Acoustic Analysis
of the Viola
By Meredith Powell
Advisor: Professor Steven Errede
REU 2012
The Viola
•
•
Fingerboard
F-holes
•
Bridge
String Instrument, larger and lower in
pitch than a violin
Tuning:
A (440 Hz)
D (294 Hz)
G (196 Hz)
C (131 Hz)
Vibration of string is transferred to
bridge, then soundpost and body, to
surrounding air.
Cross-section:
Bridge
Top Plate
2004 Andreas Eastman VA200 16” viola
Bass bar
Soundpost
Back
Plate
Goal
• Understand how body vibrates
– Resonant frequencies
• Wood resonances
• Air resonances
– Modes of vibration
Methods
• Spectral Analysis in frequency domain
– Complex Sound Pressure and Particle Velocity
– Complex Mechanical Acceleration, Velocity &
Displacement at 5 locations on instrument
• Near-field Acoustic Holography
– Vibration modes at resonant frequencies
Spectral Analysis
•
•
•
•
Excite the viola with a piezo-electric
transducer placed near bridge
Take measurements at each
frequency, from 29.5 Hz to 2030.5 Hz
in 1 Hz steps using 4 lock-in amplifiers
Measure complex pressure and
particle velocity with PU mic placed at
f-hole
Measure complex mechanical
displacement, velocity, acceleration
with piezo transducer and
accelerometer
Output Piezo and
Accelerometer
Input Piezo
5 locations of
displacement
measurement
P and U mics
P and U Spectra
Main Air Resonances @ f-holes:
–
–
220Hz (Helmholtz)
1000Hz
Open String
frequencies
Mechanical Vibration
Comparing to Violin
Violin resonances tend to lie on frequencies of open strings1
This is not the case for the viola
 Cause of more subdued, mellow timbre?
1Fletcher,
Neville H., and Thomas D. Rossing. The Physics of Musical Instruments. New York: Springer, 1998.
[Image courtesy of Violin Resonances. http://hyperphysics.phy-astr.gsu.edu/hbase/music/viores.html]
Near-Field Acoustic Holography
• Images surface vibrations at
fixed resonant frequency
XY Translation Stages
PU mic
• Measures complex pressure
and particle velocity in
proximity to the back of
instrument
–
–
–
–
Impedance: Z(x,y) = P(x,y)/U(x,y)
Intensity:
I(x,y) = P(x,y) U*(x,y)
Particle Displacement: D = iU
Particle Acceleration: A = (1/i) U
Near-Field Acoustic Holography
•
Mechanically excite viola by placing two super magnets on either side of
the top plate as close to bridge/soundpost as possible
•
A sine-wave generator is connected to a coil (in proximity to outer magnet);
Creates alternating magnetic field which induces mechanical vibrations
•
PU mic attached to XY translation stages carries out 2-dimensional scan in
1 cm steps
Magnets
Coil
Sound Intensity Level SIL(x,y) vs. Modal Frequency:
224 Hz
328 Hz
560 Hz
1078 Hz
1504 Hz
SIL(x,y) = 10 log10(|I(x,y)|/Io) {dB}
Io = 10-12 RMS Watts/m2 (Reference Sound Intensity*)
* @ f = 1 KHz
Particle Displacement Re{D(x,y)} vs. Modal Frequency:
224 Hz
328 Hz
560 Hz
1078 Hz
1504 Hz
Complex Specific Acoustic Impedance Z(x,y) vs. Modal Frequency:
224 Hz
328 Hz
560 Hz
1078 Hz
1504 Hz
Re{Z}
Im{Z}
Z(x,y) = p(x,y)/u(x,y)
{Acoustic Ohms:
Pa-s/m}
Re{Z}: air impedance associated with propagating sound
Im{Z}: air impedance associated with non-propagating sound
Complex Sound Intensity I(x,y) vs. Modal Frequency:
224 Hz
328 Hz
560 Hz
1078 Hz
1504 Hz
Re{I}
Im{I}
I(x,y) = p(x,y) u*(x,y)
{RMS Watts/m2}
Re{I}: propagating sound energy
Im{I}: non-propagating sound energy
(locally sloshes back and forth per cycle)
Acoustic Energy Density w(x,y) vs. Modal Frequency:
224 Hz
328 Hz
560 Hz
1078 Hz
1504 Hz
wrad
wvirt
wrad: energy density associated with
propagating
sound (RMS J/m3)
wvirt: energy density associated with non-propagating sound (RMS J/m3)
Summary
•
Resonant frequencies tend to lie
between the open strings frequencies
causing mellower sound.
•
Actual mechanical motion when
playing is superposition of the various
modes of vibration associated with
resonant frequencies.
•
Future work: Test multiple models of
violas, carry out same experiments on
violin/cello & compare…
Acknowledgements:
I would like to extend my gratitude to Professor Errede for all of his help and guidance
throughout this project, and for teaching me so much about acoustics and physics in general! 
The NSF REU program is funded by National Science Foundation Grant No. 1062690