RESEARCH DEPARTMENT

, INV}!;STIGATlON OF BElMHOITZ RESOUATORS

REPORT NO. B.041 Serial No. 1949/29

Research Department

J?B.rVATE Al;D COhFIDENT lA.I,

REPORT NO. B.041 Serial No. 1949729

August, 1949

. li'igs.liIos. B.041.1 to B.041.11

Htrum IGATIOh OF HELi:tIHOLTZ RESONATOPc~

.SilliIi!IARY

This report deals with a theoretical and experimental investi­gation into the characteristics of Helmholtz resonators as sound absorbers. Rschevkin and others have investigated the problem theoretically, but the experimental details given are inadequate for assessing the possible -use of such resonators 8S sound absorbers in broadcasting studios.

The present investigation shows:-

(a) The frequenc~ . of maximulTi absOl'lJt ion can be accurately cal-­culated from the dimel'lsions of the resonator, and can be varied within fairly vvide limits by simple means.

(b) Under favourable conditions absorpti,ons of 6cP/o and more can be obtained aJG resonance. The absorption peal\: is normally rather sharp, but can be broadened to some extent by Buitabl(3 design.

(c) The physical .size of the resonators can be less than that usually associated with other types of absorbers working at low frequencies.

(d) Under certain oondi tions Helmhol t-z resonators can be used. to increase the reverberation time of a room in the region around the resonance frequency.

It has to be borne in mind that some of the above advan·tages are fJmtual1y exclusive, and that in use a oompromise may have to be accepted.

1. Hi TRODUCTIOH

A HeJ~holtz resonator oonsists of a volume of air oontained in a riGid vessel, oorrmunicatinG with the outer air by means of a oomparatively narrow neck. The neck w~ be of any shape, but is usually oylindrical.

These resonators have been used in Scandinavia for many oenturies for improving the acoustics of theatres and. churches, and more recently in Denmark for broadcasting studios, but although the method of deri­vation of the resonance frequency is well Imown, there is no accepted theory of the effect produced on the aooustios of a room.

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The purpose of these experiments was to examine the behaviour of such resonators in rooms and broadca.sting studios, and to establish reliable means of predicting their effect. Recent papers on the sub­ject, (see Bibliography i and Ui)" have been mainly theoretical with li ttle experimental backing, a,nd, are aonflicting in tbeir quanti ta tive results.

In this report the theoretiaal information available ia summarised before describing the experimental work, and finally the modifications to the theory necessary in the light of' the experimental results are disaussed. Experiments are limited to frequencies below 200 c/s since sound Of higher frequencies is more easily absorbed by P01'OUS materials.

2. THEORY OF RESONAHOE

For purposes of calculation the volurne of air in the cavity of the resonator is considered as an elastia body, while the mass of the vi brating s;ystem is simply that of the air aolumn in the neck. (Refer to Appendix or Bibliography ii.) The resonance frequency is thus found to be :-

where Fo Cl

d V~ S K

_. .sL lrr!:' C ;'K

l!' h._ 2t£ ",!dVo ... - 2i iV; (l) 0 •• e •••

- resonance frequency - velocity of sound in air

.... effecti vs length of neck::': length 'C' 1.7 x rad.iua

.... volume of air in cavity '., area of aross section of neck S .. conductivity of the orifice, defined as d

3. THEORIES OF ABSORPTION

It is clear that under ideal conditions, (infinite rigidit;y of container etc. ), absorption of energy can be conSidered to take place entirely in the resistance at the neok. This resistance has two components, the viscous resistanoe of the air in the neck, and the radiation resistance, the latter being much the smaller oomponent in the case of narrow' necks. The viscous resistance ie dependent on the dimenSions of the neck and is influenced by the presence of resistive material.

The effect produced by a number of such resonators in a room has been approached from two different standpoints :-

3.1 First Approaoh (Rsohevkin, 1936)

The classical treatment of reverberation in a room, leading to Sa'bine' 8 formula, is mod.ified to include the effect of the stored

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energy in the resonators and that of the loss of energy due to friction in the neck.'

The normal Sabine formula for the untreated room in c.g.s. units :-

where V A T

'1 ::: (0.162 x 10-2 ).f-- volume of room _.. absorpt ion of room in Sabine s . reverberation time

is·modified and becomes

2 V -}- nV' -- (0.162 x 10- ). A -I llAf

•••••• (2 )

•••••• (3 i

at the resonance frequency, where n is the number of resonato11 s, and. At and V' are terms involving the dimensions of the neck and its resistanae~

From (3) it is clear that the volume of the room is effectively increased by an amount nV' due-to the presence of the resonators,and hence V' can be called the "additional volume" introduced by a single resonator. ~imilarly AI is the additional absorption.

Equation (3) also predicts that resonators in a room can act either to increase or decrease the reverberation Ume of the room, depending on the charaoteristics of the resonator and on those of the room (its volume and total absorption). If the ratio of "additional volume" to absorption, ~. , for the resonators is greater than .the same quantity for the room,

. 1.e. if 1. 62 ~ lO--3.r; is more than Tl , the reverberation time will be increased. It should be noted that this follows from the general argument :-

lot a ) :x: Then if a ::.~ fiX and b c': ny, m ) n C y

Therefore x -I. a fr'- I j}~ .~ x Y b

_. Xl -I y) Y

IncreaSing the resistance of the neck increases the absorbing power of the resonators up to a point of maximum absorption, beyond whioh increasing !lecIc resistanceresul ts· in a decrease of' absorption. This can be represented as a minimum in the curve relating reverberation time with neck reSistance, and the calculp..tecl curve for a special case is shown in Fig. l(a).

3.2 Seoond Approach

. Al ternati vo13T the problem can be considered simpl;y as one of matohing the impedance of' a banlz of resonators to a plane ;,v.ave. At the resonance frequenoy the impedance is purely resistive, being the reSistance of the ,

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neolt, and this resistance appears to the air outside to be larger than its actual value. This is because the velOCity in the neck is higher than that in the air outside, so that a transforming aotion takes place, the transformer ratio be ing the patio of tQtal neck area to the total wall and necl{ area. The e1ffective impedance Z for unit area of the resonator block is given by :-

, where A S

z

... • effec ti ve \/ijall area surround.ing eaoh hole cross-sectional area of neck

R - resistance of neclc.

Maximum absorption of energy from a plane wave talws place when the effective impedance is purely resistive and equal to the charac­teristio impedance of the air. (42 c.g.s. unil;s or "aooustic ohms" at 2000.) For an offecti ve impedance Z per uni t area of resonator block, the absorption coefficient ,X is shown in the Appendix to be 1-

~ :.: ---1.~ (42+Z)2

As with the first approach, it follows that there is a minilnum value of reverberation time at one particular value of neck resis­tance, but Fig_ l{b), which connects R with reverberation time, shows ,that the optimum neck resistance is not critical. This curve is plotted in terms of actual neck resistance to conform vvi th 1 (a), and the paint correspond.ing to a transformed impedanoe of 42 ohms is marlced. '

3.3 There are two important differenoes between these two methods of approach ;-

(1) It will be noted on reference to Fig. 1 that, althongh both theories indicate that there is a value of neck resistance for which the reverberation time is a minimum, this optimum value of resistance is \videly different in the ho cases.

(11) The first theory shO\'iis that under certain aond.itions (derJen­ding upon the neck resistance and upon the original rever­beration time of the untreated room) the roverberf.ltion time will be increased. The "matching" theory dOGS not indioate such a possibility, since it ignoros the exohange time for energy stored in the resonator.

4. DESIGN OF RESONA.TORS A.IID MEAS1JREMEHTS

The theory outJ..ined in paragra.-ph 3 applies to a container of high rigidity having tho orifice in a plane surface, and the resonators usod

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were designed to have these characteristics.

Six cylindrical tins in a,mould, each having an on one side of the blocl;;:. a total.of 96 resonators.

of 2530 ccs ~ volume were set in concrete opening li" (3.8 cm.) long and l~-" diameter Sixteen 1)f these blocks were made, providing

Equation (1) gives the frequency of each resonator as 140 c/s. The resonance frequency could be varied by inserting bushes into the necks. Two sets of bushes were made :-

(a) (b 1

neck length l~J' (3.8 cm.), diameter 1", theqretical frequency 105 c/s. neolc length~' (1.59 cm. j, diameter i", theo!,otical frequency 75 c/a.

4.1 Preliminary absorption measurements were made in the small reverbf3rn­tion chamber at Nightingale Square, but the performance of the resonators was to a oertain extent rnaslmd by the characteristics of the room. If was therefore decided that measurements on an individual resonator ahould be used as a starting. point.

4.2 Qpen air measurements

The first step was to check that· the frequency agreed "=>ith that cal­culated from theory. The arrangement originally used was as follows :-

A uni t of six resonators was placed on the ground in the oPen air wi th the holes towards a loudspeaker suru{ into a hole about G ft. away. ~ Willan's microphone was plaoed with its probe direotly in front of one of the holes, and tone was fed to the loudspeaker from a tone souroe fitted with a motor drive oapable of varying the frequency at the rate of one octave in about 3 milmtes. The output of the miorophone, suitably ampli­fied, was fed to a Neumann high speed level reoorder, giving a preasurej frequency record Similar to that shown in Fig. 2. It will be seen that the dotted curve has a double peak. This was found. to be due to~ coupling between adjacent resonators in the block, and could be prevented by inser­ting corks into the five remaining holes. The continuous gurve in Fig. 2 illustrates this condition •

.An unexpected feature of those records was the dip whioh invariably foHowed the peak, making the exaot point of resonanoe difficult to determine. A theoretical explanation (given in the Appendix 1 indicated that the peak occurred at the resonance frequency and that the dip was oontrolled by the Position of the probe. Further tests verified this 'theory.

In the arrangement eventually used the probe was sealed into a hole in the baok of the resonator, giving a measurement of pressure aoross the oapacitance, (see Ap~:?Gndix). This method produoed a simple resonance ourve, examples of which can be seen in Fig. 3.

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At this junoture the apparatus was tra.nsferred from the open air to an experimental studio with the loudspeaker 12" from the orifice and the tone kept at a relatively low level. The arrange-ment is J3hown in Fig. 9 • Results so obtained were .1del1tioal with those obtained in the open air, indicating that the effects of the room had b~en satisfactorily oliminated. .All subsequent measure­ments wore made und.er room oondi tions.

4.3 Measurement of Resistance

As it was desired to reach optimum. conditions by varying tM neck resistance artificially, it was necessary to obtain a measure"'! ment of this resistance. The resonance curves provided a simple method of mea.suring relative resistances b3T estimating the IIQII from the shape of the curve and calculating l/Q. An absolute value of resistance could also be found by mul tiplying l/Q by the calculated mass reactance of the neck.

The absolute values of resistance for the three sizes of open neck, while of the same order, were not identical with those given by os.lcula,tion. Various materiD.ls were fix~d across the neoks of the resonators to increase the resistance. It was found that the proportional change in resistance on adding a material was approxi­mately the same for all three sizes of neok. Experiments also showed that to increase resistance without appreciably ohanging the resonance frequency, 8 material having a high proportion of air, space to fabrio was neoessary. Ordinary open-weave bandage was an easily obtained material having the desired properties, and was chosen for subsequent measurements; but oertain types of Fibreglass or linen sorim would probably form a more sui table treatment in practice. The results are shown in Table .I.

In this table the value of R is oalculated as indioated above, while Z is calculated from R using the transformer ratio and having first subtraoted the value for the radiation resistanoe of the orifioe, where this is appreoiable. The last column is included for comparison purposes and gives the value of R oomputed for the open neck using the theoretical fornmla,

where r:::~ de ns i ty of air tl -_. viscosity of air ro . - radius of neck

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4.4 Reverberation Room Measurements

The measurements were marle in a reverberation ohamber, (G.ll, Nightingale Square), having a volume of about 1000 ou.ft., and 14 uni ts vvere used. cor.taining 84 individual resonators or The same disposi tion of uni t8 was maintained throughout the series. A con­trol run was first talcen with the resonators in posi tion but vii th the holes blooked with oorks, and any subsequent change in the reverberation time \vas therefore due to the aotion of the resonators themselves. The resistanoe of the resonators was gradually inoreasod until the point of maximum absorption had been reached.

The results of the tests on the 140 c/s resonators are shown in Fig. 4, It will be seen that, whereas there was negligible absorption in the oase of the open neok, ono layer of bandage across the holes produced a large l~Dk around the rosonanoe frequency. No further tests wero carried out with this size of neok for the following reasons a-

(a) The concrete neol{s were variable in size, so that the resonator frequenoies differed. Wooden bushes could be made more accurately.

(b) Resonators absol'bing at lower frequencies than 140 c/s Were considered to have more practioal applioation.

The 75 c/s bushec were therefore fitted for subsequent absorp­tion measurements made with the' necks empty, with a wad of bandage inserted, and i~i th one and 1;'1",,0 la.yers of bandage across the orifice. The curves so obtained are shown in Fig. 5 and the effect on the reverberation time of the room is given in Fig. 6.

The reverberation ourves, Fig. 6, show that the empty room unfortunately possessed a pronounced pealcat 75 c/s, which, although useful for demonstrating the action of the resonator,g,made accurate meaBurements inth18 region difficult.

Ir. dealing wi th the wads of bandage it was found to be virtually impossible to obtain absolute consistency of arrangement in individual neoks. This probably aocounts for the pronounced drop in absorption when using this partioular arrangement of the material. The ind.i­cations were that the menu resistance of the 84 resonators' was con­siderably higher than that of the sample used for resistance measure-tnent. For this reason the measurements were repeated using a type of glass wool fabrio, whioh gave a similar value' of resistance.

The values of the reverberation time at resonanoe have been plotted on Fig. 1 for comparison with the theoretical values.

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4.5 Studio Measurementa

The abnormal oondi tions of the reverberation r.oom were ideal for the foregoing measuremellts, but it was obviously desirable to oarry out tes·ts under more usu4l working oondi Hons, namely, those existing in rooms already possessing oonsid.erable absorption, keeping in mind the possibility of colouration. Tests were therefore made in tviO

rooms of this kind.

Listening :!ioom at lUghtihgale Square

This room has· a: volume of 2850 cu.ft. and is furnished in the style of an ordinary living room. The reverberation time is of the order of 0.6 seoonds. Reverberation time measurements 'were oarried out under five conditions ;-

(a. ) {b 1 (0 )

(d)

(e)

wi th resonators in the room but with the holes corlced with e4 - 75' o/s resonators - open holes wi th 84 - 75 0/8 resonators wi th two layers of bandage

(matohed oondition) wi th 84 - 75 c/s resonators wi th bandage in the neok

("overmatohed" condition) with 84 - 140 0/8 resonators - open holes.

The reverberation ourves are shown in Fig. 7 and it will be seen that the expeoted absorption took plaoe at about 75 o/s. The number of miorophone positions used for these measurements was insufficient to give accurate figupes, but the effect of the resonators ... ~as nevertheless quite apparent. In the case of the 140 o/s resonators, a1 though there wa.s a decrea.se in reverberation 'time in that region, the effect vvas so small as to be oonsid.ered insignifioant. A sp~ech test and record.ing was made under ea.ch of these conditions. Ho co1ouration vvas detectable, either in the speech test or in any of the recordings, possibly because the reverbera tion time of the room was lal'ge compared. with the decay times of the resonators.

Exper~ents in Studio 3G, Broadcasting House

A Similar series Of tests was carried out in Studio 3G whioh has a reverberation time of the order of 0.3 seaonds. The 75 c/s resonators produced a negligible effect on the l'everberation time, as '\I"ias to be el.."Pected vii th such a small area in a room containing so much absorption. The 140 c/a resonators wi th open necIes, however, produced a pronounced increase of reverberation time around that frequency (Fig. 8 l. The records from the high speed level recorder were perfectly normal in all cases, there ·being no sign of a change of slope

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at the end of thedeoay curve. In spite of this increase of time around ,the res'onance frequency, no colouration was detectable in any of the recordings, an effect for which no explanation can be give~ at present .. '

The studio was also pulsed with the 140 c/s resonators in position and,althbugh it was characterised by a ring at this, frequency, there was, contrary to expectation, no audible excitntion of the resonators by freque·ncies closely spaced on either side"

5. :QISOUSSION OF RESULTS

A oomparison between the experimental points of Fig. 1 and the two theoretical curves 1 (a) and 1 (b) makes it clear that the e~"Perimental evidence favours the second theoretical approach (Section 3.2), since the maXimmu absorption takes place when the effective impedance of the resona-tors is about equal to 42 acoustic ohms. Moreover, the optimum value of reSistance is clearly not critical, fairly large resistance changes produciug barely measurable changes in reverberation time,a result in gBneral accord with theOl1 Y. The tests were repeated at different times. and al though thoro were slight vaxiations in the actual values of abso2'p­tion, the rosul ts were consistent in indicating an optimum at 40 - 50 ohms.

5.1 The magnitude of the absorption Goefficient is in less satisfactory alS"I'eement however, since it does not rise higher than 631? by the Millington formula. This formula has been used in the Oorporation 'for many ;years now and has proved satisfactory in most cases, but an inherent disadvantage is tha,t it breaks down' for absorbents giving values near unHy, because here - log (1 - Cl} beoomes infinite and, avena square .. inch of a perfect {:lOsorber would be axpeCiiBd to reduce the reverberation time to zero. It has been observed 011 many oMasiona re0ently that both resonant anq. porous absorbers under carefully matohed oondi tions give a maximmn absorption of about 60c{0 by Millington' a formula, corresponding to nearly 100% by Sabine I s formula, and it ia evident tllat alternative methode of oaloulation should be d.eveloped for very efficient absorbers,

5.2 The tests in Studio 3Gshmved, however, that the simple matching theory of :paragraph 3.2 is inadequate since it does not account for the increase of reverberation time observed under certain oonditions. This effect, previously noted by Rschevkin, was confirmed in Studio 3G, where the reverberation time was short enough to give prominence to the effect. A modification to the simple theory was therefore proposed, using the "matching" theory but taking into' acoount also the energy which is stored in the resonators. It is assumed that when the sound is cut off, the energy accumulating intbe resonator during the steady state condition is given back to the room, and that the room and resonator act -as one system, the total energy at any

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instant being the sum of energy in the room and that in the resonators. This modified theory shows the possibility of an inorease of revel' .. beration time for low neCk resistance, the volume of the room being effectively increased by an amount proportional to the square of the HQ" of the resonators.

Fig. 1 (c) shows the relationship bet\'Ileen'resistanoe and reverberation time for the reverberation room G.ll, and it is seen that it is in better agreement with the experimental pOints than either of the others, suggesting that the modified matching theory is the most satisfactory description of the observed behaviour of the Helmholtz resonators under reverberation room conditions.

, Unfortunately the quantitative agreement between theory and experiment in the case of the incre,ase of re':er~beration time in Studio 3G was very poor, theory indicating a much smaller increase than was actually the case. Further 'Work. is therefore being carried out in an effort to establish a theory which is in better agreement with the experimental results.

6. PBAQTICAI, FORN'S OF lE1M.I-:lQL'];Z RES01{£l"TORS

All the e~,pe:rimental work described above was carried out on resonators made in the form of concrete b1001o;:s for rigidity and ease of construction. In ad.dition a uni t of six resonators wa,s made from plywood and tested for frequency and neck resistance. Thisoonstruction geve results which were identical with those from the conorete units, but had the advantages of lightness and smaller overall thiokness. The latter is of great importance vvhen dealing wi th small studios, and there is no apparent reason why the dimensions shOUld not be further reduoed. Again, for low frequency absorption it may be possible to dispense with the separating walls between individual resonators, a bloak of resonators consisting simply of a front panel sm~Jll in comparison wi th the selected wavelength and per-forated with holes of calculated diameter at a lmovvn spacing. This point has been 'Verified in connection wi th perforated oovering materials, (see Research Report B.040).

\

The wooden resonator with glass fabric serving as a resistive oovering is shown in Fig. 10, together \i9ith one of the concrete units and a wooden _busha

7. CONC1US IONS

Helmholtz resonators, suoh as those investigated in these experi-ments, should be suitable for use as sound absorbing units. For a given frequency of maximum absorption less depth is required than in the case of panel resonators so far studied, a very desirable feature for ver;y low frequency absorption. The operating frequenoy is aocurately predictable.

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The resistance may be adjusted over a useful range of seleotivity and they may either increase or decrease the reverberation time of a room acoording to the resista.nce value.

Previous theor~es have been extended to a.cco'lmt quantitatively for the conditions for maximum absorption, and quali taU vely for the inorease of reverberation time, which has been experimentally realised.

Thetria.ls which were made under actual studio conditions, though inconclusive in some other respects, showed that these resonators do not oause appreoiable colouration. It is proposed to test th{'lm under more carefully controlled conditions when opportunity arises, and an a.ssess­ment of their real value for studio treatment oompared with existing methods will then be made.

(fI. L. Kirke)

'Investigation and Report by c. L.S. Gilford A.L~ Newman F.L .. Ward

Bibliograp!l.Y

DIP

i Rschevkin, Tech. Phys. U.S.S.R. III (1936) pp. 560-576 ii Alexaflder Wood, "Aooustios", (Blackie) 1940, p. 103

ili Rsohevkin, C.R.Acad.Sc. U.S.S.R. (1938) XVIII pp. 25-30.

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APPEHDIX

Freguencyof Resonance

Fig. 11 (a.) shows the acoustic circuit of a Helmholtz resonator,' represented by its eleotrical analogue. The e .m.f. e· and ourrent i of the circuit represent the pressure and volume flow respectively in the resonator system, and the other equivalents are :-

L m/S2 ¥-0 'VI

f1-o2

R

where m effecti ve mass of air in the neolc S .. - cross-section of neck ;.-~~'

16 fS d effective length of neck do I- - :-.:::-.-- 3'i1 1t 3 f -. density of air 0 .-- voloci ty of sound in air VI ... volume of cavity Rl -. radiation resistance of neck R2 .. - frictional resistanoe of neck

V- --. viscosity of air r ill -- 21T x frequency Of incident sound

Rl is sm,all compared with R2 and may be neglocted in the ca so of resonators wi th small neck diameters such as those here considered,

The resonance

Fo"

frequency is seen to be f':C~

:1. t-=-2"1[,,/LO

o 175'-2 -rr ·l'd:Vl c·r 21f./V .

•••••• (1 )

where le is known as the oonduotivity of the orifice, equal to S. d

Response Curve of Probe Microphone in Neck

The end oorrection given above represents an effective extension of the neck beyond the wall surface, and BD in plaCing the probe near the orifice ono is effectively measuring pressure at a point inside the neclt. This condition is shown in Fig. 11 (b), where. the inductance and resistance of the neelt have been split into two. components. E ia the pressure measured by the microphone.

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Le t Zl be the impedance of the inner c ircui t LIRl C and le t Z. be the impedance of the complete oirouit LRC.

, Zll ~Rl~ 1 2):1. ... 1- (ruL1 - _j '2

(l)C j

I z I {R2 :I.

' .. ~. ( (lIL I )2)2" ( - l1il) }

and. E is given by

•• It 8 •• (2 )

We see from (2) that a maximum value of E occurs when Z is a minimum, i.e. at the resonanoe frequency of the oomplete oircuit LRC, and. similarly a minimum value of E ocours when Zl is a minimum, i.e. at the resonance frequency of a ciroui t formed by LIB-IO.

Now as L1 is always less than L, and. 0 has the same valu.e for both oiroui ts, i 1 follows that resonance will ocour for IRC at a lower frequenoy than for LIRl e, henoe the maximum value of E will always ocour at a lCilVer frequency than the minimum. As the probe is moved away from the orifioe the maximum will remain oonstant in frequenoy, 81 though becoming smaller in magnitude, while the minimum will move towards the maximum until evenutally the pressure measured ~ill be simply that of the source.

Probe Miorophone into Oavity of ResQflator

In this oondition the pressure across the capacitance C is measured, and on varying the frequency a simple series circuit response ourve will be" obtained, from vihich a valuo for the Q, factor of the oircui t can be found.

Now Q, mL R

Hence, knmring the value of L, wc obtain a value for R, the sum of the radiation and frict ional resistances.

The accuracy of the method. depend.s. on the assumption of the value of L. This value is effectively confirmed by the praotical aocuraoy of resona.nce frequencies oaloulated. from (1).

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Matohing Theory of Absorption

The room is represented in Fig. 11(0) by a consta.nt voltage generator in series wi th the characteristio impedance K, of a plane wave \'Vhere S :::.: {Jc and is numerioally equal to 42 aooust io ohms.

The transforming action,which has been postulated in section 3.2 above, is due to the inorease in velocity taking place between the plane wave front in the room and the interior of the resonator neok. Using the notation of Fig. ll(d} the transformer ratio from air to neclr will be .A/S and the effective impedance of the neck at resonance, as Been by the plane wave,· will be

z 2 1 : .. - R{A/s) • -.A

:.: -#.2 S

•••••• (3)

for unit area, since R is the impedance of the resonator at resonance. This condition is shown in Fig. l1(e).

Now the energy absorbed in the resonator

e2 This will have i ts D"Y'ea'~est value when Z :- K and energy absorbed:::: -- K

o· 4I{2

If we define this to be unity, i.e. complete absorption, the " absorption coefficient for any other value of Z is given by :-

4KZ l68Z (42 i- Z )2

for air.

It will be noted tha.t this expression, applying to the whole frontal al"ea of a resonator of which tM. neck is only a small fract ion, ia the same as that given in the Appendix of Research Report B.032 for a panel resonator in which the impedance is uniform over the whole surface.

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TABLE I

EFFECTIVE IMPEDANCES OF RESONATOR NECKS

.---------------------------------------------.---------------------------. Q R

RESONATOR 'A

lFroquency 14~~s __ T""'r::"...:a::::::n::.:s::...::f:..::·o:.:r.:::m=e.::.r__=Im::::-o:.;pe::...;d:::a=·n;:.::c~e~R::::a~t.::.i.:::.o __ 1:...::..:.... 7

Open neck 35/40 2.4 One layer of bandage 15 5.6 Open mesh saolting 13 6.7 Two layers of bandage 10 8.4 Cull urn' S 1301' im 10 8.4 , ROC1{Woo1 in cavity 9 9.3 Fibreglass over neck 8 4.3 Three layers of bandage 5 l? Bandage in neck 5 17

RESON.ATOR It

Frequency 75 c/s Transformer Impedance Ratio 147

Open neck 13 0.14 One layer of bandage ~t 0.29 T"w9 layers of bandage 0.35 Rook.Woo1 in oavity 5f- 0.35 Glass wool fabrio 3i 0.54 Bandage in neel;: 3! appl'()x. 0.54

RESONATOR C

li'reguency 105 o/a Transformer Impedance Ratio 8.5

Open neck 18 1.8 One layer of ba.nd.a.ge 8 4.0

RESONATOR D {Repeat of Resonator B in wood.) ,.

;B'r~guenoy 75 c/a Transformer Impedance Ratio 190

Open neck 13 0.14 One layer of bandage 6! 0.29

~ -lE Alteration in frequency.

Z R Acoustio Theory

ohms

3 1.4 8.5 -

10 -13 -13 -15 - H 7.3 ... ]j

'28 -28 -

22 0.08 44 -52 ... 52 -80 - JIi 80 approx. -

15 0.9 33 . -

27 0.08 57 -

- 16 -

TABLE 11

ABSORPTION COEFFICIENT

Qaloulated from'simple matohing theor~ (para. 3.2)

,,'1i '" ....• 168Z

(42 '+ Z )2

e cA Q(m

Resonator caloulated measured (Millington Formula)

A (1400/s) open neck 25% 14~

11 " I! one layer bandage 5850 53~,

B (75 c/s) .pen ne ok 85% 56%

11 tI " one layer bandage 100% 60%

" " " two layers bandage 9750 6301, le

I 11

11 " glass wool fabric 90% 53%

TABLE III

DErAILS OF RESONATORS

Volume 2530 ccs. A, B, C same ccncrete block, using different neck D of 1/IIOod, neck size as with B.

Resonat.r Calculated Neck size frequency (inches)

A 140 1 .. 5 x

B 76 0.5 x

C 105 1.0 x

D 75 0.5 x

r. = radius of neck d _ d. + 1.7 ro

1.5

0 .. 675

1.5

0.675

r. ems

1,9

0.64

1 .. 27

0.64

k = S conductivity of~rifice d

~ radiatien resistance == 0::

=

resistance of neck area of cr3SS sectien of neck actual length of neck

d. ems

3.91

1.59

3.81

1.59

Transfermer ratio = area surrounding neck S

sizes.

d S ems cms2

7.15 ll.4

2.67 1.25

5.97 5.08

2.67 1.25

k R Transformer cms bl b2 impedance

ratio

1.65 0.58 0.828 1.69

0.49 0.002 0.077 147

0.85 0.49 0.043 8.5

0.49 0.002 0.077 190 I-' ...;z

!";;<;Uf , 2.4-5-49 ; ----~

;,.. -is ~

'- -0

'" C '" ~ 0.

'" 0 ;..: ,- e L Cl- .<2 .~ .,. ~ -0

"' '" ~~: '- 0 •.

0 .<..: 0. u

~ '- ~

" ·0 ~ e c U 0 ~ 0 ~

L. ~ V ~ C> .. '" l: '-v 0

,,;;:. ~

" n.

" " ~ f L. V ., " :> ,. .~

Q. '" '0 -£ "---' "' 0

: L.. E '" ""CO >- ..c: cc C- c .~

.~ ..c: ::: ." 0 v.

'" 'u <: -0 C '" .0 :tl >-. 0

'" ~ ~

" .c 0 '- ~

..c: .. I- ... r. 0.. E

0-4 0-5 0-55. 0-6

FIG. I THEORETICAL CURVES - VARIATION OF REVERBERATION TIME

WITH NECK RESISTANCE .

INVESTIGATION OF HELMHOLTZ RESONATORS.

BSHEETS. No.1

o .,. ::. -o ()

This drawing/specification is the property of the British Broadcasting Corporation and may not be reproduced or disclosed to a third party in any !orm without the written i-'er­mission of theCorpe,ration,

FREQUENCY eis,

._----------'----'---

CJI o o

----------------------------------------------------------

--.-----r--T-"-;s-d-r-aw-j-n-g-s-p-e-c-jf1-'-ca-'-'-o-n-, -i-s-t-h-e-';~:~~~;-l- -'-"~.~------------------------- -r

~ CIJ tt>e British Broadca;ting Cor~crat!on and ,nl))'. r J: (DO not be reprOfdUCe<l (l"h d'SCi:.lS"d t,',' "1"1" •. , ! '-"'''''-'-'.''"---' _

___ pany In a"y orrn. "N:t. '.)llt trH2- ' .... r,rten ?{:r· ;

\J missi!")O of the (orporatI0r., ~ _-I. _____ ~ ____ - __ ---_-__ -_-,~,-_-_-__ ,_.J.. .. ~- ... ..,. -----

-Z < fT1

~ G)

:o~ fTl_ UlQ Oz z ~O 0" :uJ: (JlfTl

r ~ J: o et N

{Jl

II>OJ I.

gO (f.~ z­o <>'

" -G)

tu

-4

ti:i i

-bL Ui (,

IV o +' o

FREQUENCY c/s

(J1

o ID o

::i o

Vl VI C m

zp1 0 1

N ." b v' (!\ " cc <D is f\) G '1; ..., " ,0 0 c' c 0 0 0 ,i 0 0 8 c (; 0 0 0 0

8 '.

;:: r'Pq(Jer;f:)' in cvcle 5 per oecond

-------------_.

B BC IINV;-~T~~ATION OF HELMHOLTZ -r;s-~p1

RESONATORS. ~--'.---~- -----------.- -"-" ._._----_._-------

~ (,1 (j) .:-J g 0 "0 0

0 0 0 Cl 0 Cl 0

(0 !Co '0 00 0 00 0 o~ v

I\l 0 0 0 0

DEPT

(.f) l/)

C m

'---1 RESEARCH ~G-'R~'N-.~~.-----~R~E~P~O~R~T~---------

CH'D. B.041 AP'D ~ SHEETS. No. 4

This Jr'Jwlng <;pec:fI(Jt'on I$. th~ property of the 8r 1tis[1

er ?Jdca~t;ng Corporatiun .1nJ mat ;'ot be 'reproduced Of disclosed to a thIrd party If! allY form wtthOut tht"

wrlftefl perq1t~\I(jn ot {he CorporatIon

I\J o Ui '

o V1 o

<l'I o

N o o

V1 o o

(Jl '~ Cl c· 0 0 o Q 0

Freque,,<:y it, cycles per second

FIG.6.

0~

0 0 Q 0 0 0

REVERBERATION TIME CURVES FOR 75 c /s RESONATORS.

: I ! ~1: IINVESTIGAT.I_O_N __ O_F_H __ E_L_M_H __ O_LT_Z_R~E_S_O_N_A1i_O~R: OP'N. CIl'k'hI

CH'D.

" <0 0 '0 0 '00 0 C 00 -:) 00

RESEA~CH DEPT --,-----_. REPORT

B.041 8 SHEETS· No 5.

·1 I This ,jrawing/sp~(lf!cJtjon IS "the p!"'operty \;if the BI iti)h , ..

Broadcasting Corporation .l"cl ""'1 not be f~_p_r_O_d_'_I\_"_:d_'_l __ -_-·- ---- ----- ----- --- --------- _______ 1_ or dt~dosed t01.a I,hird paf'ty !n .lily !Ofill without ~hl~

wrlteen permission of the Cor·poral!on.

Fre'{uef7ry In cvc1es per second

r . I . . T---------------RESEA-RCH DEPT

lBB~IINVESTIGATION OF HELMHOLTZ RESONATORS. ~~¥+<W1-----RB~g4i I DS 41 P l~-:-t~l BSHEETS.No6_

FIG. 10 TWO FORMS OF RESONATOR & WOODEN BUSH.

FIG.9. ARRANGEMENT USE.D TO OBTAIN RESPONSE CURVES.

BBC INVESTIGATION OF HELMHOLTZ RESONATORS. I----+-~

REPORT

·a041 8 SHEETS. No. 7.

ISSUE . I

2.4-8-4~. ~ ______ ~N=E=C=K~ ________ ~~'C~A~V~IT~Y~

I I

A:: AREA SURROUNDII':"G

~ ASSOCIATED WITH EACH NECK.

S· CROSS SECTION OF

NECK ONLY.

~ 5

T

LZ

PLANE WAVE

~r--I----III L C

(a.)

c·

E L= LI~L'2. ~---------------~ R 0 1<1 -;- I< Z.

Eo

(b)

A

R RESONATOR.

L

C

K

z

FIG. 11 DIAGRAMS ILLUSTRATING APPENDIX (8)

@)

BBC INVESTIGATION OF HELMHOLTZ. ~~ RESONATORS.

REPORT

B.041 DS/1/~ 85 I-IEETS. NoS.