5.3.0 The origin of the signature modes of a violin

The text of this section is reproduced from section 3.2.3 of reference [1]: Copyright © IOP Publishing, reproduced with permission, all rights reserved.

Until recently, no very persuasive description had been given of why the low modes of a violin body should take the particular forms shown in section 5.3, or how an instrument maker might be able to influence the details (and in particular the sound radiation from them). However, an extensive series of explorations by Gough using Finite Element (FE) analysis is now giving answers to these questions. Other researchers have built FE models of the violin (for example Knott [2], Bretos et al. [3]), but their goal has generally been to reproduce the complicated structure of a violin body, and demonstrate agreement with experimental results. Rodgers went beyond this in a series of papers using FE models to explore the effects of selective wood removal in an effort to give useful guidance to instrument makers: for example [4,5,6]. But Gough has done something more fundamental: he has explored a series of models starting from very simple assumptions, then gradually adding in the complications of violin design one at a time so that the progressive evolution and emergence of the signature modes can be charted, and the relative influence of various contributory factors assessed [7,8]. The full picture revealed by Gough’s work is complicated, but an abbreviated caricature will be given here to show some key aspects.

A useful starting point is to consider a violin body with no bass bar, soundpost or neck. Furthermore, the top and back plates will be initially supposed to be identical, with no f-holes in the top. Under those conditions, the box has two planes of mirror symmetry: back-front and left-right. Every mode shape of such a box must be either symmetric or antisymmetric in each of these planes, giving four types of mode by combining the possibilities. Typical examples of the first few such modes are illustrated in Fig. 1. They do not yet look very much like the modes of the final violin, but they make a useful set of “component modes” or basis functions, which can be used in linear combinations to describe the evolving make-up of actual mode shapes when the two symmetries are successively broken as the model is made progressively more realistic.

Figure 1. Modes of a doubly-symmetric violin-shaped box, to serve as basis modes for building up the modes of a more realistic model (see text). Computed with FE analysis, reproduced by permission of Colin Gough.

A key aspect is that only one of the shapes shown here, the one labelled “breathing”, involves significant net volume change. Most of the others are antisymmetric in one or both symmetry planes, so the volume change is exactly zero. The mode labelled “Longitudinal dipole” is symmetric in both planes, but it has almost equal and opposite volume change in the upper and lower bouts. That means that the low-frequency sound radiation of each mode of the later models will be determined (predominantly) by the amount of this “breathing” shape that goes into its make-up. Familiar patterns of behaviour of coupled oscillators then lead to insights into the frequency dependence of that sound radiation.

Another aspect to notice in these shapes is that some have particularly large motion around the centre bouts (the waist of the box). The origin of this motion lies in the particular three-dimensional form of the typical violin box. British violin maker George Stoppani has produced a striking physical demonstration, by making a “floppy violin” with a normal rib garland, but very thin top and back plates made by hot forming of thin laminate sheet . The result was a structure that could be readily flexed in the fingers. With this, it was easy to discover that there is a particular deformation in which one C-bout area could be flexed and rotated: this is the motion that appears in symmetric form in the “Bending” mode and in anti-symmetric form in the “Centre bout rotation” mode of Fig 1. The fact that the deformation can be done in the hands, with plates that allow bending but are still stiff with respect to stretching deformations, strongly suggests that it is, at least approximately, an inextensional motion of the box. As Rayleigh pointed out in his pioneering work on the vibration of thin shells [9], if such an inextensional motion is kinematically possible for a given shell geometry, it is very likely to appear as the whole or a significant part of the low-frequency vibration mode shapes, because such motion represents a strong local minimum of potential energy.

The next step towards a normal violin is to add the f-holes, and allow the top and back plates to be more realistically different in details (but still with no soundpost, bass bar or neck). This breaks the front-back symmetry, but preserves the left-right symmetry. It also has the advantage of being close to a stage that most violin makers go through while building an instrument (although in reality the bass bar will always be present, but this turns out to be a relatively minor perturbation). That means that experimental data can be collected from violin bodies in this state, and compared with the FE model predictions.

Several effects are introduced at this stage. First, the addition of f-holes introduces the possibility of a Helmholtz-like resonance based on the stiffness of the air inside the cavity, and the effective mass of air flowing in and out through the f-holes. This new degree of freedom couples strongly to the breathing shape from Fig. 1, leading to a well-separated pair of modes that lie above and below the frequency of an ideal Helmholtz resonance in a cavity of the same shape but with rigid walls. This adds a useful additional low-frequency mode, usually designated A0, and also pushes the second mode of the pair significantly higher in frequency than the “breathing” mode in Fig. 1.

Depending on the details, this second mode may be pushed quite close to the frequency of the previous “bending” mode, introducing the second major change: since these shapes both have left-right symmetry they can couple together to give a pair of modes involving elements of both breathing and bending. The coupling process involves a phenomenon often called “curve veering” (see for example Perkins and Mote [10], Pierre [11]), whereby the two frequencies never become equal. Instead, the mode shapes change character through a frequency range around where one might have anticipated two equal mode frequencies, and emerge at higher frequencies having swapped characters entirely.

Now recall that of the two “coupled oscillators” being considered here, the modified “breathing” shape and the “bending” shape, only one is responsible for significant sound radiation. That means that the coupled modes of the system can display a range of sound radiation behaviour, depending upon whereabouts in the veering process a particular instrument finds itself. One mode might radiate sound much more strongly than the other, or both may have comparable sound radiation. This balance of radiation between the two modes can be quite sensitive to details of the violin structure, and it may point to a significant source of variability between instruments, and also to a significant way that a maker might exercise control over the sound. Variations in sound radiation behaviour in line with this description have been observed in real violins.

To preserve mode orthogonality, the two modes always combine the two characters, bending and breathing in this case, with opposite phases. This “sum and difference” combination is the first stage of the formation of the modes B1- and B1+ seen in Figs. 5c,d of section 5.3. Examples of the modes, compared to experimental measurements on a normal violin at this stage of construction, are shown in Fig. 2. The FE model still has many simplifications that have not been described in detail here, but nevertheless the agreement is very encouraging.

Figure 2. Signature modes of a violin body without neck or soundpost. For each mode, the left pair shows the top and back plates as measured while the right pair shows simplified FE computations. Colour scales show motion: cold and warm colours denote opposite signs. Measurements reproduced by permission of George Stoppani, FE results by permission of Colin Gough.

At this stage, the proto-violin has three low-frequency modes exhibiting significant volume change, and hence able to radiate sound effectively: A0, and the embryonic B1- and B1+. However, there is a crucial snag. When a violin body is driven by the bowed strings, the excitation force at the bridge is predominantly side-to-side: in other words, it is antisymmetric with respect to the left-right symmetry plane. That means that the strings will be unable to excite any of the three strongly radiating modes, which are all symmetric in that plane. The remaining symmetry needs to be broken. The main way this is done in a normal violin is by introducing the soundpost, although the bass bar also contributes some asymmetry.

The effect of the soundpost on these low-frequency modes is to couple a certain amount of the shapes “Centre bout rotation” and “Transverse dipole” from Fig. 1 into the modes A0, B1- and B1+, so that the nodal line on the top plate is shifted away from the centre-line to pass approximately through the soundpost position. This achieves the effect of giving those modes a significant component of rotation in the plane of the bridge, and thus allows them to be driven effectively by the bowed strings. Adding the soundpost, and adjusting its exact position, also makes further changes to the mode frequencies. These effects can still be interpreted in terms of “coupled oscillators” representing the original component motions, undergoing various veering interactions as the soundpost is adjusted. This goes some way towards explaining the sensitivity to soundpost adjustment that is found in practice: in the completed violin, the soundpost (and to a lesser extent the bridge) are the only components that remain adjustable in order to give the violin maker some fine-tuning control over the low-frequency modes, and their associated pattern of sound radiation which varies according to how the original “breathing” component shape is distributed among the modes A0, B1- and B1+.

Many details have been omitted in this account, including the role of plate curvature and the influence of anisotropy of the elastic properties of wood. One of the more surprising findings of the FE studies is that the effects of plate arching are so dominant that once fairly realistic arching is included in the model, the anisotropy of the material properties has a relatively minor effect on the frequencies and mode shapes of the signature modes. This is true despite the fact that typical spruce used for violin top plates may show a cross-grain stiffness (measured by Young’s modulus) about 10–20 times lower than the long-grain stiffness.

[1] J. Woodhouse. The acoustics of the violin: a review.  Reports on Progress in Physics 77, 115901.  DOI 10.1088/0034-4885/77/11/115901 (2014).

[2] G. A. Knott, A modal analysis of the violin using MSC/Nastran and Patran, Msc Thesis, Naval Postgraduate School, Monterey, CA (1987).

[3] J. Bretos, C. Santamaria and J. A. Moral, Vibrational patterns and frequency responses of the free plates and box of a violin obtained by finite element analysis, Journal of the Acoustical Society of America 105, 1942–50 (1999).

[4] Oliver E. Rodgers, Influence of local thickness changes on violin top plate frequencies, Journal of the Catgut Acoustical Society, 1 6–10 (1991).

[5] Gareth W. Roberts, Finite element analysis of the violin. Extract from doctoral dissertation, University College, Cardiff, reprinted in Research Papers in Violin Acoustics 1975–1993 ed C. M. Hutchins and V. Benade (Woodbury, NY: Acoustical Society of America) pp 575–90 (1997).

[6] Oliver E. Rodgers, Influence of local thickness changes on violin top plate frequencies: Journal of the Catgut Acoustical Society Series II, 2 14–6 (1993).

[7] C. E. Gough, Violin plate modes, Journal of the Acoustical Society of America 137, 139—153 (2015).

[8] C. E. Gough. A violin shell model: vibrational modes and acoustics. Journal of the Acoustical Society of America 137, 1210–1225 (2015).

[9] J. W. S. Rayleigh:The Theory of Sound (1877, reprinted by Dover, New York 1945).

[10] N. C. Perkins and C. D. Mote, Comments on curve veering in eigenvalue problems, Journal of Sound and Vibration 106 451–63 (1986).

[11] C. Pierre, Mode localization and eigenvalue loci veering phenomena in disordered structures, Journal of Sound and Vibration 126 485–502 (1988).