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onstrate and understand how the coupling between all its component parts influences the vibrational modes and their influence on the radiated sound. This has involved varying the influence of each component over a very wide range as an aid to understanding the nature of and effect of the coupling.
To give a flavor of this approach, Figure 6 illustrates the transformation of the initially freely supported individu- al plates into the modes of the empty body shell as the rib coupling strength is varied over six orders of magnitude from close to zero to a typical normal value. The highlight- ed curves illustrate how the important radiating breathing mode of the body shell is transformed from the component #5 plate mode and its extremely strong interaction with the rising frequency bouncing mode of the rigid plates that are constrained by the extensional springlike and bending of the ribs.
There are many perhaps surprising and interesting features that such computations reveal, which are described in the downloadable supplementary text Modelling Violin Modes (http://acousticstoday.org/supplementary-text-violin- acoustics-colin-e-gough/), which also gives suggestions for additional background reading. Here, I simply invite those interested to view Video 2, Video 3, and Video 4 which il- lustrate the 3-dimensional vibrations of the A0, CBR, B1−, B1+, and higher frequency dipole modes computed first in vacuum, then with coupling to the air inside the cavity via
Figure 6. Transformation of the modes of the freely supported top and back plates into those of the assembled empty shell as a function of normalized rib strength varied over six orders of magnitude.
the Helmholtz f-hole resonance, and finally with the offset soundpost added.
Such computations validate and quantify a model for the violin and related instruments treating their modes as those of a thin-walled, guitar-shaped, shallow-box shell structure, with doubly-arched plates coupled together by the ribs, cav- ity air modes, soundpost, and coupling to the vibrational modes of the neck-fingerboard assembly, the tailpiece, and strings. This model can be understood by standard coupled oscillator theory and, I believe, accounts for all known vi- brational and acoustic properties of the violin and related instruments.
Acknowledgments
I am particularly grateful to the violin makers Joseph Cur- tin, George Stoppani, and Sam Zygmuntowicz for access to their data and encouragement, to Jim Woodhouse and Evan Davis for invaluable scientific advice, and to all my col- leagues at the annual Oberlin Violin Acoustics Workshops, who have provided valuable information and feedback dur- ing the lengthy development of the present model.
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