Acoustic Analysis of the Viola

By Meredith Powell

Advisor: Professor Steven ErredeREU 2012

• 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.

2004 Andreas Eastman VA200 16” viola

The Viola

Bridge

F-holes

Finger-board

Bridge

Soundpost

Back Plate

Top Plate

Bass bar

Cross-section:

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

5 locations of displacement measurement

P and U mics

Input Piezo

Output Piezo and Accelerometer

P and U Spectra

Main Air Resonances @ f-holes:– 220Hz (Helmholtz)

– 1000Hz

Mechanical VibrationOpen String frequencies

Comparing to Violin

[Image courtesy of Violin Resonances. http://hyperphysics.phy-astr.gsu.edu/hbase/music/viores.html]

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.

Near-Field Acoustic Holography

• Images surface vibrations at fixed resonant frequency

• 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 = iU– Particle Acceleration: A = (1/i) U

PU mic

XY Translation Stages

• 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

Near-Field Acoustic Holography

Magnets

Coil

Sound Intensity Level SIL(x,y) vs. Modal Frequency:

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

224 Hz 328 Hz 560 Hz 1078 Hz 1504 Hz

Complex Specific Acoustic Impedance Z(x,y) vs. Modal Frequency:

Re{Z}: air impedance associated with propagating sound

Im{Z}: air impedance associated with non-propagating sound

Re{Z}

Im{Z}

Z(x,y) = p(x,y)/u(x,y)

{Acoustic Ohms:

Pa-s/m}

224 Hz 328 Hz 560 Hz 1078 Hz 1504 Hz

Complex Sound Intensity I(x,y) vs. Modal Frequency:

Re{I}

Im{I}

224 Hz 328 Hz 560 Hz 1078 Hz 1504 Hz

Re{I}: propagating sound energy

Im{I}: non-propagating sound energy (locally sloshes back and forth per cycle)

I(x,y) = p(x,y) u*(x,y)

{RMS Watts/m2}

Acoustic Energy Density w(x,y) vs. Modal Frequency:

wrad: energy density associated with propagating sound (RMS J/m3)

wvirt: energy density associated with non-propagating sound (RMS J/m3)

wrad

wvirt

224 Hz 328 Hz 560 Hz 1078 Hz 1504 Hz

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