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

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 (  ) Both the plane waves of a cylinder and the spherical waves a cone can be modeled using a digital waveguide. (  ) Z -L

4. 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 Theoretical Wall Loss The round trip attenuation for a tube of L is given by, 2 (  ) = e -2  L valid for diameters seen in most musical instruments.

5. R2()R2() R1()R1() 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. TerminationScattering R()R()R()R() T()T() + + T1()T1() T2()T2()

6. Open End Reflection Filter The reflection filter for an open is given by R op (  ) = Z L (  ) / Z 0 - 1 Z L (  ) / Z 0 +1 Z 0 = cc S, where and Z L (  ) is the complex terminating impedance at the open end of a cylinder, given by the expression by Levine and Schwinger. is the wave impedance,

7. j  + c/x Z 2 / Z 1 +1 Theoretical Junction Reflection The reflection at the junction is given by R(  ) = Z 2 / Z 1 - 1, * The impedance for the spherical waves is given by Z n = cc S jj This leads to a first-order, one-pole, one-zero, filter.. The impedance for plane waves is given by Zy =Zy = cc S

8. Cylinder Cylicone CylinderScatteringCone Example Waveguide Models

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

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

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 L 1 =  (  ) L 2 =  (  ) 2 (  )(1+  (  )) L 3 =  (  ) 4 (  )  (  )(1+  (  ))

15. Closed Cylinder Arrival Spectra L1L1 L2L2 L3L3

16. Speaker Reflection Transfer Function Given the arrival responses : L 1 =  (  ) L 2 =  (  ) 2 (  )(1+  (  )) L 3 =  (  ) 4 (  )  (  )(1+  (  )),  (  ) = ^  1 -  =  L1L3L1L3 (L 2 ) 2 ()() 1 +  (  ) = We are able to estimate the speaker reflection transfer function

17. Cylinder Wall Loss Transfer Function Given the arrival responses : L 1 =  (  ) L 2 =  (  ) 2 (  )(1+  (  )) L 3 =  (  ) 4 (  )  (  )(1+  (  )), and the estimate for the speaker reflection, we are able to estimate the wall loss transfer function 2 (  ) = L3L3  (  ) L 2 ^ ^

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

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

20. Arrival Responses for an Open Cylinder Y 1 =  (  ) Y 2 =  (  ) 2 (  )R op (  )(1+  (  )) Y 3 =  (  ) 4 (  )R 2 op (  )  (  )(1+  (  ))

21. Open Cylinder Arrival Spectra Y1Y1 Y2Y2 Y3Y3

22. Open End Reflection Given the second arrival for the closed tube: Y 2 =  (  ) 2 (  )R op (  )(1+  (  )), We are able to estimate the reflection from an open end R op (  ) = Y2Y2 L2L2 L 2 =  (  ) 2 (  )(1+  (  )) and the second arrival for the open tube: ^

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: R y (  ) and T y (  ), the cone: R n (  ) and T n (  )

25. Arrival Responses for Closed Cylicone A 1 =  (  ) A 2 = … A 3 = …

26. Second Arrival, Closed Cylicone A (2,1) =  (  ) y (  )R y (  )(1+  (  )) 2 A (2,2) =  (  ) y (  ) n (  )T y (  )T n (  )(1+  (  )) 22

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 N 1 =  (  ) N 2 = …

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).