Transcript Physics 106P: Lecture 1 Notes

Acoustic Impedance
Measurements
Presented by:
Brendan Sullivan
June 23, 2008
Agenda for Today
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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
What is Acoustic Impedance?
Air Pressure
P(x)
Z(x) =
U(x)
Specific Acoustic Impedance
Longitudinal
Particle Velocity
Units are Acoustical Ohms (Pa-s/m), or Ώ for short.
What Really is Acoustic
Impedance?
Take a look at this typical impedance spectrum:
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Blue lines (maxima) are
accessible frequencies
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Red lines (minima) are
inaccessible frequencies
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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)
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:
Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)
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
How Do We Measure Impedance?
Pressure Microphone
P(x)
Z(x) =
U(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
How the Microphones Work
Electret Condenser Microphone (P-mic)
d
Condenser microphone schematic
V=Ed


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
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
Placing the Microphones in a
Trumpet

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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
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
Exciting the Trumpet
Schematic of the mouthpiece
The transducer has a position that
goes as x(t) = A sin(ω t)

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
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.
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.
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.
Results: An Overview
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First time a phase-sensitive
measurement of this sort has been
made
No general theory can explain all
the data

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Even for non-phase sensitive, theory is
inaccurate
Imaginary component very small
compared to real component

Like a correction factor
Pressure vs. Frequency
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A plot of input (blue) and output (pink)
pressure versus 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
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
Pressure in the Complex Plane

A parametric plot of output (blue) and
input (pink) 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
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
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
Looking Ahead

This summer, same experiment for an
Oboe and Clarinet
Much smaller instruments make it harder
 These instruments use reeds, not metal
mouthpieces

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Data may help with a more general theory
Above: Clarinet mouthpiece
Left: Oboe reed and top of
mouthpiece
Recap
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Acoustic Impedance is defined as pressure over particle
velocity
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Relates to the accessible sounds an object can make

Measured using a DPM and U-mic

No general theory yet, though some interesting data
Special Thanks to David Pignotti,
Professor Errede, and all of you!
Questions?