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Acoustic Analysis of the Viola

This study examines the acoustic properties of the viola, focusing on its vibration modes, resonant frequencies, and the impact of wood and air on its sound. Spectral analysis, near-field acoustic holography, and measurement of complex pressure, particle velocity, and displacement are used to study the instrument. Comparisons to the violin are made to understand differences in timbre. Future work includes testing different viola models and comparing results to other string instruments.

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Acoustic Analysis of the Viola

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

  2. The Viola • 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. Finger-board F-holes Cross-section: Bridge Bridge Top Plate 2004 Andreas Eastman VA200 16” viola Back Plate Bass bar Soundpost

  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 Output Piezo and Accelerometer Input Piezo 5 locations of displacement measurement P and U mics

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

  7. Mechanical Vibration Open String frequencies

  8. 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]

  9. 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 Near-Field Acoustic Holography XY Translation Stages PU mic

  10. 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

  11. Sound Intensity Level SIL(x,y) vs. Modal Frequency: 328 Hz 1504 Hz 224 Hz 1078 Hz 560 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: 328 Hz 1504 Hz 224 Hz 1078 Hz 560 Hz

  12. Complex Specific Acoustic Impedance Z(x,y) vs. Modal Frequency: 328 Hz 1504 Hz 224 Hz 1078 Hz 560 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

  13. Complex Sound Intensity I(x,y) vs. Modal Frequency: 328 Hz 1504 Hz 224 Hz 1078 Hz 560 Hz Re{I} Im{I} 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}

  14. Acoustic Energy Density w(x,y) vs. Modal Frequency: 328 Hz 1504 Hz 224 Hz 1078 Hz 560 Hz wrad wvirt wrad: energy density associated with propagating sound (RMS J/m3) wvirt: energy density associated with non-propagating sound (RMS J/m3)

  15. 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… Summary 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

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