Physics 106P: Lecture 1 Notes

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Acoustic Impedance Measurements
Presented by Brendan Sullivan June 23, 2008
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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

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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 ? for
short.
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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)?
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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 of mouthpiece
Image modified from J. Backus, J. Acoust. Soc.
Am. 54, 470 (54)?
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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

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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

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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

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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
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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
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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(? t)?
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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.
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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.
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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.
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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

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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
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Pressure Phase vs. Frequency

• Output is mostly noise below 250 Hz
• Distinct Patterns
• Output like tan(f)?
• 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
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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
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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
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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

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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
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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

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