13 May 2007

Violin Physics, Chladni Patterns, Mysteries of Tonewoods

Rosslyn Chapel chladni patterns in ceiling ornamentation
CMT: The timbral range of most instruments is impressive. But bowing a string on a member of the violin family is even more so—the vibrations produce remarkably rich harmonics. Oh, I know the vibrations of the strings are mechanically coupled to the bridge. And the bridge transmits the vibrational energy produced by the strings to the body through its feet, to the belly, and through the soundpost to the back plate. The vibration of the body determines sound radiation and sound quality, along with the resonance of the cavity between the plates. But how is it that the richness and diversity of the sounds in the violin family is so great? Often we’re led to focus on the physics of the strings. What about the physics of the wood of the front and the back plates? Is there much new these days in terms of physics or engineering analysis of this?

Violin Crossection

DSM: There’s an increasing number of physicists who are conducting research on vibrational dynamics of stringed instruments. And the computational power to do finite-element analyses and mathematical modeling of the arched plates’ vibrational modes is finally up to the task. KTH University in Stockholm is doing exciting things, for example. Look at that video above and the images of the violin below. Metal filings placed on the violin plate’s surface will form 7 or more patterns according to the plates ‘modes’ of vibration. The pattern shows the ‘node’ areas that don’t move, while all of the areas around them are vibrating. You want to match the modes between the front plate (belly) and the back plate when these are thicknessed. If on each plate you can get modes 2 and 5 to agree within one whole step, then your instrument will vibrate as a whole unit. This will ensure the instrument plays easily. The mathematics and computation are good enough these days to clearly reveal and explain what the differences are between good instruments and not-so-good ones. The difficulty remains, though, in getting mathematical models and numerically-controlled machining to automatically create superior instruments—as contrasted to merely explaining the goodness of hand-crafted ones.

Violin Plate vibration modes, University of New South Wales
[Photos from Dept of Physics, UNSW.]

CMT: The woods traditionally employed for the construction of the violin family instruments are maple or sycamore for the back, ribs and neck, and generally sitka spruce for the front plate or belly. The variety of maple used is usually Acer pseudoplatanus or Acer platanoides. The sitka spruce used is most often the Picea abies or the Picea excelsa. It’s sometimes said that wood suitable for violins is that which has fine-grained, narrow-ring structure—grown at high altitudes or under harsh conditions such as cold weather and poor soil. Wood that’s grown too quickly in warmer environments and rich soil generally tends to be less resonant. Luthiers prefer air-dried wood (not kiln-dried). Do luthiers still bite on the wood to assess whether it will be strong enough, like they used to do in the olden days?

DSM: The whole craft seems to have become far more quantitative over the past couple of decades, especially with the growth of luthier education programs. But the qualitative lore will never go away. You know, the quality of sound is more affected by the top plate than the back plate. For that reason, a one-piece back plate is not uncommon, but a one-piece top plate is rare. Generally, each plate consists of two pieces glued symmetrically lengthwise. This orientation is preferred because the grain is wider at the bark side and narrower at the heart, in the middle of the tree. So if a single piece of wood is split and glued bark side to bark side, the acoustical properties of the wood will be approximately equal on both sides of the center line where it’s glued. Despite the two-piece, center-glued arrangement there are still only 2% of trees in today’s commercial forested stands are large enough to yield usable tonewood. And of course that tonewood-scarcity / large-tree-availability situation is even worse for the larger instruments—the violas, cellos, and double-basses.

CMT: Carleen Hutchins had an excellent article in Scientific American back in 1981, with photographs showing vibrational characteristics of wood pieces after cutting to shape, and before and after application of varnish. The pictures showed metal filings scattered on the surface of the wood before inducing vibration, and the particles collecting at the nulls or nodes in standing waves set up in each plate.

DSM: Remember, the plates of a violin bulge outward. The size and shape of these bulges, or “archings,” have an enormous influence on an instrument’s sound, as do the nature and varying thickness of the wood, and all these quantities and qualities need to be measured and modeled. What’s advanced quite a lot in the past few years is the ability (in MATLAB or in other software applications) to mathematically model the vibrational dynamics of those 3-dimensional, arched plate surfaces . . .

CMT: The front and back plates are carved mostly by hand, with ever-finer tools as the process unfolds. The tools include thimble-sized finger planes and tool-steel miniscrapers. I was interested to learn that, in making a copy of a Strad or other premier instrument, the luthier makes the plates’ outer arch match the outer arch of the original instrument. But the thickness and the inner contour is governed by the acoustics of the specific wood that the copy plate is made of. The acoustics are continually suggested evaluated by the changing sound of finger-elicited “tap tones” as the wood is gradually thinned from the inside with the finger planes and miniscrapers. The ‘copy’ is not, in other words, an exhaustive, explicit dimensional copy. Only a subset of the dimensions and contours of the original are used; the rest of the process is governed by the de facto acoustics, iteratively, by trial-and-error.

Violin Plate finger planing
DSM: The first physicist who was enthusiastic to study violin acoustical physics was Felix Savart, in the 19th Century. Boundless optimism, confronting such complexities with only rudimentary tools. Both the experimental sensors and the mathematics would not catch up with the size and scope of that challenge for 150 years! Savart experimented with the vibrational properties of violin plates, sprinkling the plates with sand and observing the patterns made when a bow was run across the edge. This technique is still used today in modified form by some luthiers to “tune” their plates.

CMT: The first American scientist to embark on ‘big science’ of the violin was Frederick Saunders, an amateur violinist who was chairman of the Physics Department at Harvard from 1926 to 1940. But there have been legions of others. University of Michigan physicist Gabriel Weinreich has now spent more than 20 years studying violin physics, including “directional tone color”—how different notes radiate from different parts of the violin’s body in different registers and at differing loudnesses. The ‘bug’ for the challenge of understanding violin physics is easy to ‘catch’, despite the difficulties that the challenge holds . . .

Violin sound production
DSM: What makes a violin belly a ‘good’ belly? The goal is to have many different vibrational modes spaced relatively uniformly. Another goal for a violin’s sound is a high “Q” (sharpness) of the resonances. With string instruments a high Q is desired because the aim is to create sound ‘character’. A cheap instrument has few resonances. Each of these has relatively low Q, and the sound dies out soon after you stop bowing. A good instrument continues to sing for some time after you stop bowing, because the resonances are sharp (high-Q) and do not dissipate the energy quickly. So a good string instrument has many high-Q resonances at favorable frequencies. Amazing, given how technology has advanced, that so much of the physics of these instruments still defies explanation. Anders Buen in Norway has some very nice MATLAB-based studies of what we’ve been discussing here. But the mesh-free ('Galerkin', 'meshless', 'MLPG', etc.) finite-element methods have not, so far as I can tell, been brought to bear on violin plate dynamics yet . . .

Anders Buen MATLAB violin study

Chladni Vibration Patterns, 3D