Acoustic Impedance Measurements

1. 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? Air Pressure P(x) U(x) Z(x) = Specific Acoustic Impedance Longitudinal Particle Velocity 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: • 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 Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)‏

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

6. How Do We Measure Impedance? Pressure Microphone P(x) U(x) Z(x) = 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 Condenser microphone schematic 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

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

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

10. Placing the Microphones in a Trumpet • Mouthpiece is much narrower than the bell • Harder to use microphones • Drill tiny holes in mouthpiece to run wires/brackets through • As tiny as possible so as not to change the instrument • 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 • A player's lips resonate at a specific frequency • Excites the instrument with nearly monochromatic sound wave • Using a function generator, drive the transducer at a specific frequency • Much like a piston • Closely recreates an actual player • Some aspects still not reproducible yet, i.e., humidity Schematic of the mouthpiece The transducer has a position that goes as x(t) = A sin(ω t)‏

12. Adding Complexion to the Measurement: Lock-in Amplifiers • We want this to be a phase-sensitive measurement • We can do this using a lock-in amplifier • How lock-in amplifiers work: • Pick out any components of the desired frequency; in this case, the function generator's frequency • Resolve vector into real (in phase) and imaginary (perfectly out of phase) parts • 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 • Magnitude of output is much less than real (output is even amplified 10x)‏ • Output component switches sign each harmonic • Output part generally increasing, real part increases then decreases • Higher notes seem louder A plot of input (blue) and output (pink) pressure versus frequency

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

18. Pressure in the Complex Plane • 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 • 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 • Distinct peaks and troughs on input we noted earlier • 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 A plot of output (blue) and input (pink) impedance versus frequency

20. How the Notes Line Up • Each data point is the frequency of output at an input impedance peak (e.g., C4 = Middle C = 261.626 Hz)‏ • Very small deviations from “accepted” notes • 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 • Acoustic Impedance is defined as pressure over particle velocity • Relates to the accessible sounds an object can make • Measured using a DPM and U-mic • No general theory yet, though some interesting data

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