1. Acoustic Analysis of the Viola By Meredith Powell Advisor: Professor Steven Errede REU 2012

2. 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:

3. Goal Understand how body vibrates –Resonant frequencies Wood resonances Air resonances –Modes of vibration

4. 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

5. 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

6. P and U Spectra Main Air Resonances @ f-holes: –220Hz (Helmholtz) –1000Hz

7. Mechanical Vibration Open String frequencies

8. Comparing to Violin [Image courtesy of Violin Resonances. http://hyperphysics.phy-astr.gsu.edu/hbase/music/viores.html]http://hyperphysics.phy-astr.gsu.edu/hbase/music/viores.html Violin resonances tend to lie on frequencies of open strings 1 This is not the case for the viola  Cause of more subdued, mellow timbre? 1 Fletcher, Neville H., and Thomas D. Rossing. The Physics of Musical Instruments. New York: Springer, 1998.

9. 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 = i  U –Particle Acceleration: A = (1/i  ) U PU mic XY Translation Stages

10. 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

11. Sound Intensity Level SIL(x,y) vs. Modal Frequency: SIL(x,y) = 10 log 10 (|I(x,y)|/I o ) {dB} I o = 10 -12 RMS Watts/m 2 (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

12. 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

13. 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/m 2 }

14. Acoustic Energy Density w(x,y) vs. Modal Frequency: w rad : energy density associated with propagating sound (RMS J/m 3 ) w virt : energy density associated with non-propagating sound (RMS J/m 3 ) w rad w virt 224 Hz 328 Hz 560 Hz 1078 Hz 1504 Hz

15. 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