Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

An investigation of acousticalproperties of the air cavity of

the violinJansson, E. V.

journal: STL-QPSRvolume: 14number: 1year: 1973pages: 001-013

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

STL-QPSR 1/1973

I. MUSICAL ACOUSTICS

A. AN ?SVESTIGATION O F ACOUSTICAL PROPERTIES OF THE AIR CAVITY OF THE VIOLIN

E. V. Jansson

Abstract

In a preliminary survey of experimental results, i t was shown that higher a i r modes in a violin-shaped cavity can be excited (l). In the present report the experimental apparatus and the findings a r e reported in detail, and a f i rs t analysis of the results a re given. It i s proved that resonance frequencies can be predicted by means of f i rs t order perturbations, the losses a r e approximately three times minimum (boundary layer losses), but considerably smaller than calculated 10s se s calculated for a typical absorption coefficient of architectual acoustics.

1. Introduction

The walls of the sounding box of the violin enclose almost entirely

the a i r volume. The total a rea of the walls i s approximately 100

times larger than that of the f-holes. The zero-order resonance of

this cavity, i.e. the Helmholtz resonance, i s well known and has

been proved to be important for the low-frequency properties of the

violin ( 2 * 3). The size of the cavity i s large enough to support many

higher resonances in the frequency range of amplification(*). Traces

of higher a i r modes have been found(5). It i s commonly accepted,

that the position of the f-holes "damps out" the higher a i r modes,

o r that these modes a r e ineffective sound radiators('). The small

a rea of the f-holes compared to the enclosing surface area suggests

that higher modes exist. Such modes may, however, still be irn-

portant a s coupling mechanisms between the places. In a prelimi-

nary presentation of this experimental work, it has been shown that .

at least between the f i rs t top plate mode and the first higher a i r mode (7) there exists such coupling .

This work was conducted to improve the understanding of the

properties of the enclosed a i r volume. The investigation is a natural

continuation of the detailed studies of plate vibrations. Although a

good understanding of the plates only was obtained, some phenomena

of the complete instrument was still not possible to explain.

The shape of the violin cavity is complex. The position and shape of the f-holes a re also complex. However, both the measures of the

STL-QPSR 1/1973

a i r volume and the f-holes a r e approximately the same in mos t in-

s t ruments , i. e. by studying one cavity we shall obtain information

significant for most violins. With this fact in mind this investiga-

tion has been conducted. The goal has pr imar i ly been to obtain a

good understanding of the physical mechanisms. The refore simple

mathematical analysis methods a r e employed to provide significant

physical information, ra ther than numbers accurately reproducing

those of the experiments. The physical analyses and the matherna-

t ical modelling a r e performed in detail enough to show that the d i f -

fe rent mechanism8 a r e qualitatively understood. In conclusion we

may say that in this investigation, a t i t s present level of sophistica-

tion, we a r e pr imar i ly interested in a qualitative understanding.

Therefore we feel f r e e to use fairly rough approximations, and to

est imate magnitudes and direction of changes by f i r s t o rder p e r -

turbation theory. The detailed numerical analysis i s left for future

work, when such i s needed.

2. Theory

F o r readily reference and discussion, let u s f i r s t summarize

most ly well known equations and formulas. The sound p res su re p

for a simple harmonic wave obeys the wave equation, which, ,with

the common time dependent pa r t removed, can be written as

where k = 2 IT f/c i s the wave number, f the frequency, and c the

velocity of the sound. F r o m the sound p res su re the particle velo-

city 6 can be calculated a s

where p is the density.

If the sound p res su re i s enclosed within the rigid walls of a

rectangular cavity and one corner is chosen a s origin and the axes

s e t paral le l with the edges, the f ~ l ~ o w i n g standing wave solution i s

obtained:

STL-QPSR 1/1973

p = p o cos(k x *x)cos(k y)cos(kZ* z ) Y

(3)

The different resonances a r e labelled by the combination of numbers

n n and n F o r cylindrical rigid boundaries with the origin in X' y' z

the center of one of the flat walls the following standing wave solution

in cylindrical coordinates i s obtained:

cos P = Po sin (my ) c o s ( ~ , z ) J , ( ~ , ~ )

The different modes a r e labelled by the combination of nZ, m , and n.

The resonance density within a frequency band df in a cavity i s ap-

proximately descr ibed by

where V i s the volume, A the a r e a of the enclosing walls, and L the

total edge length (= 4 (n ' radius + height) for the cylindrical cavity).

The total number of resonances i s obtained by integration of Eq. (7).

F o r low frequencies the separate resonances character ize the proper - t ies of a system for higher density of resonances. For frequency

( 8 , 9 ) calculations Rayleigh has worked out simple perturbation formulas

Although these a r e given under s t r i c t e r l imits , we shall feel f r e e to

use them slightly reworked to est imate effects in changes. F o r a " .

tube of a lmost constant cross-sect ion S 0

STL-QPSR 1/1973

For a rectangular room

2

- - Pthe rmal

2 mg v* p0 R

P --• s viscous - 2 2

9 Ex cy "z

Through a hole in the wall the vibrations of the inner a i r cornmuni-

cates with the outer room. The possible transmission through this

hole i s determined by the length of the hole, and the shapes of the

hole and the surrounding flanges. For thin-walled cavities the so-

called length correction because of the flanges is dominant. F o r

such complicated boundaries a s encountered in the violin i t i s quite

complex to calculate the effects of the sound holes. F o r a qpalita - tive study we shall just cite the approximative radiation impedance

3

Z of a rectangular piston a x b cmL in an infinite wall at low f r e - r (12) quencies by Morse and Ingard ,

A loss -free system oscillates a t a frequency where its average

potential energy W equals i t s average kinetic energy Wk (1 3)( 14), i. e. P

STL-QPSR 1/1973

for the same normal mode perturbed by A to the frequency k

const then 0 = Wo 2Af + d W 29 Af i f Wk = -

( ~ n f ) ~ [-TI 0

A fo W

0 ) - I and - = 1/2(1 t -

0 A W

P

where fo is the resonance frequency of the unperturbed mode. In

the case of a smal l hole perturbation (smal l radiation losses)

where Peff i s the effective p res su re over the hole a r e a Sh. The

radiation losses Pr can be calculated a s -

3. Experiments

F o r the resonance measurements a special measuring probe

was constructed (Fig. I-A- 1). Slightly off center on a plexi-glas s 615) plug an STL-Ionophone (a very high impedance source) was mounte .

li

In the same plug a B&K 1/2 microphone with three shor t and very

thin capillary tubes (lengths approximately 2 c m and each with a

diameter of ,015 cm)(I6). This sond, together with the microphone,

gave a very smooth response in the frequency range reported on here.

The sond openings and the Ionophone were placed about 1 cm apart.

Thus a probe with a smal l source and a sma l l indicator, enabling easy

measuring of resonance frequencies and Q-factors of a not too highly

damped cavity.

Four different cavities with 2 crn thick wooden walls were invest-

igated. Three of the cavities were made w.ith two f la t walls, a l l

walls of equal a r e a s , and the same distance betwe e n these two walls.

SECTION A-A

\r?

Fig . I-A-la. Sketch of m e a s u r i n g p robe with holding f i x t u r e ; sec t ion and f r o n t view. Holding f ix tu re : a bush- ing B glued to the cavi ty wall W. M e a s u r i n g probe: a p lex ig lass plug P with the STL-Ionophone S and i t s insola ted connective wir ing I, and a m i - crophone C with a th ree - tube sond M. Al l jo in ts a r e made a i r t igh t by m e a n s of O- r ings 0. '

Fig . I-A- lb . Violin-shaped cavity with the pos i t ions of bushings f o r the m e a s u r i n g p r o b e m a r k e d by c i r c l e s .

STL-QPSR 1/1973

Thus the cavities contained ;).pproximatzly the same volume, en-

closing a rea and total e e ~ e lengths. These measures were volume

1420 f 1 %, area 1 is0 cmZ f 1 %, and length 185 cm 2 1 1 %, the

per centages giving the discrepancies between the different cavities. 3

The f i rs t cavity was a rectangular cavity of 27 x 17 x 3. 1 cm , the

second a cylindrical cavity of diameter 24. 1 cm and 3.1 cm high.

The third cavity was shaped a s a violin (Fig. I-A-ib) with f lat top

and back plates. All walls were made of solid wood except from

one of the flat wzils cf each cavity v~hich was made of hard pressed

wood fiber plate. The frame (rectangular, circular, o r violin-

shaped) and the "top" and the "back" plates were made in three dif-

ferent parts. ' When screwed together for experiments the c a v i t i ~ s

were made airtight by means of greased thin rubber gaskets. The

fourth cavity was a rea l violin, which was encased in plaster. One

part of the plaster mould was covering the f-holes and the neighbor

area,and was easily removable, When this piece was removed a

reasonably correct termination by radiating f -hole s was accomplished

(the shape, the position, and the wall thickness at and of the f -holes

correct).

In the top plates of al l cavities bushings were glued. The bush-

ings were carefully machined to automatically se t the probe plug

level with the inner surfaces of the cavities. The bushings were

placed in a corner of the rectangular cavity, close to the edge of the

cylindrical cavity, and according to Fig. I-A- l b for the last two

cavities. The distances to two walls o r a corner were never larger

than a centimeter for either the sond or the Ionophone. Therefore

the input impedances measured were little influenced by probe posi-

tions, Before the acoustical measurements were started, it was

controlled that the cavity was reasonably a i r tight. Thereafter the

room temperature within the cavity was measured - in some cases

also the humidity. Thereafter a frequency response was recorded,

and the peak frequencies and halfpower bandwidths were carefully

STL-QPSR 1/1973

measured by hand. Finally the temperature in the cavity was meas -

ured again, and the cavity was taken apart . The whole se t of m e a s -

urements were thereafter repeated twice.

The three readings were averaged. The outer bounds of frequen-

cies were found to be about . 5 O/o and those of the bandwidth 10 OJo. The temperature and humidity variations were smal l enough to be

negligible.

In the violin-shaped cavity the standing wave patterns were esti-

mated. This was done by drilling sma l l holes a t different places,

and by measuring the phase and the amplitude of the sound p res su re

by means of a microphone with a short sond through the holes.

4. Results

The geometrical measures of volume, a rea , and edge length, and

Eq. (7) give the smooth lines in Fig. I-A-2. Integration of Eq. (7)

resul ts in N=24 below 4 kHz. In the experiments 2 1, 24, 24, and 25

resonances were found in the four different closed cavities, respec-

tively. With open f -holes the number of resonances in the encased

violin drops to 23. Thus we find good agreement between the theore-

tically predicted and experimentally observed resul ts . •

A closer look a t the resonances below 2 kHz reinforces the s tate-

ment that the number of resonances is fair ly independent of shape

(Fig. I-A-3a). For a more complex boundary, a s the violin bounda-

r i e s , i t i s not possible to wri te the solution to the wave-equation as

a product of three separable solutions and it is not possible to ca l -

culate the standing waves by simple means. The solution was there-

fore obtained by means of probe measurements (Fig: I-B-3b). A

brute force method a s the finite element method can always be ap-

plied, but will give little physical insight besides f rom the measure -

ments in Fig. I-A-3b, if not car r ied out into great detail. The

stability of the resul ts can, however, be tested by means of perturba-

tion theory. But before we go into that, l e t u s study the resonance

frequencies more in detail.

RESONANCE D E N S I T I E S I N RECTANGULAR and C Y L I N D R I C A L ROOM

RESONANCE D E N S I T I E S I N ENCASED V I O L I N CLOSED and OPEN F-HOLES

Fig. I-A-2a. Resonance densities in the rectangular cavity, solid lines; in the cylindrical cavity, broken lines ; and the theoretical average, the third smooth line. All cavities of approx. the same volumes, a reas , and edge lengths a s those of a violin.

b. Resonance densities in the encased violin with closed f-holes, solid lines; with open f-holes, broken lines; and the theoretical average, the third smooth line.

RESONANCES OF T H I N C A V I T I E S WITH TWO FLAT WALLS OF ECUAL AREA

I V l OL i NSHAPED

CYLINDRICAL I kHz

Fig, I-A-3a. Resonances of the rectangular cavity, the f la t violin-shaped cavity, and the cylindrical cavity - al l cavities with approx. equal volumes, a r e a s , and edge lengths. Labelling of modes in the rectangular cavity(nx, n ): (031), (1,O). (1 , l ) . (0,2), (1,2), and (0,3). Labeling of modes in dbe cylindrical cavity (rn, n): (ti, O ) , (+2 , O ) , (0, I ) , (t3,O).

THE SEVEN LOWEST MOOES OF A VIOLINSHAPED C A V I T Y

- MAX. SOUNDPRESSURE

- - - M l N . SOUNDPRESSURE

[ I PHASE 0

P M A S E n I

Fig. I-A-3b. Standing wave patterns of the seven lowest modes in the flat violin- ;

shaped cavity. Resonance frequencies 460,1040,1130,1300,1590, 1800, and 1920 Hz.

STL-QPSR 1/1973 9.

F o r the rectangular and the cylindrical cavities the measured

resonance frequencies agree closely with those calculated a s -

suming rigid wall (75 70 of the frequencies within .5 % and 98 70 within I 70 of those measured). This means that the assumptions

made regarding the frequency prediction is in good order . .

As already said, the resonance frequencies for a complex bound-

a r y i s hard to calculate f r o m the wave-equation. But by means of

perturbation and employing Eqs. (8) and (9) can the frequency shifts

f rom a s impler boundary be estimated. If the "trial" function of

the simpler boundary is close to the r ea l solution, then an accurate

estimate i s obtained; for a l e s s close a measure of the magnitude

and direction should a t leas t be obtained. This means that per turba-

tion calculations a r e physically very informative. In our case we

shall be working somewhere in between the two cases sketched. We

s t a r t by looking a t the standing wave pat terns given in Fig. I-A-3b.

We find that the f i r s t , second, third, and fifth modes a r e of longi-

tudinal type (n =nz=O, n = I , 2 , 3 rectangular boundary). The natural X Y

t r i a l function is of the type given in Eq. (3). After averaging, the vol-

ume of upper and lower end blocks with that of the adjacent cavity

a n effective rectangular cavity i s obtained giving the discrepancies

between calculated and measured frequencies marked by c i rc les in

Fig. I-A-4. By applying Eq. (8) the frequency shifts corresponding

to boundary perklrbaticns a r e obtained. Although Eq. (8) can not pos-

sibly be expected to accurately predict the frequency shifts, we find

that the la rge discrepancies a r e removed. The second and fourth

modes a r e of "transverse" type in either the upper o r the lower

half. Both modes have p res su re maxima a t the bouts and nodal

planes along the midsectional symnletry plane. The walls a r e ap-

proximately circular , i, e. , a reasonable t r ia l function i s that of

Eq. (5) with n=l, n=n =O. The corresponding resonances of a z cylindrical cavity touching the inner walls a t the bouts marked with

c i rc les a r e shown in Fig. I-A-4. The perturbations calculated f rom

FREQUENCY ESTIMATES

Fig. I-A-4. Differences between calculated and measured resonance fre - 1 quencies of the flat violin-shaped cavity. Circles correspond to the frequencies calculated from the trial functions, and tri- angles to the frequencies calculated from the boundaries of the . trial functions pertrubed to fit those of the violin-shaped cavity.

B O U N D A R Y PERTURBAT l ONS

I i

0 > 2.0 kHz

Fig. I-A-5a. Different per turbat ions of the violin-shaped boundaries - only one of the two symmet r i ca l halves a r e marked in the f igure .

FREQUENCY S H I F T S FROM BOUNDARY PERTURBATIONS

Fig. I-A-5b. Frequency shifts calculated with the boundary perturbations of F ig . I-A-5a.

J

RECTANGULAR ROOM

Q m V = 3 , 3 n a =0,33

I 1 I

loo

0 . 4

r . C Y L I N D R I C A L ROOM

-

-

- a,,= 2,3e a =0,15

I I 1 rn

2a

(.

0 1 2 3 4 kHz

F L A T V I O L I N S H A P E D ROOM

Fig. I-A-6. Measured Q-factor s of (a) the rectangular cavity, (b) the f lat violin-shaped cavity, and (c) the cylindrical cavity. Single measures marked by dots and average by the solid line.

STL-QPSR 1/1973

Estimates according to Eqs. (12 - 16) show the following. The

boundary layer losses a r e dominated by the viscous losses and

Q - + 10 J f '

i. e. an average Q, a third in magnitude of the measured. By adding

wall losses with a standard % = 0.08 i s obtained

f + 4 kHz

This results in an overestimate of the total losses of approximately

ten to three times compared to those measured. Thus we may con-

clude that the boundary layer losses a re not the whole contribution

to the losses, but the 10s ses to the walls a r e considerably smaller

than a typical Gw indicates. The results a re in agreement with ( 1 7 ) Schelleng ' s findings .

Let us so study the effects of arching and f -holes. The arching

of a violin is smooth and moderate, the greatest depth being about 1 #

cm. Therefore we do not expect any larger changes in losses, vibra-

tion patterns, and resonance frequencies because of the arching.

The measurements show that the losses a r e within the uncertainty

of calculations. The frequency shifts a r e also small, within + 3 aJo, this also being in agreement with the expectations.

The influence of the f -holes on radiation and frequencies a r e

determined by the hole geometry and the position of the holes in r e -

lation to the standing wave, which is shown by Eqs. (19 - 2 1). There - '

fore the standing wave patterns of Fig. I-A-3b, can be used to

estimate the effects of the f-holes., From the standing wave patterns

we find immediately that the third and fifth modes have high sound pres-

sures at the position of the f -hole s. Therefore these mode s should be

influenced most by the f-holes. The third mode acts in agreement

FREQUENCY DIFFERENCES ARCHED-FLAT and OPEN-CLOSED A R C H E D

Fig. I-A-7. Differences between measured resonance frequencies of the encased violin with closed f-holes, and of the flat violin- shaped cavity (circles). Differences between measured re- sonance frequencies of the encased violin with open f -holes, and with closed f -holes (squares).

2 3 4 kHz

d

ENCASED VIOLIN-CLOSED F-HOLES

-

L

- 0,,=2,9Vf U =0,42

1 1 I I

0 200

150

100

5 0

0 0 1 2 3 4 kHz

I

- ENCASED VIOLIN-OPEN F-HOLES

-

-

- 6 =0,65

t 1 I 1 I

Fig. I-A-8. Measured Q-factors of the encased violin with (a) closed f -holes and (b) open f-holes. ' Single meas - ures marked by dots and average by the solid line.

STL-QPSR 1/1973 12.

with the theoretical conclusions, the Q drops considerably and the

frequency increases. The radiation of the fifth mode i s so effective

that the mode cannot be traced with open f-holes. The relatively

large frequency shifts of the f i r s t mode is likely to stem the in- 2

crease in velocity through the f-holes due to the (1/27 f) factor, Eq.

( 20). A study of the average Q:s shows that the f-holes on the aver-

age have resulted in a clear decrease in Q-factors (Fig. I-A-8).

Furthermore we can see several examples of what previously has

been discussed. Certain modes a r e damped harder than others,

thus giving a still greater spread in the Q:s from the average. The

losses of the f i r s t resonance a re little effected by the f -holes, which

may seem contradictory. However, the predicted increase of velo-

city through the f-holes is compensated by the decrease in radia-

tion resistance, Eq. (17).

5. Conclusions

The experiments have proved that in a violin-shaped cavity two

types of resonances a r e easily found in the lower frequency range:

1) Longitudinal resonances in the complete cavity, and 2) transverse I

resonanceseither between the upper or the lower bouts. The rdso-

nances a r e moderately influenced by boundary perturbations a s arch-

ing and f-holes. The power losses within the cavity a r e shown to be

larger than the boundary layer losses (a theoretical minimum) but

considerably smaller than 10s se s predicted from dampings of wooden

walls. The effect of the f -holes follows the qualitative predictions,

the main factor being the position of standing waves in relation to

position of the f-holes.

Acknowledgments

This work was supported by the Swedish Humanistic Research

Council and the Swedish Natural Science Research Council.

STL-QPSR 1/1973

References

(1) E. Jans son: "Recent Studies of Wall and Air Resonances in the Violinff, paper QQ3 presented a t the 84th Meeting of the Acoustical Society of America, Miami Beach, NOV. 28-Dec. i , 1972 and STL-QPSR 4/1972, pp. 34-39.

(2) F. A. Saunder s: "The Mechanical Actions of Violinstf, J. Acoust.Soc. Am. 9 (1937), pp. 81 -98.

(3) J. C. Schelleng: "The Violin a s a Circuit", J. Acoust. Soc. Am. 35 (i963), pp. 326-338 and p. 1291.

(4) E. Jansson: "Analogies between Bowed-String Instruments and the Human Voice, Source Fi l te r Models", STL-QPSR 3/1966, pp. 4-6.

(5) F. A. Saunders: "Recent Work on Violins", J. Acoust. Soc. Am. 25 (1953), pp. 491 -498.

( 6 ) see , for instance, ref. (3).

(7) see ref . (I).

(8) J. W. Rayleigh: Theory of Sound, Vol. 11, pp. 66 -68 (Dover, New York 1945).

(9) J. W. Rayleigh: Theory of Sound, Vol. I, pp. 336-338 (Dover, New York 1945).

(10) L. Crerner: "eber die akustische Grenzschicht von s ta r ren Wanden", Archiv d. Elektr. Ubertragung - 2 (1948), pp. 136-139.

I

(I 1) U. Ingard: "On the Theory and Design of Acoustic Resonators", J.Acoust. Soc.Arn. 25 (1953), pp. 1037 -1061.

(12) P. M. Morse and K, U. Ingard: Theoretical Acoustics, pp. 392-394 (McGraw Hill, New York 1968).

(13) cf. J. W. Rayleigh: Theory of Sound, Vol. I, § 91 (Dover, New York 1945)-

(14) see also G. Temple and W. G. Bickley: Rayleigh' s Principle and its Applications to Engineerifis over, New York 1956).

(15) F. Fransson and E. Jansson: ro roper ties of the STL-Iono- phone Transducer", STL-QPSR 2 -3/1971, pp. 43-52.

(16) This sond was constructed by F. Fransson for measurements on flutes, but turned out to be an excellent sond for a l l .

kinds of measurements a s in the present case.

(17) see ref. (3).