1. Acoustic Impedance Measurements Acoustic Impedance Measurements Presented by: Brendan Sullivan June 23, 2008

2. Agenda for Today  What acoustic impedance is and why we are interested in it  Physical interpretations of acoustic impedance Notes on an instrument Electrical circuits  How to measure acoustic impedance First, in General Mainly, in a trumpet Phase Sensitive  Results No general theory, but some interesting data  Future Plans

3. What is Acoustic Impedance? P(x) U(x) Z(x) = Air Pressure Longitudinal Particle Velocity Specific Acoustic Impedance Units are Acoustical Ohms (Pa-s/m), or Ώ for short.

4. What Really is Acoustic Impedance? Take a look at this typical impedance spectrum: Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)‏  Blue lines (maxima) are accessible frequencies  Red lines (minima) are inaccessible frequencies  The first peak is the fundamental  Subsequent peaks are harmonics Harmonics decrease in amplitude – just as in the overtones of an instrument

5. Ohms? Impedance? This sounds like a circuit... m...because it is! m Any acoustical system creates an acoustical circuit Parts of the acoustical system behave exactly like the components of a circuit Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)‏ Z i – Mouthpiece input impedance Z – Mouthpiece output Impedance L – The inductance, or the area between the cup and tube R, C – Values determined by geometry of mouthpiece The Circuit Components:

6. How Do We Measure Impedance? P(x) U(x) Z(x) = Pressure Microphone Time-Integrated Differential Pressure Microphone  Two quantities to measure: pressure (P) and particle velocity (U)‏  For pressure, we use a pressure microphone  For particle velocity, we use a (time-integrated) differential pressure microphone

7. How the Microphones Work Electret Condenser Microphone (P-mic)‏ d V = E d  Pressure (sound) waves press against front plate, changing d, thereby inducing a voltage  Assuming elastic particle-plate collisions, conservation of momentum ensures induced voltage is linear in pressure Condenser microphone schematic

8. How the Microphones Work Fix this: Differential Pressure Microphone (DPM)‏  Measures the pressure immediately to the right and left of a particular location  Numerically integrates to find the pressure at that location Differential pressure microphone schematic

9. Placing the Microphones in a Trumpet A trumpet bell - notice the large, accessible geometry l The openness of the trumpet bell makes mounting the exit microphones easy l Microphones can be secured outside the trumpet and simply placed in l Wiring can also be done externally Schematic of the bell: the mics easily fit in the bell and can be wired/secured externally

10. Placing the Microphones in a Trumpet l Mouthpiece is much narrower than the bell m Harder to use microphones l Drill tiny holes in mouthpiece to run wires/brackets through m As tiny as possible so as not to change the instrument l Can't just run directly out of the mouthpiece because the path is blocked by a transducer... Schematic of the mouthpiece notice that the wires run through small holes in the mouthpiece

11. Exciting the Trumpet Schematic of the mouthpiece The transducer has a position that goes as x(t) = A sin(ω t)‏ l A player's lips resonate at a specific frequency m Excites the instrument with nearly monochromatic sound wave l Using a function generator, drive the transducer at a specific frequency m Much like a piston l Closely recreates an actual player m Some aspects still not reproducible yet, i.e., humidity

12. Adding Complexion to the Measurement: Lock-in Amplifiers l We want this to be a phase-sensitive measurement  We can do this using a lock-in amplifier l How lock-in amplifiers work: m Pick out any components of the desired frequency; in this case, the function generator's frequency m Resolve vector into real (in phase) and imaginary (perfectly out of phase) parts m Record the real and imaginary values separately A phasor diagram: The lock-in amplifier will pick out the blue vector and resolve it into its real (red) and imaginary (green)‏ components.

13. An overview of the setup: each microphone is connected to a lock-in amplifier which is recorded on a computer. The spectrum is obtained by sweeping a frequency range.

14. Above: A picture of the trumpet with measurements being taken. The four closed boxes are the microphones and the open box is the piezo driver Left: A picture of the measurement setup.

15. Results: An Overview First time a phase-sensitive measurement of this sort has been made No general theory can explain all the data  Even for non-phase sensitive, theory is inaccurate Imaginary component very small compared to real component  Like a correction factor

16. Pressure vs. Frequency l Magnitude of output is much less than real (output is even amplified 10x)‏ l Output component switches sign each harmonic l Output part generally increasing, real part increases then decreases m Higher notes seem louder A plot of input (blue) and output (pink) pressure versus frequency

17. Pressure Phase vs. Frequency A plot of output (blue) and input (pink) phase difference versus frequency l Output is mostly noise below ~250 Hz l Distinct Patterns  Output like tan(φ)‏  Input has defined peaks and troughs  Period increases with frequency l Indicative that something cyclical is happening with phase difference

18. Pressure in the Complex Plane l Different way to look at the last plot – the elliptical nature of the plots indicates the repeating phase shift  Bigger loops correspond to higher frequencies l No 'deeper' interpretation of this data  No general theory, yet A parametric plot of output (blue) and input (pink) pressure in the complex plane

19. Complex Acoustic Impedance A plot of output (blue) and input (pink) impedance versus frequency l Distinct peaks and troughs on input we noted earlier l Output is nearly linear (three separate lines, perhaps)‏  Relates to structure of musical notes, but we won't go into that  Can only access the output frequencies at input peaks

20. How the Notes Line Up l Each data point is the frequency of output at an input impedance peak (e.g., C4 = Middle C = 261.626 Hz)‏ l Very small deviations from “accepted” notes l Since measurement errors on experiment were ~ 5%, these notes clearly coincide with accepted notes

21. Looking Ahead This summer, same experiment for an Oboe and Clarinet  Much smaller instruments make it harder  These instruments use reeds, not metal mouthpieces Data may help with a more general theory Above: Clarinet mouthpiece Left: Oboe reed and top of mouthpiece

22. Recap l Acoustic Impedance is defined as pressure over particle velocity l Relates to the accessible sounds an object can make l Measured using a DPM and U-mic l No general theory yet, though some interesting data

23. Questions? Special Thanks to David Pignotti, Professor Errede, and all of you!