Dynamics of an Archtop Jazz Guitar

Mark FrenchDepartment of Mechanical Engineering

University of Michigan – Dearborn3901 Evergreen Rd

Dearborn, MI 48128-1491Markfrench@aol.com

AbstractThe dynamics of an archtop jazz guitar have beenexplored using a scanning laser vibrometer. Thenatural frequencies and mode shapes of theinstrument were captured. Also, the effect ofhaving two soundholes was investigatedexperimentally.

BackgroundInvestigations of the structural dynamics of guitarsand instruments of the violin family have appearedin the literature for several decades. In nearly allcases, though, the instruments were very traditionalones (like violins) or guitars with flat tops. Anapparent omission from the literature is the archtopguitar.

For the purposes of this paper, one might considerthis type of instrument to be a cross between violinsand flat top guitars. Archtops have the same typeof curved top and back plates as the violin. Theyare also traditionally made of the same materials(spruce top and figured soft maple back). Finally,they usually share geometry of the openings cutinto the top (f-holes).

However, the internal structure of the archtop isunlike either the flat top guitar or the violin. Itgenerally has a symmetrically braced top and thereis no mechanical connection between the top andthe back other than at the edges. In violins, there isa single brace placed off center and a vertical post,called a sound post, connecting the top and back(see Figure 1). Steel string folk guitars usuallyhave asymmetric bracing in a roughly X-shapedpattern while nylon string classical guitars oftenhave a symmetric fan-shaped bracing pattern.

The mode shapes of flat top guitars tend to follow apattern [1,2]. The air in the body dominates the firstmode, which is often called the Helmholz mode. Itis not strictly a Helmholz resonance since the wallsof the cavity are not rigid, but it is analogous. Inany case, the terminology is probably tooentrenched to be changed now. Since there are nonode lines either parallel to or perpendicular to theaxis of the instrument, it can also be identified as a(0,0) mode in keeping with accepted terminology ofplate modes. The second and third modes are also

(0,0) modes. One is created by the motion of thetop and back in phase and the other is due tomotion out of phase. Higher modes are typicallymore complex. Predictably, X-bracedinstruments tend to show asymmetricdeformations while symmetrically bracedinstruments (usually classical guitars) tend toexhibit more symmetric modes. Figure 2 showsthree of the lower modes of a representative flattop guitar.

Note that the first two modes (105 Hz and 208 Hzrespectively) are quite similar. These are two ofthe three lower modes generally found in flat topinstruments. The next mode (336 Hz) is alsofairly typical. It is very asymmetric with a nodeline that coincides with one of the heaviestbraces on the soundboard.

The InstrumentAlthough there is something of a renaissance inarchtop guitar making under way, most are madein small batches and largely by hand. Theinstrument used here was made by the author(see Figure 3). The top was carved frominstrument grade spruce. The back was carvedfrom soft maple and the sides were bent fromsycamore. The internal bracing follows one ofthe two traditional patterns and consists of twosymmetrically placed longitudinal braces.

Test MethodsThe instrument was freely supported from a stiffaluminum channel frame by rubber bands. Therigid body frequencies of the instrument were nohigher than a few Hertz, so no interaction wasexpected between the suspension and theflexible modes of the instrument. The structurewas excited using a small electromechanicalshaker and the response of the top wasmeasured using a scanning laser vibrometer.

The vibrometer combined with a small forcegauge on the shaker allowed transfer functions tobe measured quickly at a grid of points definedon the tops of the instrument. Individual modeshapes are, thus, easily animated. The basic testsetup is shown in Figure 4. Note that theinstrument in this figure is a flat top guitar.

Experimental ResultsThe first five modes of the top were identified usingthe scanning vibrometer. A small shaker driving theright side of the bridge provided the input force.Figures 5-9 show these modes. There are severalpoints of immediate interest. The first is that thelowest mode of the instrument is a rigid bodymotion of the tailpiece. The second mode is atypical ‘breathing’ mode and probably representsthe Helmholz mode of the instrument. The next twomodes differ fundamentally from those of most flattop guitars in that they are antisymmetric about thecenterline of the instrument.

Since one of the fundamental differences betweenthis type of guitar and flat top instruments is theconfiguration of the sound holes, it seemedreasonable to measure the effect of covering one ofthem. Figure 10 shows the frequency responsefunctions that resulted using an accelerometer atthe bridge and a hammer tap at the same point.

It is interesting to note that the 155 Hz mode isessentially eliminated. Simple discrete models [3]suggest that a reduction in the area of a singlesound hole would lower the fundamental frequency.These models do not predict the effect in Figure 10and should perhaps be modified.

Subjective Sound QualityThe link between modal test results of instrumentsand the quality of the sound produced is still beingdefined. While there are many opinions regardingthe application of structural testing, it seems to begenerally accepted that good instruments usuallyhave a fundamental frequency slightly higher thanthat of the low E string. In conventional tuning, thelowest string is tuned to a frequency of 82.4 Hz.The lowest body mode on good guitars (howeverone defines good) is usually in the range of 95-105Hz.

The top of this instrument was thicker than istypical. One effect of this change is that thefundamental frequency is relatively high – 155 Hz.In spite of this, the sound of the instrument isacceptable. The tone is even throughout theregister, though it doesn’t have a particularly stronglow end. A lower fundamental frequency wouldprobably improve the sound. Certainly, the present

value represents an upper acceptable limit for afull size instrument.

Math ModelsSimple 2-DOF and 3-DOF models for the lowfrequency behavior of stringed instruments haveappeared in the literature for some time [3]. Thetwo DOF model couples a flexible top surfacewith a Helmholz resonator as shown in Figure 11.The walls are assumed to be rigid everywhereexcept for the flexible section. Adding a flexibleback creates the 3-DOF model as shown inFigure 13.

The surprising number of antisymmetric lowermodes in the archtop hints at some basicphysical phenomena that are not captured bythese models. Figure 13 shows a possibleconfiguration that might be helpful. A morecomplete treatment of this models will beaddressed in a later work.

ConclusionsThe modal response of an archtop guitar hasbeen measured using a full-field optical method.The results are not qualitatively similar either toinstruments in the violin family or with those fromflat top guitars. This suggests that a completeunderstanding of archtop instruments cannot beobtained by simply extrapolating results fromthese other families. The results presented hereare for a single instrument. While they suggestintriguing general results, a body of data fromother archtop instruments is required.

References

1. Fletcher, N.H. and Rossing, T.D.; “ThePhysics of Musical Instruments 2nd Edition”;Springer-Verlag, 1998.

2. Elejabarrieta, M.J., Ezcurra, A andSantamaria, C.; “Coupled Modes of theResonance Box of the Guitar”; Journal of theAcoustical Society of America, Vol 111, No.5, May 2002, pp 2283-2292.

3. Christensen, Ove; “Quantitative Models ForLow Frequency Guitar Function”; TheJournal of Guitar Acoustics, Issue 6,September 1982.

Figure 1 – Cross Section of Violin

Figure 2 – Lower Modes of A Flat Top Guitar

Figure 3 – Test Instrument

BassbarSound Post

Top

Back

105 Hz 208 Hz 366 Hz

Figure 4 – Typical Test Setup

Figure 5 – Mode 1, 133 Hz Figure 6 – Mode 2, 155 Hz

Figure 7 – Mode 3, 180 Hz Figure 8 – Mode 4, 229 Hz

Figure 9 – Mode 5, 248 Hz

Figure 10 - FRF Showing Effect of Two Sound Holes

Figure 11 – Two Degree of Freedom Model

Figure 12 – Three Degree of Freedom Model

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F

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Figure 13 – Four Degree of Freedom Model

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Ap, mp, xpAh1, mh1, xh1

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