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Violin Acoustics
differences in the sounds of even the finest Stradivari and Guarneri violins. Puzzlingly, it is currently still difficult to identify which specific features of the acoustic response cor- relate strongly with differences in perceived quality – other than at low and high frequencies.
At low frequencies, Dunnewald (1991) and Bissinger (2008a) found that poor violins usually have a very weak sound out- put, whereas at high frequencies, the response of all violins is strongly influenced by the vibrating mass of the bridge. This is easily demonstrated by adding a mute to the top of the bridge, with the increased mass increasing its high-frequen- cy cut-off filtering action. This leads to a “softer,” “warmer,” and less intense sound, even for bowed notes played on the lower strings, which still involve important contributions from the higher frequency partials. The bridge mass and de- sign can therefore strongly influence the sound of an instru- ment.
At low frequencies, the bowing forces cause the bridge to rock backward and forward on the island area. The resulting asymmetric rocking then allows components of the bowing force in the rocking direction to excite both antisymmetric and symmetric volume-changing modes. In particular, it en- ables the vibrating strings to excite a single, volume-chang- ing, breathing mode primarily responsible directly and indi- rectly for almost all the sound radiated at frequencies in the signature mode frequency range (Gough, 2015b).
In addition to radiating sound directly, the b1− breathing mode excites the a0 Helmholtz f-hole resonance. The cou- pling between the component a0 and breathing modes re- sults in a pair of A0 and B1 normal modes describing their in- and out-of-phase vibrations.
Once the frequencies of the A0 and B1 modes are known, their monopole source strengths are automatically fixed. This follows from what is colloquially known as the “tooth- paste effect” or zero-frequency sum rule (Weinreich, 1985). Well below the a0 resonance, any inward flow of air into the cavity induced by the cavity wall vibrations will be matched by an equal outward flow through the f-holes. Because the source strengths of the coupled f-hole and breathing modes have to cancel at low frequencies, their contribution to the radiated sound is automatically determined throughout the signature mode frequency range, apart from the very small frequency range around their resonances when damping be- comes important.
In practice, the strongly radiating breathing mode is also 28 | Acoustics Today | Summer 2016
weakly coupled to the nonradiating bending mode of the body shell, illustrated to the right of the plot in Figure 6. This is a consequence of the different elastic properties of the arched top and back plates. When the shell breathes, the arched plate edges of the two plates move inward and out- ward by different amounts. This induces a bending of the body shell like the bending of a bimetallic strip induced by the differential expansion of the dissimilar metals. This is the origin of the coupling between the b1− breathing and b1+ bending component modes of the body shell. This results in the pair of B1− and B1+ modes, with relative radiating strengths determined by the amplitude of the component breathing mode in each (Gough, 2015b). Such a model de- scribes the dominant features of the typical low frequency acoustic response illustrated in Figure 4.
The introduction of the offset soundpost results in a lo- calized decrease and asymmetry of the shell-mode shapes across the island area between the f-holes. This and coupling to the f-hole mode result in a large increase in the compo- nent breathing mode frequency, increasing its coupling to the component bending mode. It also accounts for the asym- metric rocking of the bridge, enabling horizontal compo- nents of the bowing forces to excite the strongly radiating breathing component of any its coupled modes.
The soundpost and enclosed air also induce coupling of the breathing modes to the other nonradiating body shell modes and to the vibrational modes of all attached components like the neck, fingerboard, tailpiece, and strings. This is responsi- ble for the additional weakly radiating normal modes appear- ing as substructure in the acoustic response, as in Figure 4.
Modeling Violin Modes
A successful physical model for the resonant modes of the fully assembled violin needs to describe the relationship be- tween the modes of the assembled body shell and those of the individual plates before assembly and to show how the body shell modes are affected by their coupling to the cav- ity air modes within the shell walls, by the offset soundpost wedged between the top and back plates, the strings, and all other attached components like the neck, fingerboard, tail- piece, strings, and even the player.
Such a model is described in two recently published papers on the vibrations of both the individual plates and the as- sembled shell (Gough, 2015a,b). COMSOL 3.5 Shell Struc- ture finite-element software has been used to compute the modes of a slightly simplified model of the violin to dem-