Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Coupling of string motions totop plate modes in a guitar.

Preliminary reportJansson, E. V.

journal: STL-QPSRvolume: 14number: 4year: 1973pages: 019-038

http://www.speech.kth.se/qpsr

STL-QPSR 4/1973 2 0.

by a s t ructure of b a r s glued to i t s inner side, the so-called bracing. The

bracing s tructure i s made differently in order to obtain different tonal

qualities and static strengths. ( I ) The guitar investigated in this work

has a bracing system that i s approximately symmetric around the length

axis of the guitar. Thus i t i s representative for a la rge group of clas-

sical guitars. The end support of the s t r ings makes coupling possible

between al l kinds of string motions and the top plate. How the different

string motions couple to the different plate modes cannot be simply cal-

culated a s several small string motions a t the saddle a r e possible, - sliding, both longitudinally and t ransver sally, and rolling. However, we

can measure the deformation of the sounding box resulting from a spec-

i f ic string displacement, i. e. the coupling f rom a specific string dis-

placement to the modes of vibration. Such information i s presented in

this paper. I t provides fundamental facts on the sound producing mech-

anism and an estimate of how this mechanisms is influenced by different

constructional parameters . The information i s fur thermore used to e s -

timate how different playings influence on the tone production.

Vibration modes and deformations for specific excitations o r loads a r e

difficult and t i resome to measure by means of traditional probe techniques.

By means of the optical technique, hologram interferometry, informative

records can rapidly be obtained. ( 2 - 6 ) The technique gives records show-

ing the displacements of a whole surface, which does not have to be pol-

ished o r flat. The displacement magnitudes a r e mapped by equal displace-

ment l ines by dark and bright interference fringes, which a r e superim-

posed on an ordinary picture of the object, see Fig. 11-A-2. The double

exposure technique employs interference between two holographic images

of the same object, a) the object in "equilibrium" and b) the object with i t s

surface deformed. This experimental technique i s employed in this investi-

gation.

To avoid confusion and obtain simplicity in this paper we shall hereaf-

t e r use the word displacement to mean str ing displacement and the word

deformation to mean plate displacement.

The way of describing the sound producing mechanism and the way of

investigating i t experimentally a r e by no means limited to the guitar. The procedures a r e in fact general and can be used for any kind of sound pro-

ducing system containing vibrating walls. Therefore the work reported on

STL-QPSR 4/1973 2 1 .

can be regarded a s an investigation of a physically complex model of gen-

e ra l interest, the results of which can be applied to other specific systems

o r devices of interest.

Theory

Let us start with summarizing a few we3i known relations for later use.

As i t i s enough for our present use we shall limit our study to homogenous

strings of neglectable stiffness. The vibrations of such a string stretched

between two fix supports, can be described a s a sum of four components,

two transversal, one longitudinal and one torsional. By assuming no slid-

ing a t the plucking position of the string, the applied transversal force

where T i s string tension, a the length of the string, AqTis the displace-

ment a t the plucking position a t distances y A and (I - y ) R respectively

from the end supports. The applied force i s balanced by a force a t each

support

where (1 - Y ) i s the normalized distance plucking position - opposite support.

The same relations apply to longitudinal displacements A dr and forces t A FL , and torsional displacements A 4 and torsional moments A M by

'P 'P exchanging T for [ Em S ] and [Go I ] , where E i s the elasticity modulus,

2 4 G the shear modulus, S = rr a i s the cross-sectional arez , I = ALL and 2

a i s the radius of the string. For the homogenous string i s G = m)' The material constant IJ, i s called poisson' s ratio and equals approx.

0. 3 for most solid materials.

Large displacements A $ and A 4 give ra ise to longitudinal forces at cP

the string supports by means of increased string tension. For large trans-

versal displacements but still small angle changes, we get a string length-

ening and a corresponding increased string tension, i. e. a longitudinal

force

STL-QPSR 4/1973

a t the support distance Y R f rom the plucking position. A torque

s t re tches the cylinder generatrix, which averaged over the cross-sec-

tional a r e a of the string gives a n increased string tension, i. e. a longi-

tudinal force

a t the support y R from the plucking position.

If a s t r ing i s displaced through friction by a force with a t ransversa l

and a longitudinal component the following relations apply. The string

can "accept" a maximum force F related to the t r ansve r sa l force F La T

and the friction coefficient fk a s F La < F T . f k w h i c h l i m i t s F for La la rge angles cp , cp being the angle between total force F and F The T' following relations a r e valid between the longitudinal force F L, FTs and c p .

and FL/F = min(sin cp , fkcos cp ) (5)

Small vibrations with the displacements a s above obey the wave equa-

tion

where k = - 2 n f i s the wave number, f the frequency, and c the wave C

velocity. The velocity of the different waves corresponding to the differ-

ent kinds of displacements a r e

--a

T t ransversa l CT = \IpS

E longitudinal C L = J-;- t o r sinal

where p i s the density of the string mater ial . An open string of length R I I

has a fundamental frequency for t r ansve r sa l vibrations I

STL-QPSR 4/1973 23.

The formula applies to longitudinal and torsional waves by exchanging T

for L and cp . By means of Eq. (8), s e r i e s expansion of Eq. (7a), Eq.

( I ) for the longitudinal case with y ( I - y )=I ( ~ o o k e s law) the detuning Af

of a string and i t s slackening can be related to the elasticity as

The main pa ramete r s of the simplest oscillating system, the so-called

simple oscil lator, a r e resonance frequency and quality factor. (8) These

two pa ramete r s give the normalized vibration amplitude a s function of

frequency. To be able to calculate the absolute magnitude of the vibra-

tions we need a third parameter too, for instance the stiffness. The

stiffness can be derived f rom the displacement a t zero frequency A $ 0.

With these three pa ramete r s the behavior of the simple oscil lator is com-

pletely determined. The vibration displacement follows a simple r e so -

nance (approx. max. ) Q t imes the displacement a t zero-frequency. The

velocity, giving the source strength of a n acoustical radiator , equals the

displacement multiplied by the angular frequency.

The vibrations of a complex system a s a finite plate can be described

by a weighted sum of i t s normal modes. Every single normal mode r e -

presents a simple oscil lator and the whole system an infinite number of

~ i m p l e oscil lators. The complete system can conveniently be modelled

by an infinite number of se r i e s resonant c ircui ts , which a r e connected i n

parallel to the driving force, each circuit representing a normal mode. (9)

The weight factors come from two different phenomena. F i r s t the excita-

tion force i s split up on different normal modes depending on their im-

pedance a t the excitation frequency. Secondly, the impedance of each

normal mode must be modified by an excitation factor. Each excitation

factor depends on the position of excitation in relation to the standing

wave pattern of the normal mode. In the circuit diagram the excitation

factors enter a s simple multiplication constants, one and the same fo r

each se r i e s circuit and in general different fo r different circuits. F o r low frequencies the velocities a r e

and thus the di spla c em ent

STL-QPSR 4/1973 25.

Three se r i e s of main experiments were conducted. F i r s t a pilot

s e r i e s was undertaken to get a gr ip of the deformations involved and the

influence of the boundary conditions. Secondly the coupling plate deforma-

tions and t ransversa l forces applied to the middle of the s t r ings were in-

vestigated. Finally the corresponding forces were applied to the saddle.

Mainly the same hologram interferometer and l a s e r speckle interfero-

m e t e r a s was set up for ear l ie r experiments were used. (11, 12) The

t ime-average interferometer was used for making the double exposure

hologram-interferograms and the speckle interferometer to control the

stability and to study the vibration modes. Three consecutive recordings

were made of each experimental situation to control the reproducibility.

The resul t of each se r i e s were preliminarily evaluated to serve a s a guide

for the continued experiments.

In the f i r s t s e r i e s , the pilot s e r i e s , a ve ry loose holding of the guitar

was employed. The guitar was laid with i t s right side onto the hologram

table with i t s top plate ver t ical and i t s neck approx. horizontal. No

clamping was applied - only a counter weight slightly overbalancing the

weight of the neck. A "clamp" was screwed to and a c r o s s the neck per -

pendicular to the strings approx. a t half their lengths, see Fig. 11-A-2.

By means of a thin thread the 6th string was displaced to a stop of the

clamp. The thread was fastened in the clamp, thus giving no unbalanced

forces . The f i r s t exposure was made, then the thread was burned off

and finally the second exposure was made. The procedure gives records

corresponding to a n approx. "free egde" guitar, thus not necessar i ly

limiting the displacements to the top plate.

The pilot s e r i e s proved that the displacements were very much dom-

inant in the top plate, smaller in the back plate, and still smal le r a t the

sides of the guitar.

The experiments turned out to be difficult to perform for a single

person. Therefore i t was decided to apply the force by means of the same

arrangement that had been employed for the string displacement - force

calibrations. However, a s the force applied was not automatically ba-

lanced a f i r m e r holding of the guitar had to be employed. This was ob- tained by making a rectangular f r ame , which was screwed to the holo-

gram table. In this f r ame the guitar was held by means of wooden blocks

and thin pieces of rubber. A s the deformations a t the edges of the top

plate were very small we felt f r e e to do so. 1

STL-QPSR 4/1973 26.

In the decbnd se r i e s the force was applied to the middle of the strings.

The light intensities were adjusted to give a fairly long exposure t ime,

2 x 20 sec, which enabled the following simple experimental procedure.

F i r s t the string was displaced by means of the thx'ead and weight. Pen-

delilm motions of the weight were dahiped out by means of a light damping

pad touching the side of the weight. After a short stabilizing time the

f i r s t exposure was started. After 2 0 sec the thread was rapidly lifted

off the wheel and the weight placed on a fix support, thus giving no dis-

placement of the guitar string and the expo sure was cortinued for 20 sec.

Because of the long exposure time there was no need for splitting up the

exposure in two par ts by a shutter.

F i r s t the top plate deformations were recorded for t ransversa l string

displacements in parallel with the top plate resulting from a force of

1.96 N. Secondly the top plate deformations were recorded for t rans- I

versa1 string displacements perpendicular to the top plate. Because of

the extreme sensitivity to forces in this direction a force of 0.98 1V was

employed. Finally the third string was twisted a t i t s middle to estimate

the influence of torsional vibrations. To avoid t ransversa l displacements

the string was kept in position by a support. The deformation turned out

to be very small and furthermore difficult to reproduce.

In the second se r i e s i t was found that the same deformations were ob-

tained for forces in parallel to the top plate independent on which string

was displaced. F o r forces perpendicular to the top plate the deformations

were much dependent on which string was excited. The deformations of

the two se r i e s a r e by fa r la rges t in the top plate a r e a "belowtf the sound

hole.

The third and las t s e r i e s of experiments were conducted by exciting

the top plate through the most active coupling element, i. e. the bridge.

The forces were applied a t the saddle and were decreased to half to make

the records directly comparatible to those of the second ser ies . As the

second se r i e s had proved that only one type of deformation i s obtained

for t ransversa l forces in parallel with the top plate only one excitation

position was applied for this force (at the 6th string). Three excitation

positions were used to excite the plate perpendicularly - a t the 1 st string,

a t the middle of the saddle, and a t the 6th string. Fur thermore , "long-

itudinal" forces , i. e. additional forces in the opposite direction of the

s t r ings, were applied to the same points.

STL-QPSR 4/1973 27.

The resulting deformation f rom three forces of different magnitude,

full and approx. half standard force, were measured to t e s t the l inearity

of the force-deformation ratio.

The vibrations of the top plate of the complete guitar were pre l imimr-

i ly studied. By means of the l a se r speckle interferometer the resonan-

c e s were sought for and vibration patterns a t resonance recorded by the

t ime-average hologram-interferometer.

Results

Let u s s t a r t setting a few standards for the interpretation of the ex-

perimental resul ts . In Fig. 11-A- I a coordinate system i s defined, which

i s used for the data evaluation. The x-y-plane i s c h o ~ e n to coincide with

the top plate surface neglecting i t s smal l bulginess. Origo i s set a t the

middle of the saddle and 100 lengths units in the coordinate system a r e set

equal to the maximum width of the guitar 38.2 cm. All deformations

recorded a r e assumed to be perpendicular to the top plate and thus the

top plate deformations A z can be written a s functions of position x and y.

The assumption regarding the deformation i s well motivated a s the r e -

cording system i s by far most sensitive to such deformations (directions

of illumination and observation approx. perpendicular to the plate) and a s

the plate generally i s l i t t le deformed along i t s edges. The deformations

A z can be directly read out of tE.e interferograms a s number of white

fringes and a re tabulated in this form. Each white fringe corresponds to

an approx. increase of deformation by half a wave length of the l a se r -10 (13)

light, i. e. 3200 - 10 m ) .

The resu l t s of the string experiments a r e summarized in Table 11-A-I.

The string tension i s calculated by means of Eqs . (8) and (7a). Thus cal-

culated string tensions and Eq. ( i ) predict the measured string displace-

ments resulting f rom the t ransversa l fo rces with an accuracy of approx.

5 70, which proves that the experimental conditions a r e under control.

The elasti city E. S i s calculated f rom Eq. ( 9 ) , the presented tension and

measured string slackening f rom a 6 yo detuning. Thus obtained elastici-

t i e s a r e in reasonable agreement with those on "Kunststoff-Fasern" given

by Jahnel considering the relatively la rge uncertainty of our measure - (14)

, I

ments (approx. 20 70).

Fig. 11-A- I. Definition of coordinate system.

STL-QPSR 4/1973 28.

Before going into a more detailed analysis of the deformations let u s

shortly summarize and compare the g ross resul ts of the different ser ies .

In the f i r s t s e r i e s with approx. f ree boundaries i t i s found that the defor-

mations a r e predominantly in the top plate by a t least a factor 20. The

magnitudes and patterns of the top plate deformations of the f i r s t s e r i e s

a r e refound in the second ser ies . Fur thermore the records of the se-

cond se r i e s show the same top plate deformations independent of string

for a force - A Fx. In the third se r i e s with the forces applied to the

saddle the same deformations a r e found in the top plate a r e a below the

sound hole a s in the ear l ie r two ser ies . This a r e a i s in a l l cases the

a r e a of large deformations.

Table I1 -A -I. String data

Strings: Levin Classic Guitar Standard (807) Siver VFound on Nylon

(the fir s t three strings plain, the other strings wound)

String Tuned Freq . Diam. p . S Tension A I) E. S N o Hz cm g/cm N cm N

The f i r s t t-hree columns give the string labellings and

corresponding frequencies. The next t h r e e pre sent two

columns of measured string data and the string tensions

calculated f rom these two, the frequencies and the string

lengths 65. I cm. The las t but one column gives the

t ransversa l string displacement measured a t the middle

of the s t r ings for the t ransversa l standard f rame (1.96 N)

applied to the same point. The las t column gives the

elasticity calculated from experimental measures .

STL-QPSR 4/1973 29.

Typical examples of deformations for different forces a r e shown in

Fig. 11-A-2. F r o m this figure we can immediately real ize some funda-

mentals by remembering that the black and white fringes represent equal

displacements starting with approx. zero-displacement along the edges.

F o r a t ransversa l string force -Ex we obtain a deformation of the top

plate corresponding to a one way displacement for x > 0, and an op2osite

directional displacement for x < 0. For a t ransversa l strfng force mZ we obtain an al l one way displacement. F o r l~ngi tudina l string force - A F

Y we obtain a one way displacement for y > 0 and an opposite directional

displacement for y < O . The patterns of two way deformations a r e in ter -

preted to consist of an inward deformation in the direction of the applied

force. In the f i r s t o rde r approximation the deformations correspond to

three normal modes of vibration, 1) two antinodal a r e a s divided by a v e r -

t ical nodal line, 2) one antinodal a r e a , and 3) two antinodal a r e a s divided

by a horizontal nodal line (the same vibration patterns a s tho se recorded

by Molin and Stetson, see ref. ( I5) . The deformations of Fig. 11-A-Zb and

c a r e , however, not symmetrical with respect to the y-axis. The de-

formation i s la rges t a t the bridge approx. a t the position of the applied

force. These nonsymmetries must be taken into account in the second

order approximation.

In the evaluation of the deformation patterns the following procedure i was used. The f i r s t order approximations give three different modes of

deformations, which a r e symmetric o r antisymmetric around the y-axis.

These deformation modes normalized to the standard force a r e labelled

TDi , TD2, and TD3 (top plate deformations I , 2, and 3 respectively).

In the second o rde r approximation the deformations recorded a r e matched

by mixtures of TD I , TD2, and TD3 in accordance to the theory presented

for Eq. (10). The positions of maximum deformations give the mixing

ratio of the modes and the magnitudes a t maximum deformation give the

absolute magnitudes of the modes.

The procedure was employed for a detailed analysis for the case of

t ransversa l forces: F o r forces -mx i t i s found that the position and

magnitudes of maximum deformations a r e in good agreement in a l l cases

af ter small corrections for tiltings of the instrument, see Table 11-A-11.

The table also shows that the positions of maxima a r e close to the x-axis

and equally spaced around the y-axis. F o r forces dF2 we also find that 1

STL-QPSR 4/1973 31.

the maximum deformations a r e close to the x-axis. The magnitudes and

positions of maxima correspond excellently between the second and third

ser ies . The admixture of modes can simply be evaluated by making sec-

tions of the deformations along the x-axis and fitting weighted sums of the

different modes a s described above.

F i r s t the deformations TDI and TD2 were drawn and slightly "cor-

rected" to make them perfectly symmetr ical and antisymmetrical r e s -

pectively, see Fig. 11-A-3a. Thereafter the fitting procedure was applied

to a l l ca ses of excitation forces A F 2 . The resu l t s a r e presented in

Fig. 11-A-3, Fig. 11-A-4, and Table 11-A-111.

Table 11-A -111. Deformation reconstructions

Excitation

Second s e r i e s

Excitation Po sition

a l l s t r ings

s t r ing I

string I1

string 111

string IV

string V

string VI

s t r ing 111

Deformation

m i r d se r i e s - AFx string VI -TD2

0. 50Fz string I 0. 5 ( T ~ l - I . 6TD2) 1 1 middle O.5TDl I I s tr ing VI 0. 53(TDl+2. OTD2)

-amy string I -2(TD3- 0.4TD2) I I middle -2TD3 I I string VI - 2 ( T ~ 3 + 0 . 4 ~ ~ 2 )

The deformations corresponding to the longitudinal forces a r e a bit

hazardous to evaluate, because I ) the deformations along the e dge s a r e

considerable and 2) the deformations cannot be made up by a sum of T D i ,

TD2, and TD3. We can, however, make a f i r s t t r y by assuming that

TD2 descr ibes the deformations corresponding to the c ro ssmode - the

Fig. 11-A-3. Recorded and fitted deformations z measured in number of white fringes along the x-axis with forces applied to the bridge.

a) Deformation fo r force A Fz = 0. 49 N applied at middle of saddle (measured and approximated 0.5TDICI) and deformation for force 6 F, = -0. 98 N applied to any string a t saddle (measured and approximated TD2 A );

b) and c) Deformations for force A F, = 0 .49 N applied to strings I and VI respectively a t saddle (measured fitted combinations of TDI and TD2 0 ) .

String a I 1

String DI 1

4 String IP I

Fig. II-A-4. Recorded and fitted deformat ions A z m e a s u r e d i n number of white f r inges along the x - ax i s with fo r ce s h F, = 0. 49 applied to the middle of each s t r i ng separa te ly . (measu red fitted cornbinations of T D I and T D 2 0 ) .

STL-QPSR 4/1973 32.

mode of Fig. 5 e in (I5) - in the position range of maximum deformations

of the plate. The resu l t s a r e given in Table 11-A-111. With the same

justifications a measure of the deformations resulting f rom the torque

a r e estimated. I t must be borne in mind that these numerical resu l t s

represent hardly m o r e than qualitative measures . In Table I1 -A-IV a r e (16) volume deformations estimated a s suggested by Jansson .

The experiments proved that the force-deformation ratio in the range

investigated i s l inear within the uncertainty l imits .

The frequencies of modes T i , T2, and T3 corresponding to TDI, TD2,

and TD3 respectively were found to be 150, 215, and 420 Hz. F r o m these

frequencies and the volume deformations the source strengths of the anti-

nodal a r e a s a r e calculated and tabula?ed in Table 11-A-IV.

Table 11-A-IV. Mode Pa ramete r s - Deformation Peak . Volume Vibration F req . Source strength

Mode Deformation Deformation Mode a t resonance -6 -8 3

xO. 31x10 m x i 0 m 3 Hz ( ~ O - ~ / Q ) X I ~ /sec

TD3 -4. 5 -1 .1 T 3 42 0 -3 .0

t i . 0 t o . 14 t o . 37

Before leaving the experimental raw data, still another thing should be

pointed out. The top plate i s in general bending relatively l i t t le along the

x-axis in spite of the low stiffness of the plate in this direction. The

stiffness i s little increased with the bracing system employed. The stiff-

ening of the top plate therefore depends on the bridge which i s c lear ly de-

monstrated in Fig. 11-A-3c fo r instance.

Evaluations I

F r o m the experimental resu l t s we can evaluate in detail how different '

string displacements influence the deformations. Fur thermore we can

estimate how much the different modes a r e exci ted and the importance of I

different excitations.

I

STL-QPSR 4/1973 3 3.

The major case with t r ansve r sa l displacements i n different angles i s

qualitatively sketched in Fig. I1 -A- 5. A force Px applied in the x-

direction resu l t s in a de fo rma t ionaof TD2 along the x -ax i s . This de-

formation i s independent of s t r ing number a s shown in the upper figure.

F o r a force with a z-component we can divide the deformations in four

groups, two main groups depending on string number and two subgroups

depending on the sign of the force component A FZ. F i r s t the component

A F gives a deformation a of TD2 a s sketched in the f i r s t column. Se- x

condly the component A F gives a deformation a s sketched underneath. ~ z This second deformation can be split fur ther into a deformation + o r - - b

of TD2 and a deformation + o r - - c of TDI a s sketched in the second column.

Addition of the deformations - a and - b with sign gives the total deformation

in the mode TD2 a s sketched in the las t column. Underneath a r e the de-

formations in TDI sketched. F o r a reversing of the force applied of 1 8 0 ~ .

the deformation components a r e a l so reversed , i. e. no change in the r e -

lations between the different modes. Fig. 11-A-5 demonstrates that the

magnitude of the deformation in TD2 and the phase relations of the de-

formations TD1 and TD2 a r e much dependent on the z-component. We

find that the addition of the components occur s for a force - A FZ for the

f i r s t th ree s t r ings, but for a force t A F for the lower three strings. z This provides a tonal-quality reason for using the thumb on the lower

s t r ings and other fingers for the higher ones in almost opposite plucking

direction.

Analytically the process can be written a s

with the factors A and B given in Table 11-A-111. Definition of the angle I

Q and the course of the function above a r e given in F i g . 11-A - 6. The dia-

gram shows besides the ear l ie r mentioned phase and magnitude differences,

that the resulting exci tat io~ls of TDI and TD2 a r e ve ry sensitive to changes

of the angle 8 in the range of 8 = 0, and that the excitation of TD2 is about

zero for O equals to approx. t o r - 50°. The source strength of mode TI

var i e s f rom 0 to + 0.27 . ~ o - ~ / ( Q . f) and that of T2 from 0 to

0. 83- 10- ' / (a f) , i. e. both contributions have about equal magnitude range.

A s the resonance frequencies of T i and TZ a r e well above the lowest

string fundamental f - 82.4 Hz, ::esonance amplification of Q is en- T - counte~-ed a t the resonance frequencies.

String I-P[

String I-lI

a-b - String IP-a

a-b -

Fig. 11-A-5. Qualitative resul t of measurements . a ) upper section: a fo rce A Fx excites TD2 only; b) middle section: a force applied t ransversa l ly to the three

higher s t r ings excites ( I ) TD2 a depending on i t s x -com- ponent and (2) a combination O~-TDZ b and TDI c depending on the z-component. The direction gf the z-component de te rmines whether the magnitudes of a and b a r e sub- - - t racted o r added;

c) lower section: a force applied t ransversa l ly to the lower t h r e e s t r ings excites TDI and TD2 in the s ame way but with opposite phase re la t ions of a and b in relation to - - the z-component of the force.

Fig. 11-A-6. Resulting deformation divisions i n modes TDI and TD2 as function of direction of forces applied to different string.

STL-QPSR 4/1973 34.

F o r longitudinal string forces we obtain a deformation in TD3 with

TD2 superimposed a s given in Table 11-A-111. The magnitude of frictio.1

coefficients wae estimated to 0. 5 , which means that the string can

"accept" a longitudinal force half the magnitude of the t r ansve r sa l force,.

However, in reali ty the maximum longitudinal force component is slight-

ly lower, see Fig. 11-A-7. I t s maximum equals 45 7'0 of the total force

applied and i s obtained for 0 = 28O. This means that maximum force

f rom this excitation is about 0.45 N, i. e. 45 70 of TD3 together with a

higher mode. A 45 70 excitation of mode T3 gives two sources of strengths

f rom 0 to - 1 . 3 a n d + 0 . 1 7 . 1 0 - ~ / ( Q - f ) . The frequency of these longitudinal

vibrations s t a r t s f rom f - 2. 5 kHz calculated f rom Eqs. (8) and (7b), L - and Table 11-A-I, i. e. considerably above the resonance frequency of T3.

This means that T3 is weakly excited by this kind of string vibrations.

F o r the deformation f rom torque applied to the third string we find that

one revolution gives r i s e to a deformation of 0. l ( ~ D 3 f 0 8 T D ~ ) . This -5 corresponds to source strengths of 0.3 and 0. 037. 10 /(Q. f) , and

f 0.44- 1 o ' ~ / ( Q f ) . The deformations a r e small but quite noticeable.

Corresponding string vibrations s t a r t f rom fcp = I . 6 kHz for the third

string calculated f rom Eqs . (8) and (7c). A s f?? i s well above the r e -

sonance frequencies of T2 and T3, the vibration modes a r e weakly ex-

cited.

Let u s so estimate the second o r d e r t e r m s , giving the coupling of

t ransver sa l to longitudinal, and torsional to longitudinal displacements.

F r o m Eq. (3) we can calculate the coupling f rom t ransversa l dis-

placement to longitudinal. This gives that a t ransversa l displacement

of 0.4 c m gives r i s e to longitudinal forces of magnitudes 0.4 N, the

forces proportional to the square of the displacement. A longitudinal

force of this s ize gives a deformation of 0 .4 t imes TD3, i. e. source

strength of - I . 2 and 0.15.1 o - ~ / ( Q - I). The lower frequency limit of

corresponding vibration i s 2 . fT , i. e. 105 Hz , which means that we can

obtain resonance amplific ation.

By means of Eq. (4) and physical pa ramete r s of the third string we

obtain a longitudinal force of 0. 1 N for one revolution, which corresponds

to 0.1 TD3 measured. This fact indicates that only f 0. 08 TD2 derives

directly f rom the moment applied to the bridge, and that TD3 i s excited

Fig. 11-A-7. Accepted longitudinal s t r ing force FL a s function of the direction 6 of the applied force F in relation to a plane t r ansve r sa l to the s t r ing and of frict ion coefficient fk.

STL-QPSR 4/1973 3 6 .

1aw)i Secondly he can obtain differefit tonal qualities by playing the same

tone on different str'ings a s different s t r ings couples differently to dif-

ferent gdate modes. Thirdly the tonal quality i s dependent on the d i rec-

tion of plucking, F o r instance for a t ransversa l plucking in the range of 0 t 6 0 the f i r s t two normal modes may be changed f rom zero to maximum.

F o r the fourth by letting the string rol l differently over the plucking finger

he may set the string into elliptic motions with moving axes , which give

quite a t ime dependent starting t ransient and possibly a slowly varying

course - a living tone.

The experiments have a lso provided u s with an understanding of the

function of the bridge and how i t may influence the quality of an ins t ru-

ment. In the experimental section i t was pointed out that the bridge

provides the top plate with a "cross bracing". Therefore the top plate

should be regarded a s a plate with a supporting s t ructure - the bridge

and the bracing in combination. Thus the tuning o r the adjustments of

the top plate should involve both bracing and bridge to give wanted nor-

m a l mode patterns and frequencies. F o r instance, a stiffer bridge will

increase the frequency, a heavier will lower the frequency of T i , and

a nonsymmetrical bridge will make T2 non-anti symmetr ical and in-

c rease the radiation of this mode.

The bridge and saddle work a lso a s an element transmitting vibrations

f rom the s t r ings to the top plate. Our experiments prove that depending

on different string motions, the bridge works differently. F o r forces

perpendicular to the top plate we obtain mainly the sum of two modes

TDI and TD2 corresponding to the f i r s t two normal modes of vibration.

The excitation of the two modes depends on the bridge - a l e s s stiff

bridge along the saddle will increase the magnitude of TDI in relation

to TD2.

F o r a force in parallel to the bridge the second normal mode i s ex-

citkd because of the torque moment - force multiplied by saddle height,

i. e. a higher bridge excites m o r e effectively the second mode. F o r

longitudinal string forces the bridge again gives a moment to the top

plate, i. e. the main parameters being the height of the saddle. In this

case the experiments show that the bridge distributes the applied moment

unevenly along the bridge, i. e . the torsional stiffness of the bridge may

be an important parameter in exciting the c ro ss-mode.

STL-QPSR 4/1973 37.

The phenomena discussed h e r e in the low frequency range apply to

higher frequencies and modes a t leas t qualitatively.

Shortly summarized we have found evidence for that a player can ob-

tain different tonal qualities by choice of string and direction of plucking,

and that the instrument maker can influence the tonal production by a l -

tering height of saddle, stiffness, and m a s s distribution of the bridge.

Acknowledgments

The author is greatly indebted to AB Herman Carlson Levin in GLlte-

borg for the guitar prepared for these studies.

This work was supported by the Swedish Humanistic Research Goun-

ci l and the Swedish Natural Science Research Council.

References

(1) F o r a m o r e extensive description of guitars, see for instance Jahnel, F. : Die Gi tar re und ih r Bau, Verlag Das Musikinstru- ment , Frankfurt a m Main 1973, 2nd ed.

(2) Jans son, E. , Molin, N -E. , and Sundin, H. : "Resonances of a Violin Body Studied by Hologram Interferometry and Acoustical Methods", Physica Scripta - 2 (1970), pp. 243-256.

(3) Collier, R. J . , Burckhardt, C. B. , and Lawrence, H. L.: Optical Holography, Academic P r e s s , New York and London 1971, Chapter 15, pp. 418-453.

(4) Powell, R. L. and Stetson, K. A. : "Interferometric Vibration Anal- ys i s by Wavefornt Reconstruction", J. Optical Society of America 55 (1965), pp. 1593-1598. -

(5) Stetson, K.A. and Powell, R. L. : "Interferometric Hologram Evalua- tion and Real-Time Vibration Analysis of Diffuse Objects", J. Optical Society of America - 55 (1965), pp. 1694- 1695.

(6) Brown, G. M. , Grant, R. M. , and Stroke. G. W. : "Theorv of Holo- . . graphic lnterferometryfl , J. Acoust. S ~ C . ~ m . - 45 (1969j, pp. 11 66- 1179. I

(7) See ref. (3) , Eqs. 15.6 and 15. 18, and 515.4. 3, o r ref. (6) $11, Eq. (17) and §V Eq. (50).

(8) Mores , P. M. : Vibration and Sound, McGraw-Hill, New York 1948, 2nd ed. , Chapter 2.

(9) Skudrzyk, E. J. : "Vibrations of a System with a Finite o r Infinite ~ u m b e r of Resonances", ~ . ~ c o ; s t . Soc. Am. 30 (1958), pp. 1140: - 1151.

(10) C r e m e r , L. and Heckl, M. : Structure-Borne Sound, t ranslated and revised by E. E. Ungar, Springer-Verlag Berlin, Heidelberg, New York 1973.

STL-QPSR 4/1973 38.

(1 I ) Jansson, E. V. : "An Investigation of a Violin by L a s e r Speckle Interferometry and Acoustical Measurements", Acustica 29 - (1973), pp. 21-28.

(12) Ek, L. and Molin, N - E . : "Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry", Optics Communications 2 (1971), pp. 419-424. -

(13) See Ref. (3) $15. 3 .4 and Eq. (15. 7) o r Ref. (6) Eq. (19).

(14) See Ref. ( I ) , table on p. 221.

(15) Jansson, E. V. : "A Study of Acoustical and Hologram Interfero- m e t r i c Measurements of the Top Plate Vibrations of a Guitar", Acustica 25 (1971), pp. 95- 100. -