Guitar Acoustics 101
From a talk to the Seattle University Physics Club,
February 22, 2007
Dr. John A. Decker, Jr.
RainSong Graphite Guitars & guitarmasterworks
My training is as a Theoretical Aerodynamicist – B.Sc. & Degree of Aeronautical Engineer from M.I.T. – and as a Plasma Physicist – Ph.D. from Cambridge University – having drifted from aerodynamics to magnetohydrodynamics to plasma physics during graduate school.
My Ph.D. thesis was on magnetoacoustic waves in cesium plasmas, and I worked for several years afterwards doing plasma-stability modeling experiments in thermally-generated cesium plasmas – so-called “Q Machines” – exploring the effects of “Minimum-B” magnetic field geometries on plasma losses.
Q-Machines work by taking advantage of a quirk of nature: the ionization potential of cesium is lower than the work function of tungsten, so a cesium atom directed at a tungsten plate in a vacuum is emitted minus an electron as a positive cesium ion. If one then heats the tungsten plate hot enough so that its thermionic emission of electrons balances the positive current of cesium ions, the result is a low-pressure neutral plasma with no imbedded electrical currents – a so-called “quiescent plasma” – hence the name of the apparatus.
A Q-Machine plasma is cold – the ions are at a temperature of a small fraction of an electron-volt – but since the cesium ions are heavy theorists were convinced that such a cold, heavy cesium plasma would have the dynamics of a relativistic deuterium plasma – the sort of 100-million °K plasma in which fusion reactions should occur. And almost all Q-Machine experiments were designed to test stability predictions for proposed Controlled Thermonuclear Fusion reactor designs.
As it turned out, the experiment we at first thought proved the utility of minimum-B field geometries wound up demonstrating that what we thought were wave-driven instabilities were, instead, simple E x B particle drifts (OUT, of course) caused by thermal gradients in the white-hot tungsten ionizer plate which generated the plasma. Since my group’s work demonstrated that all the observed supposed “plasma instabilities” were, in fact, artifacts of the experimental apparatus, and that the thermal-uniformity requirements on the tungsten ionizer plate needed to avoid these artifacts was about two orders of magnitude beyond the current state-of-the-art, I was left with the choice of deciding I didn’t believe my own work, or finding something else to do with my professional life.
Since I was an experimentalist and a measurements specialist, with expertise in optical methods, I moved to running a small engineering-and-manufacturing group which made optical and x-ray instruments for sounding rockets and observatory spacecraft.
I wound up being co-inventor of binary-transform infrared spectrometry -- so-called Hadamard Transform Spectrometry, named after the 19th Century French mathematician who pioneered binary orthogonal functions. This has the signal-to-noise improvement of Fourier Transform Spectrometry but can be implemented with dispersive spectrometers and moving slit arrays, rather than the hyper-precise moving-mirror interferometers used by FTS.
This, in turn, led, by way of a decade running a small company making HTS instruments, several years as R&D Manager of a major semiconductor-equipment manufacturer and a short, trying, existence attempting to get a start-up manufacturer of photovoltaic solar panels off the ground, to being hired as Managing Director of the Air Force optical observatory on the summit of Haleakala, on Maui, Hawaii. The link was that I had become a specialist in the design and management of large and/or complex computer-operated optical systems.
Large telescopes are beautiful pieces of apparatus, and the work the Haleakala observatory was doing was exciting. But I discovered that running a government facility was really not to my liking – far, FAR too much politics – so, since I had already run two new-start-up companies, I went back to working for myself, designing & making hand-held optical systems run by small pocket computers. My company’s primary product was a fully-automatic marine sextant built around a Hewlett-Packard pocket computer, which would not only reduce sun, moon and star sights to latitude & longitude, but would identify the celestial object you sighted – “Celestial Navigation for Dummies”. This was potentially a life-saver for sailors in trouble with no electronics or power.
And, since I had started to play the classical guitar, and I had moved to the hot, humid tropics, I very quickly became aware of the effects of climate on wood musical instruments: they warp, their necks twist, they come unglued. Since I was familiar with advanced materials, primarily from their use in space instruments, I decided that “there had to be a better way” and started the long process which led to the invention of the RainSong Graphite Guitar. Which finally brings me to the topic of this paper:
Acoustics of Stringed Musical Instruments
Guitars strike most people as having nothing whatever to do with Physics. In my experience, however, Guitar Acoustics has been the most challenging physics of my whole career.
The reasons are two:
- Guitar Acoustics theory is complex to the point of being unsolvable – the theoretical description of the vibrations of a guitar soundboard leads typically to a set of a thousand simultaneous nonlinear partial differential equations; and
- Measurements tell one almost nothing useful – what one hears when a guitar is being played -- and measures on an oscilloscope or frequency analyzer – is usually dominated by the excitation mechanism, and the frequencies used by the human brain to determine “quality” look like white noise to laboratory instruments.
I’ll go into both of these factors in more detail below, but first I want to acquaint you with the “Milieu One”/”Milieu Two” dichotomy. Astrophysicists like to say that the Universe is divided into two “milieus”.
“Milieu One” is characterized by comparatively simple phenomena, elegantly described by theories based on first principles. “Milieu Two” is characterized by complex phenomena, describable only by empirical theories.
In Astrophysics, the boundary between these “milieus” is at the farthest distance to which we have sent a spacecraft. Where we haven’t visited is “Milieu One”: simple phenomena, understood on the basis of first principles. Examples are galactic formation or the Big Bang.
Places we have observed close-up are “Milieu Two”: complex phenomena, describable only on an ad hoc basis. An example is Lunar “Geology”, which we thought we understood until we brought back samples which didn’t fit the theories at all – I remember sitting in the Cornell University Faculty Club right after the Apollo sample-return missions talking with Lunar “Geologists” & Lunar “Geochemists” who felt their whole careers had just been destroyed!
Musical Instrument Acoustics is a “Milieu Two” situation: the phenomena are complex and interact nonlinearly, simple theoretical descriptions don’t fit the observations, and one is forced to fall back on empirical relationships and ad hoc descriptions.
For an example of the difficulty of applying simple acoustics theory to guitars,
Let’s consider the Physics 101 description of the vibration of a string stretched between two fixed supports: a fundamental frequency determined by string weight, string tension and the distance between the supports (called “the scale length” in guitarmaking), and an array of harmonics at integer multiples of that frequency. This is what you observe if you pluck a piece of fishing line tied tightly between two nails, and is also what you observe for the lower modes of a violin or cello, and for most solid-body electric guitars, archtop guitars & banjos.
It is not observed, however, for classical guitars or flat-top steel-string acoustic guitars or for similar instruments where the strings are anchored in the soundboard. For these instruments, the fundamental frequency is missing – the lowest frequency observed is the first harmonic.
It turns out that this is caused by the way the strings excite the soundboard. A guitar string vibrating by itself produces almost no audible sound – one can sit next to someone playing an unplugged electric guitar and not hear it. The sound of a guitar – heard by the ear or measured by a laboratory instrument – almost all comes from the soundboard, the “top” of the guitar, which serves to amplify the vibrations of the strings, matching the acoustic impedance of the instrument to the “free space” of the room in which the guitar is being played.
Antonio de Torres, who invented the modern “classical guitar” in 18th Century Spain, is generally credited with being the first to realize that essentially all the tone of a guitar comes from the soundboard – the “top” – and that the back, sides and neck play essentially neutral structural roles except in pathological situations. This is still controversial in some circles, but Torres proved his point by building a guitar with a fine spruce top and a paper-maché body. I have a recording of Spanish guitar music played on this guitar and on several of Torres’ fine rosewood-body instruments, and the tone quality is identical!!
For a guitar in which the strings are anchored to the soundboard itself – the invariable design for classical guitars, including all of my guitarmasterworks guitars, and the usual design for flat-top steel-string acoustic guitars like the RainSong – the soundboard is excited not directly by the lateral vibration of the strings but by the periodic variations of string tension created by those vibrations. And this variation in string tension occurs at twice the fundamental frequency -- a tension maxima occurs for each of the two lateral excursion maxima per vibration period.
Hence, the fundamental is not excited for classical & steel-string acoustic guitars.
A similar absence of the fundamental also occurs in Q-Machines, where it is caused by resonance between acoustic waves in the plasma and the neutral cesium atom jet onto the ionizer plate, which is why I happened to notice the effect in guitars.
An accurate theoretical description of the vibrations of a guitar soundboard needs to take into account:
- that the soundboard vibrates in two dimensions, producing two-dimensional modes which are a function of the geometry;
- that the back and sides of the guitar are also vibrating independently, as is the air inside the guitar – all of them in complex two- or three-dimensional modes;
- all of these modes – soundboard, back & body cavity – interact with each other; and
- since wood is a lossy medium (strongly so above about 1 kHz), these interactions are nonlinear.
The result for a typical situation is a set of around a thousand simultaneous nonlinear partial differential equations -- one equation for each mode, including all the harmonics. Closed-form solutions are only possible for the lowest modes of highly-simplified geometries, never for realistic representations of real guitars. Numerical solutions typically require large digital computers even for calculations of the lowest modes of somewhat simplified geometries.
The situation isn’t much better in the Measurements area.
It is actually quite easy to record the frequency spectrum of a guitar: excite it with a speaker hooked up to a frequency generator, then record the acoustic energy as a function of frequency with a microphone and an oscilloscope or frequency analyzer. Or one can simply play a guitar into a microphone and feed the output directly into a frequency analyzer.
The problem is that the lower few modes stand out cleanly, while the higher frequencies all blend into what looks like white noise on the oscilloscope – actually it is usually “pink noise”, with slightly more energy at lower frequencies.
Essentially ALL acoustic guitars display the same lower-mode spectrum – a priceless 18th Century masterpiece has essentially the same lower modes as a piece of junk bought in a Mexican flea-market. And the frequencies which the human ear-and-brain use to determine “quality” lie right in the middle of the white-noise band! To an oscilloscope or frequency analyzer, junk guitars and masterpieces look the same!
There have been numerous “blind playings” of acoustic guitars and violins, the instruments played behind a curtain by a skilled musician, the sound frequency spectra recorded by instruments in front of the curtain, and a panel of listeners also in front of the curtain. For many years, such “blind guitar tastings” were a feature of the bi-yearly conventions of the Guild of American Luthiers and the Association of Stringed Instrument Artisans. I’ve taken part in them, they are lots of fun, and they are VERY informative.
The invariable result of all such “blind tastings” is that the listeners have no trouble distinguishing a junk guitar from a fine guitar, or a Stradivarius from a junk violin, just by listening to them being played. But even Ph.D. physicists who have specialized in Musical Instrument Acoustics for their whole careers, cannot distinguish “fine” from “junk” instruments on the basis of their acoustic frequency spectra or sonograms.
The ultimate result is that luthiers – originally, persons who made lutes, but now persons who make any stringed musical instrument – depend for guidance on the trained human ear, not on acoustic theory or laboratory measurements.
As one amusing example, I remember a question following a paper on the theory of body cavity modes at a guitar-acoustics seminar, where the presenter happened to be both a physicist and a luthier. He was asked about the correct relationship between the size of the soundhole and the dimensions of the guitar body cavity – a topic about which there is a large body of theory in the literature. His reply was “Well, if the soundhole is too small, you can’t get your hand inside to work on the innards of the guitar, and if it’s too large, it looks funny.”
The situation isn’t totally hopeless for the application of physics to guitarmaking, as careful study of the equations can lead to scaling laws and approximate solutions which can be quite useful, and, in fact, which were essential in the development of the RainSong Guitar.
Similarly, one can measure the 2-dimensional shapes of the soundboard modes, by supporting the soundboard horizontally from the edges while exciting it with a speaker and frequency generator, sprinkling a light contrasting-color powder on the soundboard, and noting that the powder collects along the node lines where the vibration amplitudes have minima. These patterns can be photographed, can provide valuable information to the soundboard designer, and were the basis for a “breakthrough” in the design of the RainSong’s soundboard, which I use in my guitarmasterworks guitars.
LONG Excursis: the development of the RainSong Graphite Guitar
When I first looked at the problem of making an acoustic guitar out of composite materials, I did what any good physicist would do: I looked up all the back copies of
The Journal of Guitar Acoustics and The Journal of the CatGut Acoustical Society and wrote down the equations for the vibrations in a guitar soundboard.
And I recognized them!
These were the same equations I had struggled with years before in graduate school,
working on magnetoacoustic waves in plasmas and I had some understanding of their properties. They described acoustic waves which propagated at different speeds in different directions – for plasma waves the magnetic field lines defined the preferred direction, for a guitar soundboard the direction was defined by the wood grain.
Since my starting point was to duplicate, as closely as possible, the acoustics of a fine wood guitar, I used these equations to develop a scaling law – a formula which related the properties of a fine spruce soundboard to the properties required of a composite-materials soundboard. In specific, it related the stiffness and density needed by a composite soundboard to the stiffness and density of a spruce soundboard.
The stiffness in the formula was to be matched both along the wood grain and across the grain, which was comparatively straightforward. Density matching proved to be more interesting. In the plasmas I had studied, the “thickness” – the third dimension – was comparable to the other dimensions, so one had to use the full 3-dimensional form of the equations. As a guitar soundboard, however, is quite thin in comparison to its length and width, the terms describing vibrations within the thickness of the soundboard are far smaller than the terms describing vibrations along the surface of the soundboard, so I could use a much-simplified two-dimensional form of the equations. This, in turn, meant that I needed to match, not the bulk densities (grams/cubic centimeter), but the area densities (grams/square centimeter).
This sounds obvious, but let me assure you that it was NOT obvious in advance. The published scholarly papers and a great deal of work by some major guitar manufacturers’ R&D labs all assumed that a composite-materials soundboard must match the bulk properties of a wood soundboard. And I wound up with a US Patent based on this “area” scaling law.
Armed with this scaling law, and a great deal of materials data from the library on both the woods used in musical instruments and a wide variety of man-made materials, I compiled a list of candidate materials from which a soundboard might be made. Aluminum has roughly the right stiffness and density, and aluminum acoustic guitars were made back in the 1950s, but the thermal expansion of aluminum is so high that aluminum guitars would expand out of tune under stage lights. Steel was ruled out because it is several times too heavy for a given stiffness. And most other metals were simply not stiff enough.
The most common “composite materials” – that is usually defined as a material like reinforced concrete where the tension is taken by one element (the “rebar”) and the compression by another (the concrete) – materials like fiberglass, micarta and particle board, were discarded because they, too, were too heavy for their stiffness. The bulk densities of most carbon-fiber-based composite-materials – carbon fibers in a resin matrix, generally called “graphite” in the trade – are also too high to match spruce, but the area densities can be made to be an excellent match.
I then made up around five hundred “coupons” – 4” by 6” thin panels – using various combinations of resins and fibers, chosen to match the area density and stiffness
required by my scaling law. By this time I had talked a master luthier, Señor Lorenzo Pimentel, of Albuquerque, New Mexico, to teach me how to make a guitar. He taught me how to check the “tap tone” of a panel – how to hold it by a corner as I tapped it with a knuckle – and what sort of tone I needed to hear to be able to make a fine-sounding guitar top. Lorenzo routinely selected the wood for his soundboards using “tap tones”, and I used “tap tones” to select the three or four best candidate materials – all variations on carbon fibers in an epoxy-resin matrix, as it turned out – from the hundreds of “coupons” I had made.
I then made prototype soundboards from these best-candidate materials, went out and bought some old beat-up guitars at the swap meet, cut off their soundboards and replaced them with the “graphite” soundboards, producing a handful of prototype “graphite guitars”. Based on others’ efforts to make a composite-materials guitar soundboard, I was expecting these prototype guitars to sound really BAD – tinny and thin, and not loud enough to be useful -- which would have allowed me to abandon the project with only a hundred dollars or so down the drain.
I was flabbergasted when our very first prototype sounded really GOOD, with the clear trebles for which “the graphite sound” has become known, and a full, rich tone throughout its whole range.
It took another ten years of R&D and Manufacturing Engineering – and an incredible amount of money -- before we had a product with the sound we wanted which we could produce reliably for a reasonable cost. But despite all the years of constant R&D, the first production RainSong Graphite Guitar to come out of our Kihei, Maui, factory in the spring of 1995 had essentially the same soundboard design of our first junk-guitar prototype back in 1985. It looked a lot better, was easier to play, and had more refined tone. But our first analytical/empirical guess at the required materials properties turned out to be “spot on”.
As I mentioned above, mode pattern measurements provided the basis for a major “breakthrough” in the soundboard design used both by RainSongs and by my guitarmasterworks guitars.
All classical and acoustical guitars – and all of the violin-family instruments – have braces on the inside of the soundboard whose primary function is to keep the soundboard from buckling under string tension. Originally – and still for the violin family – these were simple longitudinal bars -- “bass bars” for the violin, as they are on the bass side of the soundboard.
Later they evolved into ladder-like structures combining longitudinal and lateral bars, as are still used on lutes, ouds and similar “ancient instruments”. Starting with Torres and his followers with classical guitars, and continuing with Martin and the other developers of the modern steel-string acoustic guitar, the soundboard bracing became much lighter and was arrayed in fans or webs.
Luthiers universally believe that the size and location of the braces significantly effect the tone of the instrument, and, while some designs – like Martin’s “X-braces” for flat-top steel-strings – entered into the common literature, most luthiers consider their bracing designs as proprietary trade secrets. Lorenzo Pimentel, for instance, made me swear – wanted to have me sign in blood – that I would never share his classical-guitar bracing pattern which he graciously allowed us to use on our first classical RainSongs.
One of the more interesting bracing designs is the “Kasha” system, named after a physical chemist from Florida who first proposed that the soundboard vibrated in parabolic patterns, and that the braces should have parabolic shapes – and be arrayed in parabolic fans – to optimize the guitar’s tone. Some very fine guitars have been made using “Kasha bracing”, most notably by the late Richard Schneider, of Port Townsend, WA, and by one of my mentors, an Israeli woman named Gila Eban who has lived and worked on Long Island, NY, for some years.
I was planning to make a wood guitar using this system as a prototype for a “Kasha braced” classical RainSong, and as part of the design process did an intensive literature search on Kasha soundboards. I eventually found a paper – I believe it was in the Journal of Guitar Acoustics – which had photographs of the two-dimensional vibration modes measured for a Kasha soundboard made by Richard Schneider.
The more I looked at these mode photographs, the more puzzled and disturbed I became.
Mode photographs basically record the two-dimensional node lines – the lines of minimum vibration amplitude – at a particular excitation frequency. These node lines vary with increasing frequency, of course, as the soundboard goes from vibrating as a single plate (the fundamental) to vibrating in two halves, with the mode line roughly at the middle of the top,
through increasingly complex 2- and 3-dimensional modal geometries, to a high-frequency limit where the motion is so complex no clear nodes are observed.
The Kasha bracing pattern was claimed to encourage high-frequency modes on the treble side of the guitar, and to encourage lower-frequency modes on the bass side of the guitar.
If this was, in fact, true, one would expect to see small nodes on the treble side and widely-spaced nodes on the bass side.
This was not what was observed.
In fact the photographs of the modes of the Kasha-braced top were identical to those of the modes for an unbraced top plate. Further, if the braces glued to the back of the soundboard had, in fact, a causal effect on the mode structure, one would expect the node lines to at least roughly follow the brace pattern since the heavy braces would be expected to locally constrain the vibration of the parts of the soundboard to which they were glued.
In fact, the node lines crossed the location of the braces at arbitrary angles for all frequencies, which meant that the braces were having essentially NO effect on the vibration modes, and were serving simply to provide structural strength and prevent the top from collapsing under string tension.
After thinking about this for a while, my training as an aircraft designer kicked in, I calculated Nyquist Diagrams – amplitude vs frequency plots for resonant systems – for the guitar as a dynamic system and I decided that a guitar soundboard really was quite similar to an airplane wing or a speaker cone, and simply needed to be
- as stiff as possible,
- as light as possible,
- and to have no resonances in the audible frequency range.
This led directly to a lightweight bracefree soundboard design for the RainSong, which we call “Projection-Tuned Layering™”, and on which we have a US Patent. It relies on modern aerospace structural design approaches, and consists of a “sandwich” structure
- a thin carbon fiber-epoxy face plate or “top”,
- backed up by a “spacer” layer of lightweight foam,
- with a second even-thinner carbon fiber-epoxy “backing plate”.
The resulting guitars have full warm tone, and graphite’s “signature” clear treble range.
Projection-Tuned Layering™will work with wood soundboards, but it requires extremely difficult craftsmanship, because wood is orders of magnitude less stiff than graphite for a given weight, and wood sandwich soundboards constructed with normal thicknesses usually sound “overbuilt” and dull. To make the technique work, it is necessary to make the top and particularly the backing plate as thin as possible, to the point where the wood is translucent (see photo 1).
I have licensed RainSong’s patented PTL technology, and have used it in all my recent guitarmasterworks guitars.
guitarmasterworks soundboards (see photos 2, 3 & 4) typically have
- a spruce or cedar top thinned down to 1.5 mm (1/3 of normal thickness),
- a 1/8”-thick Nomex honeycomb spacer, and
- a spruce or maple backing plate thinned down to around 0.3 mm (see photo 1).
I also make a point of making the attachment of the soundboard to the sides of the guitar as flexible as possible, to come as close as possible to achieving the “free plate” vibration of theory, by moving the honeycomb and backing plate away from the sides, and attaching only the top plate to the rest of the guitar (see photo 5).
Finally, I would like to note that “The Carbon Sound” page of RainSong’s web site www.rainsong.com/sound/
contains a bibliography on musical instrument acoustics.