Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Recent studies of wall and airresonances in the violin

Jansson, E. V.

journal: STL-QPSRvolume: 13number: 4year: 1972pages: 034-039

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

STL-QPSR 4/1972

C. RECENT STUDIES OF WALL AND AIR RESONANCES IN THE VIOLIN",^"

E . Jansson

Abstract

It i s often assumed that higher a i r -modes in violins a r e non-existent because of the F-hole s. Experiments with cavities with rigid walls show, however, that this i s not t rue. The higher modes a r e in general only moderately affected by the F-holes. These modes a r e of two kinds, lon- gitudinal standing waves in the whole cavity and t ransverse standing wave s in the cavity on either side of the c-bouts. Detailed studies of the f i r s t a i r -mode above the Helmholtz mode in r ea l violins have been performed. These studies show that this a i r-mode falls in the frequency range of the main wood resonance and it i s excited f rom the bridge, i .e. the coopera- tion between this mode and the walls may be very important.

I. Introduction

The resonant box of a violin consists of a top plate and a back plate

glued to the r ibs , thus enclosing an a i r volume. In the top plate two sound

holes a r e cut, i. e. the F -hole s . The total a r e a of the two sound holes

taken together i s about 12 cm2 and that of the total a r e a of the enclosing 2

walls about 1200 cm . Thus the rat io between the two a r e a s i s about 1/106.

This smal l ratio implies that only a small portion of the stored energy of

standing waves se t up into the a i r volume i s likely to radiate through the

F-holes and that several higher a i r -modes may be expected in the en-

closed a i r cavity of the violin. However, it i s usually assumed that the

shape and the position of the F-hole s a r e such that higher a i r -modes can-

not effectively influence the sound producing mechanism of a violin, see

for example a theoretical study of the violin by ,Schelleng('). In ea r i i e r

studies by Saunders t r aces were found of higher a i r -modes , but these

( 2 ) modes seemed to give little influence on the total behavior of the violin . Pre l iminary experiments with a rectangular box gave that the energy

leakage through the F-holes does not remove higher a i r -modes in general.

This means that the acoustical inimpedance seen f r o m the plates may be

important although the sound radiation through the sound holes i s negli-

gible.

This paper was given a t the 84th meeting of the Acoustical Society of America, Miami Beach, Fla. , USA, Nov. 28 -Dec. 1, 1972

>L 46 The optical investigations a r e performed in cooperation with the Institute of Optical Research , KTH.

STL-QPSR 4/1972 3 5.

Therefore the present study was s tar ted to remove the ambiguity of

the above summarized r e su l t s and to give an under standing of the higher

air-modes. In this paper we shall l imit the discussion to the frequency

range below 2 kHz.

11. Experiments with violin- shaped cavities

The f i r s t main question to answer is: Does it o r does it not exis t

several higher a i r -mode s in a violin - shaped cavity with "F -hole s" ? To

find the answer of this question we measured the acoustical input imped-

ance of the a i r volume of a violin encased in p las te r thus blocking the

motion of the walls and allowing examination of the a i r cavity in isolation.

The impedance was measured by means of a specially designed measure - ment probe containing an STL-ionophone and a B&K 4 1 3 3 microphone with

a short and thin sond (length about 2 cm and effective diameter about

0 .025 cm). The impedance was measured both with closed and with open

F-hole s. When the F-hole s were open, then the a r e a of top plate between

the c-bouts (at the waist) and around the F-holes was f ree f rom plas te r ,

so that the radiation a t the F-holes should be the same a s in playing. The

resul t i s exemplified by two measured impedance curves, the upper one

with closed and the lower one with open F-holes ( ~ i g . 111-C-I). The up-

pe r and the lower diagrams show grossly the same number of peaks but

the lower impedance curves have slightly l e s s marked peaks.

The peak frequencies and the - 3 dB bandwidths were accurately

measured by means of a frequency counter for a l l resonances. F r o m

the se measurements the Q -factor s were calculated. The Q -factor s ob-

tained in this way a r e plotted a s function of peak frequencies in Fig.

111-C-2. F o r simplicity the modes a r e numbered starting f rom ze ro for

the Helmholtz mode. F r o m the diagram we find that the peak frequencies

a r e only slightly changed by the F-holes. The Q-factors a r e moderately

lowered by the F-hole s in a l l modes but two, namely the third and the

sixth mode. In the third mode the Q-factor is considerably lowered and

in the sixth mode no t r aces of a peak shows up. These resu l t s support

the hypothesis previously introduced; there a r e several resonances and

these resonances a r e in general only ~nodera te ly affected by the F-hole s.

? 2 3 kHz 4

0

10

20

30

40

50

60

70

80 0 1 2 3 kHz 4

Fig. 111-C - I. Acoustical inimpedance measured close to the end button of a violin encased in plaster ,

Top curve - closed F-holes Lower curve - open F-holes

1 kHz

Fig. III-C-2. Q-factors and frequencies of the f i r s t seven a i r modes of a violin encased in plaster.

STL-QPSR 4/1972 3 6 .

Next question we asked ourselves was: Why a r e just the third and

the sixth modes so affected by the F-hole s ? The oscillation modes of

a cavity can be greatly affected by holes in the cavity walls. Not only

the size but a l so the place of the hole a r e important, a s pointed out by

Schelling('). A hole dril led a t a sound p res su re maximum can affect

the standing wave considerably while a hole a t a sound p res su re min-

imum will have little effect on the standing wave. Thus we should f i r s t

record the standing wave patterns. Thereafter we can see if these pat-

t e rns explain what happens when the F-holes a r e opened. To simplify

our measurements we made these experiments with a cavity shaped like

the inside of a violin but with flat top and back plate. The resonance

frequencies of this cavity approximate within a few percent those of the

violin encased in plaster . This indicates that the arching of the top and

back plates i s not very important to the seven lowest modes. The dif-

ferent standing waves were excited and the sound p r e s s u r e s a t different

positions were measured through small holes dril led in the walls. The

different standing wave pat terns estimated a r e presented in Fig. 111-C-3.

The f i r s t , the third, and the fifth modes correspond to the f i r s t three

modes of a pipe closed in both ends although moderately perturbed by

the swelling and shrinking of the cross-sect ional a rea . The second and

the seventh modes a r e resonances of the cavity below the narrowing

section between the c-bouts. The fourth mode i s the m i r r o r image of

the second mode in the cavity pa r t above the c-bouts. The sixth mode

i s made up by a combination of vertical and horizontal standing waves.

The resu l t s allow the volume of a violin to be regarded a s consisting of

two coupled cavities. At some resonances the coupling between these I

cavities i s strong and the energy i s stored in the whole a i r volume, a t

some other ones weak and the enerby i s solely stored in e i t h e r of the

upper or lower cavity.

The F-holes a r e grossly situated between the c-bouts. This is a t

about the border a r e a between the upper and lower cavity estimated from

Fig. 111-C-3, i. e. an a r e a of low sound p res su re for the second, the

fourth, and the seventh modes. These modes will therefore be little

affected by the sound holes. Fur thermore , the f i r s t and fifth modes

have sound p res su re minimum between the c-bouts and a r e thus little

affected. Only the third and the sixth modes have sound p res su re max-

I:nlum in the vicinity of the F-hole s , which explains why their Q -factor

Fig. III-C -3 . Standing wave patterns of the seven lowest modes of a violin-shaped flat cavity.

MAX. SOUNDPRESSURE

- - - M I N . SOUNDPRESSURE

L-] PHASE 0

STL-QPSR 4/1972 37.

drop considerably when the F-holes a r e opened. The ru le regarding

the position of holes in relation to sound p r e s s u r e maximum'has thus

proved to give resu l t s in agreement with the experiments.

To summarize , we may say that regarding the resonances of a cavity

shaped like a violin and with F-hole s, higher air -modes exis t and most

of them a r e moderately affected by the F-hole s. The propert ies of the

a i r -mode s obey a t lea st qualitatively the simple rule s regarding holes

in cavities with standing wave s.

111. Experiments with violins

So far we have studied the modes of a cavity with rigid and heavy

walls. In a r ea l violin this is not a good approximation. F r o m ea r l i e r

studie s by 3 ans son, Molin, and Sundin, we know that plates a r e vibrating

with little motion a t the r ibs a t leas t for higher frequencies(3). The air-

modes of the vio1j.n shaped cavities have sound p res su re maximum gen-

eral ly a t the "ribs", i. e. a t the places where the plate motion is small.

The fact that the places of maximum sound p res su re and maximum plate

motion a r e different, indicates that the coupling between plate motion

and a i r -modes i s weak and that the higher a i r -modes a r e sti l l p resent

in the violin.

In our experiments , yet not finished, we have begun with a detailed

study of the air-mode I , which i s a t about 500 Hz and thus close to a

major resonance peak of violins - the so-called main wood peak. We

built a new measuring probe consisting of an STL-ionophone and a B&K

1/4 inch microphone with a sond. These we mounted on and into an "end

button" of plexi-glass. By means of this special "end button" we were

able to record the a i r vibrations with a well defined acoustical input on

violins, even stringed and tuned instruments.

With this device the acoustical input impedance a t the end button of

six violins was measured in the frequency range of air-mode 1. A clear

peak showed up in a l l instruments. Test with a sond microphone in dif-

ferent positions inside the instrument verified that this was the air-mode.

The frequencies and Q-factors of these peaks a r e displayed in Fig. III-C-4.

The frequencies a r e lowered compared to those of the violin shaped cav-

ity with F -hole s and fall close in frequency to A sharp, i. e. just in the 4

Fig. 111-C-4. Q-factors and frequencies of the first air-mode in six complete violins.

Q

60

50

40

30

20 b

400 t 450 t 500 Hz

' I t 1 I I I I I I

0 - I

- 0 @@@ 0

- I

- - 0

- - I I I 1 I I 1 I 1 b

A I R MODE 1

0 Mean value 1 HS 71 2 SK 21 3 Magg ( N C T ) 4 Guer ( N C T ) 5 CSSR (DNP) 6 EJN 52

STL-QPSR 4/1972 3 8.

region of the main wood peak generally found in good violins. The Q-

factor averaged for s ix violins i s only slightly lower than the Q-factors

for the violin shaped cavity, i. e. the losses through the walls a r e mod-

erate . Thus we have proved that air-mode 1 i s set up in complete violins.

Our next question is: Can this air-mode be excited by the plate vibra-

tions ? We excited the bridge electro-magnetically and measured the f r e - quency response by the microphone in the end button in the frequency

range of a i r -mode 1. Probe measurements proved that the a i r was 0 s -

cillating in a i r -mode 1. The pressure minimum of the air-mode was

found to be roughly in the place of the bridge. A peak was sti l l generally

found (5 out of 6) corresponding to the air-mode. Fur thermore the inside

of the lower par t of the top plate was found to be coupled in phase with

the sound pressure of the a i r -mode.

IV. Conclusion

In a previous study of our test instrument HS 7 1, it was found that

the wall vibrations a r e mainly in the f i r s t top plate mode in the region

of the main wood peak(4). In the present study we have found that the

f i r s t air-mode above the Helmholtz mode fal ls in the same region. The

experiments have proved that the two modes a r e coupled. Standing wave

pat terns of the two modes a r e drawn for comparison in Fig. 111-C-5.

The entire top plate moves in phase, whereas the sound p ressu re in the 0 upper and the lower p a r t s a r e 180 out of phase. Thus a simple and

direct coupling between the sound p ressu re of the a i r -mode and the

vibrations of top plate i s not possible. However, an estimate of the

volume displacement by the top plate above and below the bridge, i. e.

the pressure minimum, gives a difference of about 10 70, the lower pa r t

giving the grea ter displacement. The phase relations of such a coupling

agree with the experimentally determined phase relations, the sound

p ressu re of the a i r mode 1 being direct coupled to the lower pa r t of the

top plate vibrations.

Thus we have proved, that a t least one a i r -mode above the Helm-

holtz mode is present in violins, that this mode is coupled to wall vibra-

tions. We have a lso given an explanation of the coupling mechanism.

Although the a i r -mode does not radiate through the F -holes, the acous-

t ical load on the top plate vibrations may be important. Especially a s

Fig. III-C-5a. The f i r s t a i r -mode (470 Hz , Q = 65), and

b. the f i r s t top plate mode of violin HS 7 i ( 4 8 0 Hz). ( ~ i g . III-C -5b i s f r o m Jansson-Molin-Sundin, Phys i ca Scr ipta , Vol. 2 , pp. 243-256, 1970.)

STL-QPSR 4/1972

the mode fal ls in a frequency range of great interest , the range where

violins have a peak in their acoustical output and where the so-called

wolfnote i s to be found.

Our investigations a r e continued to study the importance of the

higher a i r -modes and how the relation between the top plate mode and

the a i r -mode influence s the main wood peak, the wolfnote, and thus the

quality of the instruments.

Acknowledgments

This work was supported by the Swedish Humanistic Research Council

and the Swedish Natural Science Re search Council.

References

(1) Schelleng, 3 . C. : "The Violin a s a Circuit", J . of the Acoust. Soc.Am. - 35 (1963), pp. 326-338 and 1291.

(2) Saunder s , F. : "Recent Work on Violins1', J. of the Acoust. Soc. Am. - 2 5 (1953), pp. 491-498.

( 3 ) Jansson, E . , Molin, N - E . , andsundin, H.: "Resonances o f a Violin Body Studied by Holograrn Interferometry and Acous- t ical Methods", Physica Scripta - 2 (1970), pp. 243-256.

(4) Jansson, E . : l ' f in Inves t iga t ionofaVio l inbyLase r Speckle Interferometry and Acoustical Measurements", Acustica (in the p res s ) .