Observing the Effects of Waveguide Model Elements in Acoustic Tube Measurements

1. Observing the Effects of Waveguide Model Elements in Acoustic Tube Measurements Tamara Smyth tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University Jonathan Abel abel@batnet.net Universal Audio Inc. and Stanford University (CCRMA) Meeting of the Acoustical Society of America, Honolulu, Hawaii December 2, 2006

2. Outline • Digital waveguide theory • Technique for measuring an impulse response from acoustic tube structures • Observing waveguide theory in measured responses • Comparing model and measurement

3. A Digital Waveguide Section Z-L () () Z-L • Both the plane waves of a cylinder and the • spherical waves a cone can be modeled using • a digital waveguide.

4. Theoretical Wall Loss • The effects of viscous drag and thermal conduction along the bore walls, lead to an attenuation in the propagating waves, determined by () = 2 x 10-5  / a valid for diameters seen in most musical instruments. • The round trip attenuation for a tube of L • is given by, 2() = e-2L

5. + + Termination and Scattering A change of impedance, such as a termination or connection to another waveguide section, will require additional filters to account for reflection and possibly transmission. Termination Scattering T() T1() R() R() R1() R2() T2()

6. Open End Reflection Filter • The reflection filter for an open is given by ZL ()/ Z0 - 1 Rop() = , ZL ()/ Z0 +1 c where Z0 = is the wave impedance, S and ZL() is the complex terminating impedance at the open end of a cylinder, given by the expression by Levine and Schwinger.

7. Theoretical Junction Reflection The reflection at the junction is given by Z2 / Z1 - 1 , R() = * Z2 / Z1 +1 The impedance for plane waves is given by c Zy= S The impedance for the spherical waves is given by c j . Zn = j + c/x S This leads to a first-order, one-pole, one-zero, filter.

8. Example Waveguide Models Cylinder Cylicone Cylinder Scattering Cone

9. Four Measured Tube Structures Cylinder, speaker-closed Cylinder, speaker-open Cylicone, speaker-open Cylicone, speaker-closed

10. + Obtaining an Impulse Response from an LTI System Measurement noise n(t) s(t) h(t) r(t) Measured response Test signal LTI system The impulse is limited in amplitude and has poor noise rejection

11. Impulse Response Using a Swept Sine • The sine is swept over a frequency trajectory (t) effectively • smearing the impulse over a longer period of time. • Since higher frequencies go into the system at later times, they must be realigned to recover the impulse response.

12. Our Measurement System

13. Cylinder, Speaker-Closed From the first measurement we observe: • The speaker transfer function, () • The speaker reflection, () • The round trip wall losses for a cylinder, 2()

14. Arrival Responses for a Closed Cylinder L1 = () L2 = ()2()(1+()) L3 = ()4() ()(1+())

15. Closed Cylinder Arrival Spectra L1 L2 L3

16. Speaker Reflection Transfer Function • Given the arrival responses: L1 = () L2 = ()2()(1+()) L3 = ()4()()(1+()), L1L3 () = =  (L2)2 1 + () • We are able to estimate the speaker reflection transfer function  ^ () = 1 - 

17. Cylinder Wall Loss Transfer Function • Given the arrival responses: L1 = () L2 = ()2()(1+()) L3 = ()4()()(1+()), and the estimate for the speaker reflection, we are able to estimate the wall loss transfer function L3 ^ 2() = ^ () L2

18. Estimated and Theoretical Propagation Losses ^ ()

19. Cylinder, Speaker-Open From this measurement we observe • the reflection from an open end, Rop().

20. Arrival Responses for an Open Cylinder Y1 = () Y3 = ()4()R2op() ()(1+()) Y2 = ()2()Rop()(1+())

21. Open Cylinder Arrival Spectra Y1 Y2 Y3

22. Open End Reflection • Given the second arrival for the closed tube: L2 = ()2()(1+()) • and the second arrival for the open tube: Y2 = ()2()Rop()(1+()), • We are able to estimate the reflection from an open end ^ Y2 Rop() = L2

23. Cylinder Open End Reflection

24. Cylicone, Speaker-Closed We consolidate this measurement with the theoretical reflection and transmission filters at the junction: • the cylinder: Ry()and Ty(), • the cone: Rn() and Tn()

25. Arrival Responses for Closed Cylicone A1 = () A2 = … A3 = …

26. Second Arrival, Closed Cylicone 2 2 A(2,2) = ()y() n()Ty()Tn()(1+()) A(2,1) = ()y()Ry()(1+()) 2

27. Measured and Modeled Closed Cone Arrival

28. Cylicone, Speaker-Open From this measurement, we observe the behaviour of the reflection filter from the cone’s open end.

29. Arrival Responses for Open Cylicone N1 = () N2 = …

30. Second Arrival, Open Cylicone

31. Closed Cylinder Comparison

32. Open Cylinder Comparison

33. Closed Cylicone Comparison

34. Open Cylicone Comparison

35. Summary • We observed the behaviour of the following waveguide filter elements, from measured impulse responses: • Open end reflection (cylinder and cone) • Propagation losses (cylinder and cone) • Junction reflection and transmission (cylicone) • We confirmed that the impulse response • measurements matched the responses • of the waveguide models.

36. Conclusions • We observed and verified theoretical waveguide filter elements using our measurements. • The measurement system yields very good data at relatively low cost. • The validation of the measurement system implies it can be extended to any tube structure.

37. Acknowlegements • We would like to thank Theresa Leonard and the Banff Centre for Performing Arts. • Natural Sciences and Engineering Research Council (NSERC).