thi ection, we preent a method to calculate the tranmiion matrix T ob j of an object from the FEM. Thi method i ueful to characterize an object that i part of a waveguide, i.e. which ha an input and an output plane. It can be ued to obtain the TM of any type of dicontinuity in a waveguide. One requirement i that the evanecent mode occurring near the dicontinuity mut be ufficiently damped at the input and output plane of the imulated model. In general, cylindrical egment are thu required before and after a tonehole. The tranmiion matrix obtained from the imulation i given by T = T cyl T ob j T cyl where the TM of a cylindrical duct wa defined in Eq. 3. The effect of the cylinder i removed by calculation uing the invere of the cylinder TM: T ob j = T 1 cyl TT 1 cyl. (25) The object under tudy in thi article being a tonehole with TM defined in Eq. 1 and 4, we may extract the two impedance from the finite element imulation reult with: Z = 1/C, (26) Z a = 2(A 1)/C. (27) A tranmiion matrix T contain four frequency-dependant complex-valued parameter relating input quantitie to output quantitie: pin T11 T = 12 pout. (28) Z 0 U in T 21 T 22 Z 0 U out In order to obtain thee four parameter from finite element imulation reult, we need to imulate the problem two time with different boundary condition. By combining the reult for the two imulation cae (ubcript 1 and 2), we can write a ytem of linear equation to olve for the four parameter of the TM: p out1 Z 0 U out1 0 0 T 11 p in1 0 0 p out1 Z 0 U out1 T 12 Z p out2 Z 0 U out2 0 0 = 0 U in p out2 Z 0 U out2 T 21 T 22 p in2 Z 0 U in2 (29) The model of a tonehole on a cylindrical tube i ymmetric (revering the input and output condition lead to the exact ame ytem) and we take advantage of thi feature to olve only one quarter of the geometry. One ymmetry plane i perpendicular to the intrument body axi and i located on the center of the tonehole (A) wherea the econd ymmetry plane i defined by the axi of the intrument body and the axi of the tonehole (B) (ee Fig. 1). On the econd ymmetry plane, the boundary condition i a null normal acceleration (ymmetry). On the firt ymmetry plane we define alternatively a null normal acceleration for the ymmetric cae (cae 1) and a null preure for the anti-ymmetric cae (cae 2). From the value of the preure and normal velocity on the input plane of the model, we can deduce the value on the output plane for both imulation cae: VALIDATION p out1 = p in1, (30) Z 0 U out1 = Z 0 U in1, (31) p out2 = p in2, (32) Z 0 U out2 = Z 0 U in2. (33) The reult of our imulation are compared with the experimental data obtained by Dalmont et al. (2002) and by Keefe (1982a). Dalmont et al. meaured the hunt and erie length correction of a flanged tonehole a a function of frequency for two tonehole on a tube of radiu a = 10 mm: (1) = 0.7, t/b = 1.3 and (2) = 1.0, t/b = Both tonehole were flanged at their open end. We alo compare our imulation reult with data obtained by Keefe, who meaured the hunt and erie length correction for two unflanged tonehole on a cylinder of radiu a = 20 mm: (1) = 0.66, t/b = 0.48 and (2) = 0.32, t/b = The hunt length correction obtained from our imulation i diplayed in Fig. 2 and 3 in comparion to the experimental reult found in the literature. Our imulation reult are in good general agreement with the experimental reult of ISMA 2010, aociated meeting of ICA

4 25-31 Augut 2010, Sydney and Katoomba, Autralia Proceeding of ISMA t (o) [mm] 30 t (mm) t (o) [mm] ka ka Figure 3: Shunt length correction t (o) a a function of ka for the two tonehole tudie by Keefe: = 0.66 and t/b = 0.48 (bottom curve), = 0.32 and t/b = 3.15 (top curve). FEM reult: for = 0.66 (filled circle) and for = 0.32 (filled quare). Experimental data from Keefe (marker with error bar) and theoretical reult with Eq. 6 (dahed) ka Figure 2: Shunt length correction t (o) a a function of ka for the two tonehole tudied by Dalmont et al.: = 0.7 and t/b = 1.3 (top graph), = 1.0, t/b = 1.01 (bottom graph). FEM reult (filled circle), experimental data from Dalmont et al. (olid line) and theoretical reult with Eq. 6 (dahed). Dalmont et al. (2002) but a few intereting obervation are worth mentioning: (1) the experimental data reveal a larger hunt length correction at low frequencie for both tonehole, compared to both the theoretical formula and our imulation reult, which match; (2) the length correction predicted by our imulation reult matche the experimental data for the larger diameter tonehole in the higher frequency range, predicting a larger length correction than the current theory. In the cae of the unflanged tonehole tudied by Keefe (1982a), we found good agreement between the theoretical value, our imulation and hi experimental data for the tonehole of tall height. For the hort tonehole, there are dicrepancie: the experimental data and our imulation reult give larger length correction for the higher frequencie compared to the theory. For the reult in Table 3, our FEM imulation for the maller tonehole ( = 0.7) agree with the value predicted by the theoretical formula but diagree with the experimental value obtained by Dalmont et al. (2002). In their article, Dalmont et al. (2002) ued t a (o) = 0.28b 2, in reference to an article by Nederveen et al. (1998), a a theoretical formula for the erie length correction. A previouly mentioned, we believe the erie length correction varie a b 4. And in fact, Dalmont et al. (2002) found a fairly high error with repect to that meaurement. For the larger tonehole ( = 1.0), our imulation agree with the experimental data provided by Dalmont et al. (2002) and with all of the theoretical formula except Eq. 14 from Keefe. The agreement with the reult in Table 4 i atifactory. RESULTS AND DISCUSSION The ingle open tonehole wa imulated uing the FEM for a wide range of geometric parameter ( = b/a from 0.2 to 1.0 by tep of 0.5, t/b from 0.1 to 0.3 by tep of 0.05 and from 0.3 Tonehole t/b Decription a [mm] FEM 0.50 Dalmont et al. 0.95±0.3 Eq Eq Eq Eq FEM 2.90 Dalmont et al. 2.8±0.3 Eq Eq Eq Eq Table 3: Serie length correction t a (o) in mm. Comparion between imulation, theorie, and experimental data for the tonehole tudied by Dalmont et al. to 1.3 by tep of 0.2 and ka from 0.1 to 1.0 by tep of 0.05 with an additional low frequency point at ka = 0.01). The lowet frequency imulated wa 55Hz. For each of thee parameter, the four term of the tranmiion matrix were obtained and the hunt and erie length correction calculated uing the procedure previouly decribed. For the low frequency value of the hunt length correction (t e ), we were able to obtain a data-fit formula that matche the complete et of reult. There i no data obtained for tonehole with < 0.2 becaue, for uch mall-diameter tonehole, we were not able to obtain precie reult. In order to enure that the data-fit formula be valid for all value of, we added the two theoretical contraint expreed in Eq. 8 and 9. The equation that we obtained i: with t e /b = lim t (o) /b = t/b + [1 + f ()g(,t/b)]h(), (34) k 0 f () = , g(,t/b) = 1 tanh(0.778t/b), h() = ISMA 2010, aociated meeting of ICA 2010

5 Proceeding of ISMA 2010 Tonehole t/b Decription a [mm] FEM 0.78 Keefe 0.8±0.2 Eq Eq Eq Eq FEM Keefe not meaurable Eq Eq Eq Eq Table 4: Serie length correction t a (o) in mm. Comparion between imulation, theorie and experimental data for the tonehole tudied by Keefe. The open hunt impedance a a function of frequency i then evaluated a: Z (o) = j tankt e, (35) which, in the low frequency limit, become Z (o) = jkt (o). Thi expreion work relatively well when ka < 0.2. More work i required to develop a formula that matche the imulation data up to ka = 1. ti/b Augut 2010, Sydney and Katoomba, Autralia 0.1 Figure 5: Comparion of Eq. 36 (olid curve) for the inner length correction i /b with equation from the literature: Eq. 19 (dah-dot), Eq. 20 (dahed), Eq. 21 (dotted). Thi formula i compared with the formula from the literature in Fig. 5. (t (o) t)/b (t (o) t)/b Figure 4: Difference between the hunt length correction and the tonehole height t divided by the tonehole radiu b a a function of : FEM reult for tall (quare) and hort (circle) tonehole. Data fit formula (dotted). Current theory with Eq. 6 (dahed). In Fig. 4, the imulation reult are hown for the two extreme cae of hort (circle) and tall (quare) tonehole a well a the data-fit formula (dotted) and the theoretical Eq. 6. Thi figure how the um of the radiation length correction and the inner length correction. A expected, for the tonehole of hort height, thi length correction i larger than for tall tonehole, becaue the unflanged tonehole ending become gradually flanged by the body of the intrument. Even for the tonehole of larger height, the new data-fit formula doe not match exactly with the current theory, uggeting that the inner length correction found with our imulation i different. We can obtain the inner length correction t i by ubtracting the unflanged pipe radiation length correction t r = b and the matching volume length correction t m, Eq. 7, from Eq. 34 with t : t i /b = (36) kb Figure 6: Difference between the hunt length correction and the tonehole height t divided by the tonehole radiu b a a function of kb for three value of (0.2, 0.5 and 1.0, from top curve to bottom curve) and a value of t/b = 0.1 obtained from FEM imulation (filled circle). Current theory with Eq. 6 (dahed), new reult with Eq. 35 (dotted). The mot important dicrepancy between current tonehole theorie and our imulation reult concern the frequency dependence of the hunt length correction for tonehole of hort height, which i diplayed in Fig. 6 for three tonehole with t/b = 0.1: (1) = 0.2, (2) = 0.5 and (3) = 1.0. For each of thee tonehole, the hunt length correction increae with frequency more than predicted. Equation 35 from thi paper better predict the frequency dependence compared to current theory but dicrepancie remain. Similar reult were obtained by Keefe (1982a) (ee Fig. 3). One conequence of thi behavior i that the higher reonance of an intrument with hort chimney height are lower in frequencie than predicted by the current theory. Thi effect tend to hrink the ratio of higher reonance relative to the fundamental. For conical intrument, thi counteract the natural preading of the reonance that occur in truncated cone. For the low frequency value of the erie length correction, we obtained the following data-fit formula: a /b 4 = f (,t/b)g(), (37) where [ ] f (t/b) = 1 + ( ) 1 tanh(2.666t/b), ISMA 2010, aociated meeting of ICA

6 25-31 Augut 2010, Sydney and Katoomba, Autralia Proceeding of ISMA t a (o) /b t (c) [mm] Figure 7: Serie length correction a /b 4 a a function of. FEM reult: limit for large t/b (filled quare), limit for mall t/b (filled circle). Theoretical formula: Eq. 14 (dah-dot), Eq. 15 (dahed) and Eq. 16 (dotted) Figure 9: Shunt length correction t (c) a a function of with t/b = 0.1 (bottom) and t/b = 2.0 (top). FEM reult (filled circle). Theoretical value (t +t m ) where t m i calculated uing Eq. 7 (dahed). t a (o) /b t/b Figure 8: Serie length correction t a (o) /b 4 a a function of t/b for = 1.0. FEM reult (filled circle). Data fit formula Eq. 37 (dotted). Theory: Eq. 16 ( dahed), Eq. 14 (dah-dot). and g() = Figure 7 diplay the reult of our imulation for two extreme cae: hort chimney height (circle) and large chimney height (quare), in comparion to theoretical formula from the literature and an experimental data point from Keefe (1982a). Thi figure reveal that none of the theoretical equation i valid for all value of and chimney height. In the cae of large chimney height, Eq. 15 provide a good approximation. The dependence of the erie length correction on the tonehole height i diplayed in Fig. 8 for a tonehole with = 1.0, which reveal that neither Eq. 16 nor Eq. 14 match our FEM reult. The reult of our imulation for cloed tonehole confirm the validity of the low frequency limit of the hunt length correction. Figure 9 how that the low frequency value of the hunt length correction i very well repreented by the length t +t m, that i, by the volume of the tonehole. The cotangent term in Eq. 13 tend toward infinity when k(t +t m ) 0, conequently, the influence of an inner length correction i expected to be maximal when k(t +t m ) π/2 and negligible when it goe toward zero. A an example, for a tonehole height of 5 mm, the maximal influence of the inner length correction i above 20 khz wherea, for a tonehole of 5 cm chimney height, thi occur above 2 khz. Therefore, thi term ha a negligible influence even in the higher frequency range of woodwind intrument except poibly for intrument with very tall tonehole uch a the baoon (for which t varie between 5 to 40 mm). Neverthele, to tudy thi term, it i ueful to define the impedance of the cloed ide hole a: Z (c) = j cotk(t +t m +t (c) i ), (38) where t m i the matching volume length correction defined in Eq. 7 and t (c) i i the inner length correction (located inide the cotangent term rather than outide, thu it i not equivalent to the value from the literature). We can obtain the value of t (c) i from our imulation reult with: ( ) t (c) i = 1 1 k tan 1 jz (c) t t m. (39) Thi value may be compared with the current theoretical value by applying the previou equation to the calculated impedance of the cloed ide hole uing Eq. 13. Thi i hown in Fig. 10. Dicrepancie between the imulation reult and the theoretical value exit. In the cae of the hort-height tonehole (top graph), the magnitude of the inner length correction i very mall but it i remarkable that it value i negative for the two larger-diameter tonehole ( value of 0.8 and 1.0). Dicrepancie are alo apparent for the large-height tonehole (bottom graph). In thi cae, the dicrepancie are alo mot important for the larger-diameter tonehole. Further reearch i required to fully characterize thi effect. For intrument with normally ized tonehole (flute, clarinet, axophone) thi i likely negligible a explained previouly. In the cae of the erie length correction, we obtained a new formula that take into account more preciely the height and radiu of the tonehole: where t a (c) = f (,t/b)g(), (40) b 4 f (,t/b) = 1 [ ][1 tanh(2.385t/b)], g() = In Fig. 11, we conider the low frequency limit of the erie length correction t a (c) for hort and tall tonehole height compared to previou theorie. The reult for the tall tonehole are the ame a for an open hole (ee Fig. 7). When the tonehole are hort in height, the erie length correction term diminihe in magnitude. Figure 12 preent thi length correction a a function of the ratio t/b for one tonehole ( = 1.0) in comparion with current theorie. 6 ISMA 2010, aociated meeting of ICA 2010

7 Proceeding of ISMA Augut 2010, Sydney and Katoomba, Autralia t (c) i /b t a (c) /b kb Figure 11: Serie length correction t (c) a /b 4 a a function of. FEM reult: limit for large t/b (filled quare), limit for mall t/b (filled circle). Theoretical formula: Eq. 14 (dah-dot), Eq. 15 (dahed) and Eq. 16 (dotted). t (c) i /b kb Figure 10: Inner length correction t (c) i /b for cloed tonehole a a function of kb for = 0.2,0.5,0.8,1.0 obtained from FEM imulation (filled circle) compared to theory (dahed). Top: t/b = 0.5, bottom: t/b = 2.0. The dotted line i a viual aid. t a (c) /b Impact of Conicity A tonehole on a conical bore i no longer ymmetric. In thi ituation, we propoe to modify the model repreented by Eq. 4 with: 1 + Zau /Z T hole = Z au + Z ad + Z au Z ad /Z, (41) 1/Z 1 + Z ad /Z where Z au i the erie impedance for the uptream half of the tonehole and Z ad of the downtream half. In a imilar manner a for tonehole on cylindrical bore, we obtained the TM of the tonehole on a conical bore uing Finite Element imulation. The tranmiion matrix T hole of the tonehole i obtained from the tranmiion matrix T of the imulated ytem by multiplying thi matrix by the invere of the TM of the two egment of truncated cone, T coneu and T coned : T hole = T 1 cone u TT 1 cone d, (42) where the TM of a conical waveguide i (Fletcher and Roing 2008): [ ] rt T cone = out in(kl θ out ) jr inkl, jrt in t out in(kl θ out + θ int ) rt in in(kl + θ in ) (43) where x in and x out are repectively the ditance of the input plane and output plane from the apex of the conical frutum and where r = x out /x in, L = x out x in, θ in = arctan(kx in ), θ out = arctan(kx out ), t in = 1/inθ in and t out = 1/inθ out. We are intereted in determining whether or not the hunt impedance Z i different from that derived for a cylindrical bore and to determine the effect of the aymmetry on the value of Z au and Z ad. The tonehole parameter were obtained for two conical waveguide with taper angle of 3 and 6 degree. A t/b Figure 12: Serie length correction t a (c) /b 4 a a function of t/b for = 1.0. FEM reult (filled circle). Data fit formula Eq. 37 (dotted). Theory: Eq. 16 (dah-dot), Eq. 14 (dahed). for tonehole on a cylindrical bore, we developed a data-fit formula for the hunt equivalent length of the open tonehole from the imulation data (with the ame et of parameter). Then we calculated the difference between the two fit formulae and determined that the maximal difference i b in both cae. Thi i a very mall difference and we are confident to conclude that the hunt length correction are unchanged relative to their value on a cylindrical bore. A concluion for the erie length correction i more difficult. For the imulation of tonehole on cylindrical bore, we olved only half the ytem, thu forcing the ymmetry. In the cae of the conical bore, the complete ytem i olved and the TM i not forced to be ymmetric, which eem to augment numerical error. A can be een in Fig. 13, the uptream and downtream value of the erie length correction are very cloe to one another even though they become lightly different for maller value of. The impact of the erie length correction being relatively mall and increaingly le important a the tonehole become maller, thi i likely to be negligible. From thi analyi, we conclude that the ue of TM parameter developed for tonehole on cylindrical bore are valid for conical bore, at leat up to an angle of 6 degree and probably for wider angle a well. CONCLUDING REMARKS We obtained new formula for the hunt and erie impedance of an unflanged tonehole that i valid for tonehole height com- ISMA 2010, aociated meeting of ICA

8 25-31 Augut 2010, Sydney and Katoomba, Autralia Proceeding of ISMA t a (o) [mm] Figure 13: Serie length correction t a in mm for a tonehole on a conical bore with taper angle of 3 degree: limit for long (quare) and hort (circle) tonehole uptream part (filled), downtream part (unfilled). mon in woodwind intrument. For an open tonehole, the hunt length correction i expreed in Eq. 34 and the erie length correction in Eq. 37. For a cloed ide hole, the erie length correction i expreed in Eq. 40 wherea the hunt length correction from the literature, Eq. 11 and 12, are valid. We verified that the hunt length correction of a tonehole on a cylindrical bore can be ued a well on a conical bore. ACKNOWLEDGEMENTS The author would like to thank Jean-Pierre Dalmont for providing experimental reult. They alo wih to acknowledge the upport of the Natural Science and Engineering Reearch Council of Canada, the Canadian Foundation for Innovation, and the Centre for Interdiciplinary Reearch in Muic Media and Technology at McGill Univerity. The firt author gratefully acknowledge the Fond Québécoi de la Recherche ur la Nature et le Technologie for a doctoral reearch cholarhip. REFERENCES Dalmont, Jean-Pierre et al. (2002). Experimental Determination of the Equivalent Circuit of an Open Side Hole: Linear and Non Linear Behaviour. Acutica 88, pp Dubo, V. et al. (1999a). Theory of Sound Propagation in a Duct with a Branched Tube Uing Modal Decompoition. Acutica 85, pp Dubo, Vincent et al. (1999b). Theory of Sound Propagation in a Duct with a Branched Tube Uing Modal Decompoition. Acutica 85, pp Fletcher, Neville H. and Thoma D. Roing (2008). The Phyic of Muical Intrument. 2nd ed Springer. Keefe, Dougla H. (1981). Woodwind Tone Hole Acoutic and the Spectrum Tranformation Function. PhD thei. Cae Wetern Reerve Univerity. (1982a). Experiment on the ingle woodwind tonehole. J. Acout. Soc. Am. 72.3, pp (1982b). Theory of the ingle woodwind tonehole. J. Acout. Soc. Am. 72.3, pp (1990). Woodwind air column model. J. Acout. Soc. Am. 88, pp Nederveen, C. J. et al. (1998). Correction for Woodwind Tone-Hole Calculation. Acutica 84, pp Nederveen, Corneli Johanne (1998). Acoutical Apect of Woodwind Intrument. Revied. DeKalb, Illinoi: Northern Illinoi Univerity Pre, p Pierce, Allan D. (1989). Acoutic, An Introduction to It Phyical Principle and Application. Woodbury, New-York: Acoutical Society of America, p ISMA 2010, aociated meeting of ICA 2010