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A note on the interaction between a Helmholtz resonator and an acoustic mode of an enclosure
Journal of Sound and Vibration (1980) 72(3), 365-378
A NOTE
ON THE
RESONATOR
INTERACTION
AND AN ACOUSTIC
BETWEEN
A HELMHOLTZ
MODE OF AN ENCLOSURE
F. J. FAHY AND C. SCHOFIELD? Institute of Sound and Vibration Research,
University of Southampton, Southampton SO9 SNH. England
(Received 18 August 1979, and in revised ,form 14 April 1980)
A single Helmholtz resonator is coupled to an enclosure and tuned to the natural frequency of one of its low order acoustic modes. The effect on the free, and forced, vibrations of the fluid in the enclosure is analyzed. The conditions necessary for the resonator to increase the damping of the two resultant modes, and to control the room response to excitation at frequencies within the range embracing both natural frequencies, are investigated. A simple design graph is presented. 1. INTRODUCTION The motivation for the analysis presented in this paper was the apparent lack of agreement between the authors of various books and papers on the effect of coupling a Helmholtz resonator to a larger enclosure, such as a room. Indeed, the authors of many elementary text books completely neglect to discuss the influence of the system parameters on the resonator performance, and thereby give the impression that resonators are, variously, powerful means of increasing room absorption, of increasing reverberation time, and of suppressing standing waves excited at resonance by sources in the room. The classical analysis of the behaviour of a bafhed resonator coupled to a half-space (see, e.g., reference [l]) is not relevant to the low frequency room problem because the radiation resistance of the resonator mouth, which features prominently in the optimization equation, is strongly affected by room resonance. and by the location of the resonator mouth within the enclosed space. It is assumed herein that a resonator would normally be employed in an attempt to modify room behaviour in the “low” frequency range, in which individual room modes and resonance frequencies are clearly distinguishable: the average separation between resonance frequencies is assumed significantly to exceed the average modal bandwidth. Hence the general multi-mode problem is reduced to one of coupling between two oscillators: one is a room mode, the other the resonator mode. In this sense it is analogous to the problem of the dynamic absorber used on mechanical structures [2], but the coupling is gyroscopic, and not elastic as in the mechanical case. In 1960, van Leeuwen [3] presented an analysis of the interaction problem treated herein, for which purpose he applied an electrical circuit analogy. Although some of the results of the present analysis are in agreement with those of van Leeuwen, the measured reduction in amplitude of a room mode effected by the introduction of a very lightly damped resonator tuned to the room mode frequency is far greater than his chart would predict. However this apparent discrepancy may arise from the ambiguity of his term t The
work was partly
performed
by this author
as a third year honours
project
in Acoustical
Engineering
under the supervision of the first author. 365 0022-460X/80/190365
+ 14 $02.00/O
@ 1980 Academic
Press Inc. (London)
Limited
366
F. J. FAHY
AND
C. SCHOFIELD
“amplitude reduction”; he does not indicate whether this refers to the uncoupled or coupled mode frequencies. In the present paper the effects at both coupled and uncoupled mode frequencies are indicated. More recently Britz and Pollard [4] presented a rather abstract study of the general coupled oscillation problem. The present analysis, which was in progress when reference [4] appeared, can be considered to be a practical example of the general case. The analysis is not claimed to be novel, but it is hoped that the very simple, explicit forms of the results will be of value to practitioners. 2. FREE VIBRATION
The equation which governs the behaviour of the fluid in the enclosure is v2* -(l/C2)fJ
= s(r, t),
(1)
where I&is the acoustic velocity potential, q is the volume velocity source strength density distribution within the volume and on its surface, and indicates differentiation with respect to time (there is usually no need, in what follows, to distinguish between partial and “total” time differentiation). The resonator equation is pof’S~+SRi~+(poC*S*/VR)5=-P(rR)S,
(2)
where 4 is the particle velocity out of the neck, I’is the effective length of the neck, which accounts for the reactive field local to the mouth, Ri is the resistance of the resonator excluding the external radiation resistance, V, is the resonator volume and P(rR) is the acoustic pressure at the mouth of the resonator. Also, ~(r, t) = -p&r,
t).
(3)
In free vibration the volume velocity directed out of the resonator mouth constitutes an effective source in the room, q = &S(r -rg), provided that the mouth perimeter is very small compared with an acoustic wavelength. This equivalence provides the roomresonator coupling mechanism. Equation (1) can be expressed in terms of a series expansion of 1/1into the room modes $N: f/h, t) = EN ~N(fhbN(r). These modes, rewctively, satisfy the equation (C2V2*~
-AL+!&)?P~J
=O,
(4)
subject to the room boundary conditions; AN is the complex natural frequency of mode N. For convenience of analysis it is assumed that the room boundary conditions are not strongly frequency dependent (except at the mouth of the resonator), and that the normalized impedance of the walls is much greater than unity, so that the modes $N may be assumed to form an orthogonal set of real functions [5]. Where lightly damped modes exist in a room, these assumptions are likely to be valid. Hence one may isolate each single room mode (N) in the usual manner; VAN(ti;,-h:ly,)=-c*iSj
$N(r)ti(r-rR)
dV= -c2S$N(r,)&
(5)
V
where V is the room volume and AN is the mode normalization factor, given by AN = V-’ j f&r) d V. It is convenient at this stage to replace the complex frequency AN by an eqUiVaht red frequency ON and an equivalent ad hoc damping coefficient cNt where hN = hN - CN. Using equations (2), (3) and (5) gives, for the coupled equations of the room mode and the
HELMHOLTZ
RESONATOR
AND
367
AN ENCLOSURE
resonator,
where u’, = c2S/l’Vr is the undamped natural frequency of the resonator. Note that the resonator is coupled to all the acoustic modes of the room, not just to the mode N under consideration, through the term on the right-hand side of equation (7). It is now assumed that the resonator frequency oR is made equal, by design, to the natural frequency ON of the particular room modes under consideration; further, the coupling to all other room modes is assumed to be negligible by virtue of their remoteness in frequency from @R; a minimum of two bandwidths’ separation would seem to be adequate. The validity of this assumption is open to experimental verification. Equations (6) and (7) then become ~N+~CN~N+~~~N=-(C’S~NI~R)/VAN)~,
(6)
5+R5’+wf5=(1/I’)~NSlr?J(rR),
(8)
in which w, = OR = ON, and R = Ri/pol’. the two complex natural frequencies of the coupled system can now be sought by letting pN(f) = @N e*’ and t(r) = ,$e*‘. Substitution of these expressions into equations (6) and (8) yields the frequency equation (A2+2cNA +wE)(A’+RA
+o~)=-h2~,~IL2N(rR)VR/ANV
(9)
It is of interest first to find the coupled natural frequencies in the absence of dissipative mechanisms: i.e., R = cN = 0. 3. FREE VIBRATION 3.1. UNDAMPED
FREE VIBRATION
When there is no damping, the frequency equation (9) reduces to (A*+w~)~+A*w~E~=~, where e2 = 4&(rR) VR/A,V; E is termed the coupling parameter system behaviour. The solutions to equation (10) are A*=-w:[1+(~*/2)(1*(4/&1)~‘*)].
(10) and is central to the
(11)
Now 0 < &, < 1, and $> AN > f, the smaller limit corresponding to a “three-dimensional” mode and the larger to a “one-dimensional” mode (oblique and axial modes of rectangular rooms). Since VR c A i and V b A:, where A0= 27rc/w,, for the models to be valid, then E c 1. Hence equation (12) can be written to a first approximation as
or Ai= *iw,(l-~./2)
= fiwi,
A2==fiw,(l+c/2)=rti~2.
(12a, b)
Since E is real and positive, one natural frequency of the coupled system is greater than w,, and one is less, as expected. The choice of positive or negative signs in expressions (12a) and (12b) is purely a formality. The separation between natural frequencies is given by
368
F. J. FAHY
AND
C. SCHOFIELD
Aw 2: &cooand is dependent upon the ratio of resonator to room volume, for a given resonator position and mode. Clearly the influence of the resonator is greatest when it is situated at a modal pressure anti-node, and is greater for three-dimensional modes than for one-dimensional modes. The relative magnitudes and phases of the room mode and resonator co-ordinates !PN and 6 in each coupled mode can be found by substituting the solutions PN = ?&,,,1 eA1’, @NzeAzLand & eA1’,& eh2’ into either of equations (6) and (S), excluding the damping terms: e.g., (w: = w:)& = iwrPN&&)lI’,
or
&/!f+N, =i[l/c
+$](&&)/w,l’).
(13, 14)
Similarly hl+~~ = -iDl.s
-41Wdrd/d).
(15)
It is interesting, and initially surprising, to note that i/ @ is only weakly dependent upon resonator position because E cc $N(rR) and normally l/e of. No coupling occurs when the resonator is at an acoustic mode pressure node: i.e. lJIN(rR)= 0. Equations (14) and (15) indicate that
lid~&m
=-1.
(16)
This result is physically reasonable because the undamped natural frequency of the resonator lies approximately mid-way between the coupled mode natural frequencies and hence, relative to a given pressure at the mouth, p = -po@&&), the phases of undamped resonator response ,$will be zero and V, and the amplitudes will be similar. This result also suggests that the ratio of energy dissipated by the resonator to that dissipated by the room will be similar in both modes: i.e., the modal damping ratios will be similar. 3.2. DAMPED FREE VIBRATION The general quartic frequency equation (9) has, when damping is present, exact solutions of very complex algebraic form [6] for which it has been found difficult to derive simple expressions. It was therefore found more convenient to derive the approximate damping factors of the coupled modes by an indirect method. The loss factor n of an oscillator may be defined by the steady state equation G/at
= -7p&
+ 97 = 0,
(17)
in which ,?? is the time averaged energy of an oscillator of natural frequency wg which is subject to a power input 7r. The loss factor is related to the Q-factor, the damping ratio [ and the reverberation time T of the oscillator as follows: n=Q-l
= 25 = 6/woT loglo e = 13.8/woT.
Free, damped oscillation is assumed to occur at one of the coupled, undamped natural frequencies, with equation (17) holding with r = 0. The short-time-averaged energies of the room and resonator respectively are then LXn = &0v&?I%V1*/2&
A??,,,= o:p01’s]~]2/2.
(18119)
The rates of energy dissipation are a/at(E,,,,)
= -2 ch&ovAnr/~Br12/2c2,
a/a@‘,,,> = -o:R$)i]*/2.
(20321) Now, from equation (17),
HELMHOLTZ
RESONATOR
AND
AN
ENCLOSURE
369
Substitution in equation (22) of the expressions in equations (18)-(21), together with elimination of the ratio \&\/\!@~~jby using equation (14), yields
It is now convenient to express the result in terms of the Q-factors of the room, QN, the resonator, QR, and the coupled mode, Q1, by using the relationships OR = wcp,l’/Ri and QN = 42~3,. Note that QR is not the Q-factor of the uncoupled resonator because it does not account for radiation losses. One thus obtains
QIIQN=
32 - &)2
32+4*
l+(l-P)(QN/QR)’
Q2’QN’~l+(l-~)(QN/QR)’
(%a, b)
Since, normally, &CC1,
Ql= Q2 = ~QNQRI(QN +
OR).
(25)
The following special cases are of interest.
(i) QR <
(26)
(ii) QR = ON: the resonator damping ratio equals the room mode damping ratio. Then Q~=Qz=QN=QR.
(27)
(iii) QR D Q: the resonator damping ratio is much less than the room mode damping ratio. Then Q~==Qz=~QN.
(28)
The general results are presented in graphical form in Figure 1. These results suggest that the room mode decay rates may be increased by introducing a well coupled, highly damped resonator, but that the decay rates can at most be halved by introducing a very lightly damped resonator. If a lightly damped resonator, having a natural frequency not very close to an uncoupled room mode frequency, is introduced into a room, it is likely that the extra coupled mode introduced will exhibit a decay rate very close to that corresponding to the resonator Q factor, but it is likely that such a mode will involve very little room energy, and hence will not be objectively, or subjectively, important.
4. FORCED
VIBRATION
4.1. MODAL RESPONSE As an example of forced response, it is assumed that a simple harmonic point source in the room, of strength &(r - I,) e’“‘, excites the coupled system into steady state vibration. Equations (6) and (7) then become (W~+2iWcN-W2)~N=-i(Wc2S61/N(rR)/V/1N)i-c2~~~(rS),
(~0;+iwR -w’)i=
i(o/l’) 1 +.&~(r~). M
(29) (30)
370
F. J. FAHY
AND
C:. SCHOFIELD
2
-$
--T,
=T2= TN
1.5 --T, = T2=I.2 T,, I --T, =T2=I.5T, 0.5
o’2v/‘I-T, 0 0.2
= T2= 2T” 0.5
I
I.5
2
I
CON Figure 1. Theoretical reverberation time ratios T,, T, = reverberation times at frequencies o1 and w2 with resonator. TN = reverberation time at frequency wN in absence of resonator.
Solving for @N yields [(~~N+~~wcN-w’)(w~R+~wR -w’)-w’c~S~~(~~)/VA~~‘]~~ - (02c2S,h(rR)/
VANI’) C
@.&A&)
= --~~&~(r~)(w~ +iwR -w2).
(31)
MfN
If now one neglects the presence of room modes other than N, on the basis that the average separation of room mode natural frequencies greatly exceeds the average modal bandwidth, and lets the uncoupled room mode and resonator frequencies coincide (wn = ON = wC), equation (31) becomes I PN = -c”&N(,,)(& +ioR -W”)/[(W: +2iWCN -W’)(CO~+iwR -U2)-W2&2]. (32) In the absence of the resonator, the room “response” at resonance is given by @‘K= ic2&N(rs)/2uNcN. Hence the room response may be normalized to remove the dependence upon source strength distribution, as follows: @Nf@k = -2iwNcN(w: +iwR -
W’)/[(~2
+2&N
-w’)(wf
+ioR --~*)--~*w~F~].
(33)
One can now consider the normalized response at three different forcing frequencies: (i) the original room mode frequency WN; (ii) and (iii), the new coupled mode undamped natural frequencies w1 and w2 (see equations (12)). For these three cases, the following results are obtained from equation (33): case(i)(@=w,=WjV=WR):
HELMHOLTZ
RESONATOR
AND
AN ENCLOSURE
371
case(ii)(w=wi=w,(l-c/2)): ~lir~/,l~~2~[1+~2Q~]/[l+(Q~+Q~)2~2];
(35)
case (iii) (0 = w2 = w,(l + e/2)): ~P2,/~o,~2=~!P~/v/~o,12.
(36)
It has been assumed that E CC1 in equations (35) and (36). Note that w1 and o2 are not exactly the frequencies of maximum modal response because the damped natural frequencies are different from the undamped natural frequencies, but for most practical purposes the differences are negligible. The results are oresented in generalized form in Figure 2.
Curve of maximum power absorption at wN
-,
Figure 2. Theoretical differences between SPLs at o N, w,, and ol and SPL at wN in absence of resonator. SPL at oN; ----, SPL at o, and ~2.
4.2. POWER ABSORBED BY THE RESONATOR The acoustic power absorbed by the resonator is given formally by W = (-l/T)jtj,pJJ dt ds, where pR and U are the acoustic pressure and particle velocity distributions over the mouth of the resonator; U is directed cutwards from the neck into the room. From equation (3), the pressure phasor is p”= iwpOp&&). When the room is excited at frequency w = wN = wR = wCequation (34) gives @R =6~0/(1
+E~QNQR),
(37)
where fin0 is the a_mplitude of pressure at zR in the absence of the resonator. The velocity phasor is U = iwt; using equations (8) and (29) gives fi = drLN(rs)~~(r~)Q~Q~l~~~‘(~
+ QNQRE~)
(38)
or fi = -~RoQR/~owJ'(~ + QNQRE~) = -(QR/Po~‘>$R
= -Ri@Re
(39)
372
F. J. FAHY
AND
C. SCHOFIELD
Hence w=$
~~Ro~~ QR ds
I PockNI’(l+ QNQRE) ’ 22
S
’
(40)
where kn = wN/c.It is convenient to normalize this result on the incident power carried by a normally incident plane wave of amplitude &,. One thus obtains WS(~~~01~/2p,c) = I%‘= Qdkid')(l+ QNQRE~)*.
(41)
For a given value of QN and E, * is maximized when QNQ~c2 = 1; the maximum value is given by w,,,,, = p,,c/4Ri.
(42)
This value is completely different from that corresponding to a baffled resonator coupled to a half space, which is dependent on the acoustic wavelength at resonance [l]. It is of interest to note that equation (34) indicates a corresponding reduction in room SPL of 6 dB.
5. DISCUSSION OF THEORETICAL
RESULTS
5.1. FREE VIBRATION As previously stated, the coupling of a resonator to a room introduces an extra mode of vibration, and when the uncoupled resonator is tuned to the natural frequency of an uncoupled and isolated room mode the effect is to produce two coupled modes of which the natural frequencies lie on either side of the original common frequency. The degree of separation of natural frequencies is proportional to the magnitude of the coupling factor, which increases with the ratio of volumes of the resonator and room, and with the degree of proximity of the resonator mouth to an acoustic mode pressure anti-node, and is greatest for an oblique mode and least for an axial mode. The relative proportions of energy stored in the resonator and in the room mode are very similar in both coupled modes, but the relative phases of the modal pressure and resonator neck velocities are of opposite sign. Consequently the damping ratios and decay rates of the two modes are similar, and this fact, combined with the relatively small frequency separation, is likely to produce very non-exponential amplitude decays following the cessation of excitation by an arbitrary source of frequency in the region of either natural frequency, because beating will normally occur. Hence it is unlikely that reverberation time measurements would truly reveal the effect of introduction of a tuned resonator. The variation of modal decay rates with resonator and room parameters is shown in Figure 1. For convenience, equivalent reverberation time ratios have been presented. 5.2.
FORCED
VIBRATION
results of section 4 are presented graphically in Figure 2, which apparently differs significantly from van Leeuwen’s corresponding Figure (2). They indicate that, if excitation is confined to the frequency UN of the original uncoupled room mode, the maximum reduction for a given value of QN is obtained by maximizing E and QR ; i.e., by placing a very lightly damped resonator of large volume at a modal pressure anti-node (see equation (34)). This result implies that the free space radiation resistance of the resonator has, in this case, nothing to do with the resonator performance; hence Gilford’s conclusions [7] regarding the influence of room edges and corners on the quantity do not necessarily apply. But, if the excitation frequency is not confined to that fixed frequency, as in the case of The
HELMHOLTZ
RESONATOR
AND
AN
ENCL.OSURE
373
speed dependent excitation frequencies as in vehicles, or in the case of multiple harmonic, or aperiodic excitation, there may be significant excitation at the resonance frequencies of the coupled modes, w1 and o2 (see equations (35) and (36)),* in such cases a maximization of e2QR may produce resonant pressure amplitudes at WI and 0~2comparable with that of the original uncoupled room mode when excited at resonance. Hence some indication of the relative reductions of pressure amplitude caused by the presence of the resonator when the room is excited at the original frequency tiN, and at the coupled frequencies WI and ~02, is desirable. Using equations (34), (35) and (36) gives
By analogy with den Hartog’s optimized dynamic absorber [2], one may require the ratio to be unity, in which case &QN =~(EQR)~/(~-(EQR)*-_(FQR)~).
(44)
The line corresponding to this condition is termed the line of “constant reduction” and is indicated on Figure 2. This figure may be used to optimize the design of an absorber, once the room mode damping and type (oblique, tangential, axial) is known. In principle, considerable reductions of steady state sound pressure level can be obtained over the “broad band” range 01 to 02, but there are clearly physical limitations to be considered. For example, for an oblique room mode of QN = lo2 (T of about 2 s at 100 Hz), with a ratio of resonator volume to room volume of lo-’ (E* = 8 x 10p4), the value of the parameter (l/ QNE ) is approximately l-1 ; reference to Figure 2 shows that it is impossible to achieve a significant “broad band” SPL reduction, although a reduction of about 6 dB could be achieved at WN if (~/EQR) could be made very low (zO.9); this would require a value of QR of about 125, which is almost impossible to achieve in practice. (Note &2 = &(rR) vR/ANv.> With a larger resonator volume, having a typical linear dimension approximately one-tenth of a typical room dimension (V/ VR = 10”) then E = 9 x 10d2 and (l/QN&) = 0.1. In this case a desirable value of QT(would be about 11, which is practicable. The conclusion from these considerations is that only relatively large resonators can effect significant pure tone or “broad band” reductions of SPL in an enclosure. 5.3.
POWER
ABSORPTION
BY THE RESONATOR
At first sight it would appear that the conditions for maximum power absorption by the resonator at the original mode frequency UN would correspond to maximum SPL reduction at that frequency. This is not however true because the effect of the presence of the resonator in creating two new coupled modes with natural frequencies different from UN is to “unload” the volume velocity source so that it actually injects less power into the enclosure. The maximum absorbed power line is indicated on Figure 2 and corresponds to the line of 6 dB reduction at oN.
6. EXPERIMENTS Measurements of SPL reduction and absorbed power were conducted in a small rectangular room of dimensions 2.10 x 2.52 x 2.50 m (V = 13.3 m3). A resonator, having a 306 mm internal diameter cylindrical body with a depth variable within a range O-100 mm (volume range O-8 x low3 m3), and a neck of 102 mm internal diameter and length of 150 mm, was introduced into the room. Two Briiel and Kjaer Type 4135 L’ microphones were placed in the neck at a separation of 100 mm. A 20 cm diameter loudspeaker was placed in one corner of the room.
374
F. J. FAHY
AND
C. SCHOFIELD
The frequency response of the room, as measured by microphones placed in various positions, was used to indicate the frequencies and types of acoustic modes; a typical trace is shown in Figure 3(a). The (l,l,l) mode at 130 Hz was chosen as being sufficiently well isolated from its neighbours at 110 and 138 Hz. The air temperature was measured to be very stable, variations from day to day corresponding to a range of kO.3 Hz. The decay rate of this mode was measured at various microphone positions, by using a level recorder; the curves were linear and reproducible. The corresponding reverberation time was 0.59 s (ON = 34.6).
(a) __~_~_.___._
Frequency (Hz)
Frequency MI)
Frequency biz)
Frequency (Hz)
,60
Figure 3. Effect of resonator on room response. (a) No resonator; (b) empty necked resonator at position B (low value of (e&-l); (c) damped resonator at position B (line A); (d) damped resonator at position C (high value of (EQR)-‘).
The resonator had to be tuned to the same frequency as the room mode under anechoic conditions, because tuning when coupled to a room is naturally impossible. The bandwidth of the resonator resonance was measured under the same conditions to yield the Q-factor. This could be set at the different values by means of introducing one or two sheets of loudspeaker grille cloth (r = 11 mks Rayls) at the base of the neck. The corresponding values of OR were 75, 16 and 9.7. With the resonator at various positions within the room including one at a modal pressure node, measurements were made of SPL reductions at the original mode natural frequency and at the frequencies of the two new coupled modes. The corresponding values are presented in Table 1. The soun_d of the coupling parameter E, (eQx)-’ and (e&-l intensity in the neck was estimated from the expression I = -PI$z sin c5/4hpow, where PI and p2 are the presssure amplitudes at the microphones in the neck, 2h is the microphone
HELMHOLTZ
RESONATOR
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AN ENCLOSURE
375
1
TABLE
Coupling parameter and Q-factor values Resonator position and condition Empty
1
(EQR)~
F
(EQNI-’
neck
A B C D
0.058 0.048 0.028 0
0.23 11.28 1J.48 cc
0.50 0.61 1.10 cc
One piece of material in neck
A B C D
0.058 OsQ47 0.027 0
1.10 1.33 2.30 co
0.51 0.62 1.10 co
Two pieces of material in neck
A B C D
0.056
l-85
0.52
0.046 0.027 0
2.24 3.88 00
0.63 1.10 co
(I+cwnc
22
)
Figure 4. Theoretical and measured normalized intensities in the resonator neck. -, Theoretical prediction; 13, experimental measurements (number of pieces of material in neck, resonator position).
376
F. J. FAHY
AND
C. SCHOFIELD
separation, and 9 is the phase angle between the two pressures [8]; the uniformity of intensity over the cross section of the neck was verified, the frequency being far less than the lowest cut-off frequency of the neck. Typical pressure response traces are presented in Figure 4, and a table of measured and calculated frequencies and SPL reductions is presented in Table 2. An attempt was made to measure the decay ratio of the coupled modes for comparison with the theoretical results presented in Figure 1. However, as expected, the beating between the coupled modes precluded accurate measurement, and the application of sophisticated discrimination techniques was not possible within the time scale of the project. The comparisons in Table 2 suggest that the present theory adequately indicates the effect of system parameters on resonator performance. The agreement between measured and calculated absorbed power is good, but somewhat fortuitous, because the likely error is as much as f 1 dB (*25%), in spite of the use of the microphone reversal technique to reduce errors due to phase mismatch.
7. CONCLUSIONS
When a resonator, tuned to have the same natural frequency as an acoustic mode of an enclosure, is coupled to the enclosure, two new coupled modes are produced of which the natural frequencies normally lie on either side of the original frequency. The separation between the new frequencies is proportional to the value of a coupling parameter, which increases with the ratio of enclosure volume to resonator volume, and with the modal pressure amplitude at the position of the resonator, and is largest for oblique modes and smaller for axial modes. The decay rates of the coupled modes are very similar for practicable values of the coupling parameter. Because of the closeness of the coupled mode frequencies the decay process is likely to exhibit the beating phenomenon and therefore will be difficult to measure by conventional means. In principle the reverberation time of each coupled mode can be reduced indefinitely by the presence of a weakly coupled, highly damped resonator. This has yet to be demonstrated experimentally. On the contrary the presence of a well coupled, lightly damped resonator can no more than double the original reverberation time. Optimization of a resonator design depends upon a knowledge of the type of acoustic enclosure mode involved and the decay rate, reverberation time, or Q-factor of that mode. A choice has to be made between (a) producing a large reduction in SPL at the resonance frequency of the original mode, which involves a risk of greatly increasing the SPL at the coupled mode resonance frequencies; or (b) producing a modest and approximately equal reduction of SPL at both coupled and original frequencies. The former approach is only suitable when it is known that the excitation frequency is closely confined to the original mode frequency. The latter strategy is better suited to excitation by variable frequency (or speed), aperiodic or multiple harmonic excitation which may excite the coupled modes at resonance. It has been shown that the condition of maximum power absorption by the resonator at the original mode frequency does not necessarily correspond to maximum effectiveness in terms of SPL reduction at these frequencies because the shift of natural frequencies “unloads” the acoustic source. The power absorption coefficient does not depend upon the matching of the radiation and internal resistances, as in the case of the resonator coupled to free space, but is simply inversely proportional to the internal resistance of the resonator. This is because the resonator acts like a secondary source in the room and the “re-radiated” and “incident” pressure fields are not independent.
neck
A B C D
: D
A
D
C
B
A
127.0 (126.0) 127.0 (126.5) 127.5 (127.5) 129.5 (129.5)
w2
Measured natural
133.0 (133.0) 133.0 (133.0) 132.0 (132.5) 129.5 (129.5)
WI
(theoretical) frequencies
t It is not clear whether this figure applies to oN or w1 and 0~2.
Two pieces of material in neck
One piece of material in neck
Empty
Resonator position and condition
TABLE
2
(i)
(Z)
.
(z:;)
(Z)
*
wz-6J1
modal (Hz)
5.0 4.5 2.7 -
6.7 57 3.0 -
-
6.8
11.0
11.6
Measured
6.2 4.7 1.9 -
8.9 6.9 3.0 -
-
9.5
16.7
19.7
Theoretical
SPL reduction --
6 5 4 -
7 6 4 -
-
4t
4t
4t
Reference
at ON (dB)
Coupled mode natural frequencies and SPL reductions
[3]
5.5 5.3 3.6 -
5.8 6.1 4.5 -
-
4.2
4.3
3.8
Measured WI
4.5 4.5 3,2 -
5.2 5.0 4.2 -
-
4.5
5.3
6.7
w2
with no resonator)
w?
7.4 6.4 3.5 -
7-o 6.2 3.9 -
-
2.9
3.0
3.1
WI,
Theoretical
(dB)
(SPK at mq and WA-(SPL at
wN
z
3 s 2
5 m
g
$
z
?I 5
8
ro
z
z F
378
F. J. FAHY AND C. SCHOFIELD
It is believed that the present theory and design chart usefully complement those of van Leeuwen because the physics of the model is more explicit and the experimental results are in close agreement with the former. In principle the same approach could be used to analyze the coupling of a panel or membrane absorber to an enclosure, when itis likely that the same general forms of behaviour and design criteria will obtain. ACKNOWLEDGMENT The advice of Professor P. 0. A. L. Davies regarding the design of the resonator, and the technical assistance of Mr D. C. Howell are gratefully acknowledged.
REFERENCES 1. C. ZWIKKER and C. W. KOSTEN 1949 Sound Absorbing Materials. New York: Elsevier. 2. J. P. DEN HARTOG 1956 Mechanical Vibration. New York: McGraw-Hill, fourth edition. 3. F. J. VANLEEUWEN 1960 European Broadcasting Union Review, A-Technical 62, 155-161. The damping of eigen-tones in small rooms by Helmholtz resonators. 4. H. R. BRITZ and H. F. POLLARD 1978 Journal of Sound and Vibration 60, 305-307.
Computational analysis of coupled resonators. 5. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill. 6. T. PEARSON 1974 Handbook of Applied Mathematics. New York: Van Nostrand Reinhold. 7. C. GILFORD 1972 Acoustics of Radio and Television Studies, I.E.E. Monograph: Series II.
London: Peter Peregrinus. 8. R. J. SAGE 1975 B.Sc. (Honours) Dissertation, University of Southampton. An investigation of the two-microphone method of acoustic intensity measurement.
A NOTE
ON THE
RESONATOR
INTERACTION
AND AN ACOUSTIC
BETWEEN
A HELMHOLTZ
MODE OF AN ENCLOSURE
F. J. FAHY AND C. SCHOFIELD? Institute of Sound and Vibration Research,
University of Southampton, Southampton SO9 SNH. England
(Received 18 August 1979, and in revised ,form 14 April 1980)
A single Helmholtz resonator is coupled to an enclosure and tuned to the natural frequency of one of its low order acoustic modes. The effect on the free, and forced, vibrations of the fluid in the enclosure is analyzed. The conditions necessary for the resonator to increase the damping of the two resultant modes, and to control the room response to excitation at frequencies within the range embracing both natural frequencies, are investigated. A simple design graph is presented. 1. INTRODUCTION The motivation for the analysis presented in this paper was the apparent lack of agreement between the authors of various books and papers on the effect of coupling a Helmholtz resonator to a larger enclosure, such as a room. Indeed, the authors of many elementary text books completely neglect to discuss the influence of the system parameters on the resonator performance, and thereby give the impression that resonators are, variously, powerful means of increasing room absorption, of increasing reverberation time, and of suppressing standing waves excited at resonance by sources in the room. The classical analysis of the behaviour of a bafhed resonator coupled to a half-space (see, e.g., reference [l]) is not relevant to the low frequency room problem because the radiation resistance of the resonator mouth, which features prominently in the optimization equation, is strongly affected by room resonance. and by the location of the resonator mouth within the enclosed space. It is assumed herein that a resonator would normally be employed in an attempt to modify room behaviour in the “low” frequency range, in which individual room modes and resonance frequencies are clearly distinguishable: the average separation between resonance frequencies is assumed significantly to exceed the average modal bandwidth. Hence the general multi-mode problem is reduced to one of coupling between two oscillators: one is a room mode, the other the resonator mode. In this sense it is analogous to the problem of the dynamic absorber used on mechanical structures [2], but the coupling is gyroscopic, and not elastic as in the mechanical case. In 1960, van Leeuwen [3] presented an analysis of the interaction problem treated herein, for which purpose he applied an electrical circuit analogy. Although some of the results of the present analysis are in agreement with those of van Leeuwen, the measured reduction in amplitude of a room mode effected by the introduction of a very lightly damped resonator tuned to the room mode frequency is far greater than his chart would predict. However this apparent discrepancy may arise from the ambiguity of his term t The
work was partly
performed
by this author
as a third year honours
project
in Acoustical
Engineering
under the supervision of the first author. 365 0022-460X/80/190365
+ 14 $02.00/O
@ 1980 Academic
Press Inc. (London)
Limited
366
F. J. FAHY
AND
C. SCHOFIELD
“amplitude reduction”; he does not indicate whether this refers to the uncoupled or coupled mode frequencies. In the present paper the effects at both coupled and uncoupled mode frequencies are indicated. More recently Britz and Pollard [4] presented a rather abstract study of the general coupled oscillation problem. The present analysis, which was in progress when reference [4] appeared, can be considered to be a practical example of the general case. The analysis is not claimed to be novel, but it is hoped that the very simple, explicit forms of the results will be of value to practitioners. 2. FREE VIBRATION
The equation which governs the behaviour of the fluid in the enclosure is v2* -(l/C2)fJ
= s(r, t),
(1)
where I&is the acoustic velocity potential, q is the volume velocity source strength density distribution within the volume and on its surface, and indicates differentiation with respect to time (there is usually no need, in what follows, to distinguish between partial and “total” time differentiation). The resonator equation is pof’S~+SRi~+(poC*S*/VR)5=-P(rR)S,
(2)
where 4 is the particle velocity out of the neck, I’is the effective length of the neck, which accounts for the reactive field local to the mouth, Ri is the resistance of the resonator excluding the external radiation resistance, V, is the resonator volume and P(rR) is the acoustic pressure at the mouth of the resonator. Also, ~(r, t) = -p&r,
t).
(3)
In free vibration the volume velocity directed out of the resonator mouth constitutes an effective source in the room, q = &S(r -rg), provided that the mouth perimeter is very small compared with an acoustic wavelength. This equivalence provides the roomresonator coupling mechanism. Equation (1) can be expressed in terms of a series expansion of 1/1into the room modes $N: f/h, t) = EN ~N(fhbN(r). These modes, rewctively, satisfy the equation (C2V2*~
-AL+!&)?P~J
=O,
(4)
subject to the room boundary conditions; AN is the complex natural frequency of mode N. For convenience of analysis it is assumed that the room boundary conditions are not strongly frequency dependent (except at the mouth of the resonator), and that the normalized impedance of the walls is much greater than unity, so that the modes $N may be assumed to form an orthogonal set of real functions [5]. Where lightly damped modes exist in a room, these assumptions are likely to be valid. Hence one may isolate each single room mode (N) in the usual manner; VAN(ti;,-h:ly,)=-c*iSj
$N(r)ti(r-rR)
dV= -c2S$N(r,)&
(5)
V
where V is the room volume and AN is the mode normalization factor, given by AN = V-’ j f&r) d V. It is convenient at this stage to replace the complex frequency AN by an eqUiVaht red frequency ON and an equivalent ad hoc damping coefficient cNt where hN = hN - CN. Using equations (2), (3) and (5) gives, for the coupled equations of the room mode and the
HELMHOLTZ
RESONATOR
AND
367
AN ENCLOSURE
resonator,
where u’, = c2S/l’Vr is the undamped natural frequency of the resonator. Note that the resonator is coupled to all the acoustic modes of the room, not just to the mode N under consideration, through the term on the right-hand side of equation (7). It is now assumed that the resonator frequency oR is made equal, by design, to the natural frequency ON of the particular room modes under consideration; further, the coupling to all other room modes is assumed to be negligible by virtue of their remoteness in frequency from @R; a minimum of two bandwidths’ separation would seem to be adequate. The validity of this assumption is open to experimental verification. Equations (6) and (7) then become ~N+~CN~N+~~~N=-(C’S~NI~R)/VAN)~,
(6)
5+R5’+wf5=(1/I’)~NSlr?J(rR),
(8)
in which w, = OR = ON, and R = Ri/pol’. the two complex natural frequencies of the coupled system can now be sought by letting pN(f) = @N e*’ and t(r) = ,$e*‘. Substitution of these expressions into equations (6) and (8) yields the frequency equation (A2+2cNA +wE)(A’+RA
+o~)=-h2~,~IL2N(rR)VR/ANV
(9)
It is of interest first to find the coupled natural frequencies in the absence of dissipative mechanisms: i.e., R = cN = 0. 3. FREE VIBRATION 3.1. UNDAMPED
FREE VIBRATION
When there is no damping, the frequency equation (9) reduces to (A*+w~)~+A*w~E~=~, where e2 = 4&(rR) VR/A,V; E is termed the coupling parameter system behaviour. The solutions to equation (10) are A*=-w:[1+(~*/2)(1*(4/&1)~‘*)].
(10) and is central to the
(11)
Now 0 < &, < 1, and $> AN > f, the smaller limit corresponding to a “three-dimensional” mode and the larger to a “one-dimensional” mode (oblique and axial modes of rectangular rooms). Since VR c A i and V b A:, where A0= 27rc/w,, for the models to be valid, then E c 1. Hence equation (12) can be written to a first approximation as
or Ai= *iw,(l-~./2)
= fiwi,
A2==fiw,(l+c/2)=rti~2.
(12a, b)
Since E is real and positive, one natural frequency of the coupled system is greater than w,, and one is less, as expected. The choice of positive or negative signs in expressions (12a) and (12b) is purely a formality. The separation between natural frequencies is given by
368
F. J. FAHY
AND
C. SCHOFIELD
Aw 2: &cooand is dependent upon the ratio of resonator to room volume, for a given resonator position and mode. Clearly the influence of the resonator is greatest when it is situated at a modal pressure anti-node, and is greater for three-dimensional modes than for one-dimensional modes. The relative magnitudes and phases of the room mode and resonator co-ordinates !PN and 6 in each coupled mode can be found by substituting the solutions PN = ?&,,,1 eA1’, @NzeAzLand & eA1’,& eh2’ into either of equations (6) and (S), excluding the damping terms: e.g., (w: = w:)& = iwrPN&&)lI’,
or
&/!f+N, =i[l/c
+$](&&)/w,l’).
(13, 14)
Similarly hl+~~ = -iDl.s
-41Wdrd/d).
(15)
It is interesting, and initially surprising, to note that i/ @ is only weakly dependent upon resonator position because E cc $N(rR) and normally l/e of. No coupling occurs when the resonator is at an acoustic mode pressure node: i.e. lJIN(rR)= 0. Equations (14) and (15) indicate that
lid~&m
=-1.
(16)
This result is physically reasonable because the undamped natural frequency of the resonator lies approximately mid-way between the coupled mode natural frequencies and hence, relative to a given pressure at the mouth, p = -po@&&), the phases of undamped resonator response ,$will be zero and V, and the amplitudes will be similar. This result also suggests that the ratio of energy dissipated by the resonator to that dissipated by the room will be similar in both modes: i.e., the modal damping ratios will be similar. 3.2. DAMPED FREE VIBRATION The general quartic frequency equation (9) has, when damping is present, exact solutions of very complex algebraic form [6] for which it has been found difficult to derive simple expressions. It was therefore found more convenient to derive the approximate damping factors of the coupled modes by an indirect method. The loss factor n of an oscillator may be defined by the steady state equation G/at
= -7p&
+ 97 = 0,
(17)
in which ,?? is the time averaged energy of an oscillator of natural frequency wg which is subject to a power input 7r. The loss factor is related to the Q-factor, the damping ratio [ and the reverberation time T of the oscillator as follows: n=Q-l
= 25 = 6/woT loglo e = 13.8/woT.
Free, damped oscillation is assumed to occur at one of the coupled, undamped natural frequencies, with equation (17) holding with r = 0. The short-time-averaged energies of the room and resonator respectively are then LXn = &0v&?I%V1*/2&
A??,,,= o:p01’s]~]2/2.
(18119)
The rates of energy dissipation are a/at(E,,,,)
= -2 ch&ovAnr/~Br12/2c2,
a/a@‘,,,> = -o:R$)i]*/2.
(20321) Now, from equation (17),
HELMHOLTZ
RESONATOR
AND
AN
ENCLOSURE
369
Substitution in equation (22) of the expressions in equations (18)-(21), together with elimination of the ratio \&\/\!@~~jby using equation (14), yields
It is now convenient to express the result in terms of the Q-factors of the room, QN, the resonator, QR, and the coupled mode, Q1, by using the relationships OR = wcp,l’/Ri and QN = 42~3,. Note that QR is not the Q-factor of the uncoupled resonator because it does not account for radiation losses. One thus obtains
QIIQN=
32 - &)2
32+4*
l+(l-P)(QN/QR)’
Q2’QN’~l+(l-~)(QN/QR)’
(%a, b)
Since, normally, &CC1,
Ql= Q2 = ~QNQRI(QN +
OR).
(25)
The following special cases are of interest.
(i) QR <
(26)
(ii) QR = ON: the resonator damping ratio equals the room mode damping ratio. Then Q~=Qz=QN=QR.
(27)
(iii) QR D Q: the resonator damping ratio is much less than the room mode damping ratio. Then Q~==Qz=~QN.
(28)
The general results are presented in graphical form in Figure 1. These results suggest that the room mode decay rates may be increased by introducing a well coupled, highly damped resonator, but that the decay rates can at most be halved by introducing a very lightly damped resonator. If a lightly damped resonator, having a natural frequency not very close to an uncoupled room mode frequency, is introduced into a room, it is likely that the extra coupled mode introduced will exhibit a decay rate very close to that corresponding to the resonator Q factor, but it is likely that such a mode will involve very little room energy, and hence will not be objectively, or subjectively, important.
4. FORCED
VIBRATION
4.1. MODAL RESPONSE As an example of forced response, it is assumed that a simple harmonic point source in the room, of strength &(r - I,) e’“‘, excites the coupled system into steady state vibration. Equations (6) and (7) then become (W~+2iWcN-W2)~N=-i(Wc2S61/N(rR)/V/1N)i-c2~~~(rS),
(~0;+iwR -w’)i=
i(o/l’) 1 +.&~(r~). M
(29) (30)
370
F. J. FAHY
AND
C:. SCHOFIELD
2
-$
--T,
=T2= TN
1.5 --T, = T2=I.2 T,, I --T, =T2=I.5T, 0.5
o’2v/‘I-T, 0 0.2
= T2= 2T” 0.5
I
I.5
2
I
CON Figure 1. Theoretical reverberation time ratios T,, T, = reverberation times at frequencies o1 and w2 with resonator. TN = reverberation time at frequency wN in absence of resonator.
Solving for @N yields [(~~N+~~wcN-w’)(w~R+~wR -w’)-w’c~S~~(~~)/VA~~‘]~~ - (02c2S,h(rR)/
VANI’) C
@.&A&)
= --~~&~(r~)(w~ +iwR -w2).
(31)
MfN
If now one neglects the presence of room modes other than N, on the basis that the average separation of room mode natural frequencies greatly exceeds the average modal bandwidth, and lets the uncoupled room mode and resonator frequencies coincide (wn = ON = wC), equation (31) becomes I PN = -c”&N(,,)(& +ioR -W”)/[(W: +2iWCN -W’)(CO~+iwR -U2)-W2&2]. (32) In the absence of the resonator, the room “response” at resonance is given by @‘K= ic2&N(rs)/2uNcN. Hence the room response may be normalized to remove the dependence upon source strength distribution, as follows: @Nf@k = -2iwNcN(w: +iwR -
W’)/[(~2
+2&N
-w’)(wf
+ioR --~*)--~*w~F~].
(33)
One can now consider the normalized response at three different forcing frequencies: (i) the original room mode frequency WN; (ii) and (iii), the new coupled mode undamped natural frequencies w1 and w2 (see equations (12)). For these three cases, the following results are obtained from equation (33): case(i)(@=w,=WjV=WR):
HELMHOLTZ
RESONATOR
AND
AN ENCLOSURE
371
case(ii)(w=wi=w,(l-c/2)): ~lir~/,l~~2~[1+~2Q~]/[l+(Q~+Q~)2~2];
(35)
case (iii) (0 = w2 = w,(l + e/2)): ~P2,/~o,~2=~!P~/v/~o,12.
(36)
It has been assumed that E CC1 in equations (35) and (36). Note that w1 and o2 are not exactly the frequencies of maximum modal response because the damped natural frequencies are different from the undamped natural frequencies, but for most practical purposes the differences are negligible. The results are oresented in generalized form in Figure 2.
Curve of maximum power absorption at wN
-,
Figure 2. Theoretical differences between SPLs at o N, w,, and ol and SPL at wN in absence of resonator. SPL at oN; ----, SPL at o, and ~2.
4.2. POWER ABSORBED BY THE RESONATOR The acoustic power absorbed by the resonator is given formally by W = (-l/T)jtj,pJJ dt ds, where pR and U are the acoustic pressure and particle velocity distributions over the mouth of the resonator; U is directed cutwards from the neck into the room. From equation (3), the pressure phasor is p”= iwpOp&&). When the room is excited at frequency w = wN = wR = wCequation (34) gives @R =6~0/(1
+E~QNQR),
(37)
where fin0 is the a_mplitude of pressure at zR in the absence of the resonator. The velocity phasor is U = iwt; using equations (8) and (29) gives fi = drLN(rs)~~(r~)Q~Q~l~~~‘(~
+ QNQRE~)
(38)
or fi = -~RoQR/~owJ'(~ + QNQRE~) = -(QR/Po~‘>$R
= -Ri@Re
(39)
372
F. J. FAHY
AND
C. SCHOFIELD
Hence w=$
~~Ro~~ QR ds
I PockNI’(l+ QNQRE) ’ 22
S
’
(40)
where kn = wN/c.It is convenient to normalize this result on the incident power carried by a normally incident plane wave of amplitude &,. One thus obtains WS(~~~01~/2p,c) = I%‘= Qdkid')(l+ QNQRE~)*.
(41)
For a given value of QN and E, * is maximized when QNQ~c2 = 1; the maximum value is given by w,,,,, = p,,c/4Ri.
(42)
This value is completely different from that corresponding to a baffled resonator coupled to a half space, which is dependent on the acoustic wavelength at resonance [l]. It is of interest to note that equation (34) indicates a corresponding reduction in room SPL of 6 dB.
5. DISCUSSION OF THEORETICAL
RESULTS
5.1. FREE VIBRATION As previously stated, the coupling of a resonator to a room introduces an extra mode of vibration, and when the uncoupled resonator is tuned to the natural frequency of an uncoupled and isolated room mode the effect is to produce two coupled modes of which the natural frequencies lie on either side of the original common frequency. The degree of separation of natural frequencies is proportional to the magnitude of the coupling factor, which increases with the ratio of volumes of the resonator and room, and with the degree of proximity of the resonator mouth to an acoustic mode pressure anti-node, and is greatest for an oblique mode and least for an axial mode. The relative proportions of energy stored in the resonator and in the room mode are very similar in both coupled modes, but the relative phases of the modal pressure and resonator neck velocities are of opposite sign. Consequently the damping ratios and decay rates of the two modes are similar, and this fact, combined with the relatively small frequency separation, is likely to produce very non-exponential amplitude decays following the cessation of excitation by an arbitrary source of frequency in the region of either natural frequency, because beating will normally occur. Hence it is unlikely that reverberation time measurements would truly reveal the effect of introduction of a tuned resonator. The variation of modal decay rates with resonator and room parameters is shown in Figure 1. For convenience, equivalent reverberation time ratios have been presented. 5.2.
FORCED
VIBRATION
results of section 4 are presented graphically in Figure 2, which apparently differs significantly from van Leeuwen’s corresponding Figure (2). They indicate that, if excitation is confined to the frequency UN of the original uncoupled room mode, the maximum reduction for a given value of QN is obtained by maximizing E and QR ; i.e., by placing a very lightly damped resonator of large volume at a modal pressure anti-node (see equation (34)). This result implies that the free space radiation resistance of the resonator has, in this case, nothing to do with the resonator performance; hence Gilford’s conclusions [7] regarding the influence of room edges and corners on the quantity do not necessarily apply. But, if the excitation frequency is not confined to that fixed frequency, as in the case of The
HELMHOLTZ
RESONATOR
AND
AN
ENCL.OSURE
373
speed dependent excitation frequencies as in vehicles, or in the case of multiple harmonic, or aperiodic excitation, there may be significant excitation at the resonance frequencies of the coupled modes, w1 and o2 (see equations (35) and (36)),* in such cases a maximization of e2QR may produce resonant pressure amplitudes at WI and 0~2comparable with that of the original uncoupled room mode when excited at resonance. Hence some indication of the relative reductions of pressure amplitude caused by the presence of the resonator when the room is excited at the original frequency tiN, and at the coupled frequencies WI and ~02, is desirable. Using equations (34), (35) and (36) gives
By analogy with den Hartog’s optimized dynamic absorber [2], one may require the ratio to be unity, in which case &QN =~(EQR)~/(~-(EQR)*-_(FQR)~).
(44)
The line corresponding to this condition is termed the line of “constant reduction” and is indicated on Figure 2. This figure may be used to optimize the design of an absorber, once the room mode damping and type (oblique, tangential, axial) is known. In principle, considerable reductions of steady state sound pressure level can be obtained over the “broad band” range 01 to 02, but there are clearly physical limitations to be considered. For example, for an oblique room mode of QN = lo2 (T of about 2 s at 100 Hz), with a ratio of resonator volume to room volume of lo-’ (E* = 8 x 10p4), the value of the parameter (l/ QNE ) is approximately l-1 ; reference to Figure 2 shows that it is impossible to achieve a significant “broad band” SPL reduction, although a reduction of about 6 dB could be achieved at WN if (~/EQR) could be made very low (zO.9); this would require a value of QR of about 125, which is almost impossible to achieve in practice. (Note &2 = &(rR) vR/ANv.> With a larger resonator volume, having a typical linear dimension approximately one-tenth of a typical room dimension (V/ VR = 10”) then E = 9 x 10d2 and (l/QN&) = 0.1. In this case a desirable value of QT(would be about 11, which is practicable. The conclusion from these considerations is that only relatively large resonators can effect significant pure tone or “broad band” reductions of SPL in an enclosure. 5.3.
POWER
ABSORPTION
BY THE RESONATOR
At first sight it would appear that the conditions for maximum power absorption by the resonator at the original mode frequency UN would correspond to maximum SPL reduction at that frequency. This is not however true because the effect of the presence of the resonator in creating two new coupled modes with natural frequencies different from UN is to “unload” the volume velocity source so that it actually injects less power into the enclosure. The maximum absorbed power line is indicated on Figure 2 and corresponds to the line of 6 dB reduction at oN.
6. EXPERIMENTS Measurements of SPL reduction and absorbed power were conducted in a small rectangular room of dimensions 2.10 x 2.52 x 2.50 m (V = 13.3 m3). A resonator, having a 306 mm internal diameter cylindrical body with a depth variable within a range O-100 mm (volume range O-8 x low3 m3), and a neck of 102 mm internal diameter and length of 150 mm, was introduced into the room. Two Briiel and Kjaer Type 4135 L’ microphones were placed in the neck at a separation of 100 mm. A 20 cm diameter loudspeaker was placed in one corner of the room.
374
F. J. FAHY
AND
C. SCHOFIELD
The frequency response of the room, as measured by microphones placed in various positions, was used to indicate the frequencies and types of acoustic modes; a typical trace is shown in Figure 3(a). The (l,l,l) mode at 130 Hz was chosen as being sufficiently well isolated from its neighbours at 110 and 138 Hz. The air temperature was measured to be very stable, variations from day to day corresponding to a range of kO.3 Hz. The decay rate of this mode was measured at various microphone positions, by using a level recorder; the curves were linear and reproducible. The corresponding reverberation time was 0.59 s (ON = 34.6).
(a) __~_~_.___._
Frequency (Hz)
Frequency MI)
Frequency biz)
Frequency (Hz)
,60
Figure 3. Effect of resonator on room response. (a) No resonator; (b) empty necked resonator at position B (low value of (e&-l); (c) damped resonator at position B (line A); (d) damped resonator at position C (high value of (EQR)-‘).
The resonator had to be tuned to the same frequency as the room mode under anechoic conditions, because tuning when coupled to a room is naturally impossible. The bandwidth of the resonator resonance was measured under the same conditions to yield the Q-factor. This could be set at the different values by means of introducing one or two sheets of loudspeaker grille cloth (r = 11 mks Rayls) at the base of the neck. The corresponding values of OR were 75, 16 and 9.7. With the resonator at various positions within the room including one at a modal pressure node, measurements were made of SPL reductions at the original mode natural frequency and at the frequencies of the two new coupled modes. The corresponding values are presented in Table 1. The soun_d of the coupling parameter E, (eQx)-’ and (e&-l intensity in the neck was estimated from the expression I = -PI$z sin c5/4hpow, where PI and p2 are the presssure amplitudes at the microphones in the neck, 2h is the microphone
HELMHOLTZ
RESONATOR
AND
AN ENCLOSURE
375
1
TABLE
Coupling parameter and Q-factor values Resonator position and condition Empty
1
(EQR)~
F
(EQNI-’
neck
A B C D
0.058 0.048 0.028 0
0.23 11.28 1J.48 cc
0.50 0.61 1.10 cc
One piece of material in neck
A B C D
0.058 OsQ47 0.027 0
1.10 1.33 2.30 co
0.51 0.62 1.10 co
Two pieces of material in neck
A B C D
0.056
l-85
0.52
0.046 0.027 0
2.24 3.88 00
0.63 1.10 co
(I+cwnc
22
)
Figure 4. Theoretical and measured normalized intensities in the resonator neck. -, Theoretical prediction; 13, experimental measurements (number of pieces of material in neck, resonator position).
376
F. J. FAHY
AND
C. SCHOFIELD
separation, and 9 is the phase angle between the two pressures [8]; the uniformity of intensity over the cross section of the neck was verified, the frequency being far less than the lowest cut-off frequency of the neck. Typical pressure response traces are presented in Figure 4, and a table of measured and calculated frequencies and SPL reductions is presented in Table 2. An attempt was made to measure the decay ratio of the coupled modes for comparison with the theoretical results presented in Figure 1. However, as expected, the beating between the coupled modes precluded accurate measurement, and the application of sophisticated discrimination techniques was not possible within the time scale of the project. The comparisons in Table 2 suggest that the present theory adequately indicates the effect of system parameters on resonator performance. The agreement between measured and calculated absorbed power is good, but somewhat fortuitous, because the likely error is as much as f 1 dB (*25%), in spite of the use of the microphone reversal technique to reduce errors due to phase mismatch.
7. CONCLUSIONS
When a resonator, tuned to have the same natural frequency as an acoustic mode of an enclosure, is coupled to the enclosure, two new coupled modes are produced of which the natural frequencies normally lie on either side of the original frequency. The separation between the new frequencies is proportional to the value of a coupling parameter, which increases with the ratio of enclosure volume to resonator volume, and with the modal pressure amplitude at the position of the resonator, and is largest for oblique modes and smaller for axial modes. The decay rates of the coupled modes are very similar for practicable values of the coupling parameter. Because of the closeness of the coupled mode frequencies the decay process is likely to exhibit the beating phenomenon and therefore will be difficult to measure by conventional means. In principle the reverberation time of each coupled mode can be reduced indefinitely by the presence of a weakly coupled, highly damped resonator. This has yet to be demonstrated experimentally. On the contrary the presence of a well coupled, lightly damped resonator can no more than double the original reverberation time. Optimization of a resonator design depends upon a knowledge of the type of acoustic enclosure mode involved and the decay rate, reverberation time, or Q-factor of that mode. A choice has to be made between (a) producing a large reduction in SPL at the resonance frequency of the original mode, which involves a risk of greatly increasing the SPL at the coupled mode resonance frequencies; or (b) producing a modest and approximately equal reduction of SPL at both coupled and original frequencies. The former approach is only suitable when it is known that the excitation frequency is closely confined to the original mode frequency. The latter strategy is better suited to excitation by variable frequency (or speed), aperiodic or multiple harmonic excitation which may excite the coupled modes at resonance. It has been shown that the condition of maximum power absorption by the resonator at the original mode frequency does not necessarily correspond to maximum effectiveness in terms of SPL reduction at these frequencies because the shift of natural frequencies “unloads” the acoustic source. The power absorption coefficient does not depend upon the matching of the radiation and internal resistances, as in the case of the resonator coupled to free space, but is simply inversely proportional to the internal resistance of the resonator. This is because the resonator acts like a secondary source in the room and the “re-radiated” and “incident” pressure fields are not independent.
neck
A B C D
: D
A
D
C
B
A
127.0 (126.0) 127.0 (126.5) 127.5 (127.5) 129.5 (129.5)
w2
Measured natural
133.0 (133.0) 133.0 (133.0) 132.0 (132.5) 129.5 (129.5)
WI
(theoretical) frequencies
t It is not clear whether this figure applies to oN or w1 and 0~2.
Two pieces of material in neck
One piece of material in neck
Empty
Resonator position and condition
TABLE
2
(i)
(Z)
.
(z:;)
(Z)
*
wz-6J1
modal (Hz)
5.0 4.5 2.7 -
6.7 57 3.0 -
-
6.8
11.0
11.6
Measured
6.2 4.7 1.9 -
8.9 6.9 3.0 -
-
9.5
16.7
19.7
Theoretical
SPL reduction --
6 5 4 -
7 6 4 -
-
4t
4t
4t
Reference
at ON (dB)
Coupled mode natural frequencies and SPL reductions
[3]
5.5 5.3 3.6 -
5.8 6.1 4.5 -
-
4.2
4.3
3.8
Measured WI
4.5 4.5 3,2 -
5.2 5.0 4.2 -
-
4.5
5.3
6.7
w2
with no resonator)
w?
7.4 6.4 3.5 -
7-o 6.2 3.9 -
-
2.9
3.0
3.1
WI,
Theoretical
(dB)
(SPK at mq and WA-(SPL at
wN
z
3 s 2
5 m
g
$
z
?I 5
8
ro
z
z F
378
F. J. FAHY AND C. SCHOFIELD
It is believed that the present theory and design chart usefully complement those of van Leeuwen because the physics of the model is more explicit and the experimental results are in close agreement with the former. In principle the same approach could be used to analyze the coupling of a panel or membrane absorber to an enclosure, when itis likely that the same general forms of behaviour and design criteria will obtain. ACKNOWLEDGMENT The advice of Professor P. 0. A. L. Davies regarding the design of the resonator, and the technical assistance of Mr D. C. Howell are gratefully acknowledged.
REFERENCES 1. C. ZWIKKER and C. W. KOSTEN 1949 Sound Absorbing Materials. New York: Elsevier. 2. J. P. DEN HARTOG 1956 Mechanical Vibration. New York: McGraw-Hill, fourth edition. 3. F. J. VANLEEUWEN 1960 European Broadcasting Union Review, A-Technical 62, 155-161. The damping of eigen-tones in small rooms by Helmholtz resonators. 4. H. R. BRITZ and H. F. POLLARD 1978 Journal of Sound and Vibration 60, 305-307.
Computational analysis of coupled resonators. 5. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill. 6. T. PEARSON 1974 Handbook of Applied Mathematics. New York: Van Nostrand Reinhold. 7. C. GILFORD 1972 Acoustics of Radio and Television Studies, I.E.E. Monograph: Series II.
London: Peter Peregrinus. 8. R. J. SAGE 1975 B.Sc. (Honours) Dissertation, University of Southampton. An investigation of the two-microphone method of acoustic intensity measurement.