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The entrainment phenomenon and thermoacoustic engines P. S. Spoora) and G. W. Swift Condensed Matter and Thermal Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ͑Received 14 September 1999; accepted for publication 1 May 2000͒ The earliest known reference to the mode-locking, or entrainment, of two maintained oscillators is ’ description of two ‘‘falling into synchrony’’ when hung on the same wall. We describe an analogous phenomenon in —the mode-locking of two thermoacoustic engines which have their cases rigidly welded together, but which are otherwise uncoupled. This ‘‘-coupling’’ might compete with acoustic coupling when the latter is used to enforce antiphase mode-locking in such engines, for cancellation. A simple theory relating the phase difference between the engines in the locked state to the corresponding ratio of their pressure amplitudes is in excellent agreement with theory and numerical simulations. The theory’s prediction relating the phase difference to the engines’ natural difference is qualitatively confirmed by experiment, despite larger experimental uncertainties. The mass coupling is relatively weak compared to the aforementioned acoustic coupling, and in general occurs in antiphase, so we conclude that mass coupling will not interfere with vibration cancellation by acoustic coupling in most circumstances. © 2000 Acoustical Society of America. ͓S0001-4966͑00͒01708-2͔ PACS numbers: 43.35.Ud, 43.40.Vn ͓HEB͔

INTRODUCTION ing ends through a narrow tube a half-wavelength long, thus encouraging the in the two engines to be in an- Entrainment, or mode-locking, whereby two or more tiphase. If the engines are attached rigidly enough to move as self-maintained oscillators lock together in frequency and a unit, however, they are of necessity coupled through their phase, exists in a fascinating array of physical and biological mutual structure as well as through the acoustic tube. Hence, systems. The earliest known reference to this phenomenon in our particular goal in the was to discover how the annals of was made by Christiaan Huygens, who efforts to cancel external case in two rigidly at- 1 wrote to his father in a letter dated 26 February 1665: tached thermoacoustic engines, by coupling them acousti- . . . I have noted an impressive effect which no one cally, might be helped or hindered by the Huygens-style has yet been able to explain. This is that two clocks, structural coupling. We refer to this coupling as ‘‘mass cou- hanging side by side and separated by one or two feet, pling,’’ since the coupling occurs through on a shared keep between them a consonance so exact that the two mass ͑the case͒ and is stronger if the oscillating gas is heavy pendula always strike together, never varying. and the case is . Therefore, to increase the mass coupling we used a pressurized, heavy gas—xenon—in the lightly In this work we describe the equivalent of the Huygens phe- 3 nomenon for acoustic . Two cylindrical-duct reso- constructed engines used for studies of acoustic coupling. nators are welded together side by side, the oscillating gas Without acoustic coupling, we found that the engines would inside the resonators acting the part of the pendula and tend to mass-lock in antiphase, as desired for vibration can- the duct cases acting the part of the wall upon which the cellation. For the range of mass-locked states we could clocks were hung. The resonators exhibit mode-locking in achieve, we measured the dependence of the amplitude ratio the same way that Huygens’ clocks synchronized their mo- of the two engines and the phase difference between them as tion. The oscillations in the ducts are maintained thermoa- a function of their natural frequency difference. Even with 3 coustically, although to observe the Huygens phenomenon, bars of xenon gas in our engines, the mass-coupling was any method of producing spontaneous oscillations will do. relatively weak; locked states existed only over a very nar- ͑ A thermoacoustic engine of a size and density row ‘‘capture bandwidth’’ of frequency differences frac- ͒ suitable for heavy industrial applications ͑such as large-scale tional differences of order 0.001 . This article provides a natural gas liquefaction2͒ experiences tons of axial dynamic quantitative explanation of these measurements, through on its structure from the pressure oscillations of the gas both theory and numerical simulations. inside it, potentially inducing intense, damaging vibrations in this structure. In a separate paper3 we show how these vibra- I. ESSENTIAL OF MODE-LOCKING tions can be canceled by rigidly attaching two engines to- Two self-maintained oscillators, such as pendulum gether, side by side, and acoustically coupling two neighbor- clocks, organ pipes, or thermoacoustic engines, may alter each other’s enough to lock in frequency and a͒Present address: Clever Fellows Innovation Consortium, 302 10th St., phase if they are sufficiently coupled. When the ‘‘detuning’’ Troy, NY 12180. ͑the difference in natural uncoupled frequencies͒ is large, or

588 J. Acoust. Soc. Am. 108 (2), August 2000 0001-4966/2000/108(2)/588/12/$17.00 © 2000 Acoustical Society of America 588 k m ␻2ϭ ͩ 1ϩ ͪ . ͑5͒ 0 m M For a straight cylindrical half- acoustic whose case of mass M is free to move, the corresponding expression is ͓see Sec. II B, especially Eqs. ͑32͒ and ͑36͔͒ ␲2a2 8 m ␻2Ӎ ͫ ϩͩ ͪ gͬ Ӷ ͑ ͒ 0 1 , mg M, 6 l2 ␲2 M where ␲a/l is the natural of a half-wave resonator with a and length l,andmg is the total mass of gas in the resonator. Thus one might conclude that FIG. 1. Mass–spring model of mass-coupled oscillators. the effective mass of the oscillating gas in a half-wave reso- ϭ ␲2 Ӎ nator is m (8/ )mg 0.8mg in the present context. the coupling is weak, the oscillators ‘‘’’ at the difference frequency, just like coupled linear passive oscillators. As the II. THEORY detuning is decreased, or as the coupling is increased, the beating slows down and eventually stops, at which point the A. Mass-and-spring model oscillators are locked. The change in frequency of one oscil- Before discussing coupled acoustic engines, we will lator depends on its phase relative to the other, so the locked consider a simple system of and springs, arranged to state at a particular detuning is characterized by a corre- resemble two mass-coupled acoustic resonators. Figure 1 sponding phase difference between the oscillators. In gen- shows a pair of mass–spring oscillators that attach to the eral, the amplitudes in the oscillators also depend on this same moving mass. The oscillators are identical except for a phase difference, and there may be a net flow of from slight difference in frequency; they have masses m and one oscillator to the other if the amplitudes are not equal. In ␻ springs kI and kII , hence also natural frequencies I an intermediate state, where the coupling is not quite strong ϭͱ ␻ ϭͱ 4 kI /m and II kII /m. Following Pippard and Van der enough to lock the oscillators, they beat with a nonsinusoidal Pol,5 the feedback mechanism that maintains the oscillations envelope—the beat period elongates when the frequencies is modeled by a resistance that can be either positive or nega- have a close approach. tive, depending on the amplitude. We let We use the following notation for two oscillators that ␻ n are mode-locked or nearly mode-locked: m I X0 R ϭ ͩ 1Ϫ ͪ , ͑7͒ I ␻ ϩ␾ Q n ␺ ͒ϭR ⌿ ͒ i[ t I(t)] ͑ ͒ XI I͑t ͕ I͑t e ͖ , 1 ␻ ϩ␾ where XI is the amplitude in oscillator I and X0 is the steady- ␺ ͑ ͒ϭR͕⌿ ͑ ͒ i[ t II(t)]͖ ͑ ͒ II t II t e , 2 state amplitude to which the oscillator relaxes. Recall from where R͕͖ indicates taking the real part. Here ␺ could be Sec. I that the amplitude XI is assumed to vary slowly with I,II ␻ displacements, pressures, , or any other similar compared to . The equation of for oscillator I variables of interest that depend on time t. We assume that in the absence of coupling is then ⌿ ␾ the amplitudes I,II and the phases I,II are real and slowly ␻ Xn ␻ ⌿ у 2 I 0 varying compared to the angular frequency ,and I,II 0. x¨ ϩ␻ x ϩ ͩ 1Ϫ ͪ x˙ ϭ0, ͑8͒ I I I Q n I The locked state is characterized by a relative phase between XI the oscillators and a ratio of their amplitudes, which are where xI(t) is the of the mass m from equilib- ⌿ rium, and the dot indicates a time derivative. We use this ␾ϭ␾ Ϫ␾ ␨ϭ II ͑ ͒ II I , ⌿ . 3 particular form because if X0 is set to zero, one obtains the I equation for an ordinary damped oscillator, with Q akin to ␻ the factor of the oscillator. We assume that oscillator I has natural frequency I and ␻ The exponent n depends on the mechanism maintaining oscillator II has II when I and II are uncoupled, so that the difference in natural frequencies, or detuning, is the oscillations. Here, we argue that a thermoacoustic engine is governed by nϭ2 when comparing steady-state ampli- ⌬␻ϭ␻ Ϫ␻ ͑ ͒ II I. 4 tudes at different operating points. In the mass-and-spring model, the power delivered or absorbed by the oscillator ͑the We let the compromise frequency ␻ that the oscillators pos- ͒ ˙ ϭϪ ␻ Ϫ n n ͉˙͉2 sess when they are coupled be a constant in time; the appar- ‘‘load’’ is WL (m /Q)(1 X0/X ) x . In a thermoa- coustic engine, such things as resonator dissipation and stack ent instantaneous difference in frequency between resonators 6 power are also proportional to amplitude squared, so the ץ ␾ץ␻ ϭ⌬ II and I is simply inst. / t. ˙ ϭ␤ 2ϭ␤ 2ϩ ˙ The obvious mechanism for frequency shift in our sys- power produced by the stack is Wstack P0 P WL , tem is the moving case. If a simple oscillator of where P0 is the unloaded pressure amplitude and P is the ˙ ϭ␤ 2Ϫ␤ 2 ˙ mass m and stiffness k is anchored to an unfettered mass M, loaded amplitude. Then we have WL P0 P ,orWL ϭϪ␤ Ϫ 2 2 2 the natural frequency is shifted according to (1 P0/P )P .

589 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 589 In practice, we find that only the ratio Q/n, and not Q or ␻␻ 2 i I X0 ␾ Ϫ␾ n separately, can be measured experimentally ͑and therefore, ͑␻2Ϫ␻2͒X ϩ ͩ 1Ϫ ͪ X ϭ␻2ei( c I)X , ͑9͒ I I Q 2 I I c for our purposes, Q/n is all that matters͒. In principle, one XI should be able to determine both Q and n from transient ␻␻ 2 i II X0 ␾ Ϫ␾ measurements; however, the thermal mass of a thermoacous- ͑␻2 Ϫ␻2͒X ϩ ͩ 1Ϫ ͪ X ϭ␻2 ei( c II)X , ͑10͒ II II Q 2 II II c tic engine causes the growth and decay time constants to be XII artificially long. The ratio Q/n that governs the steady state ␾ Ϫ␾ ␾ Ϫ␾ Ϫ␻2X ϭ␮␻2͑ei( I c)X ϪX ͒ϩ␮␻2 ͑ei( II c)X ϪX ͒, can be measured by comparing states at various loads, but if c I I c II II c ͑11͒ we consider only small loads ͑i.e., PϷP ,orXϷX ) such as 0 0 ␮ϭ the engines experience when they are coupled, it is difficult where m/M. Referring to Fig. 1, note that Xc is the mo- to distinguish Q or n separately. tion amplitude of the common mass, and hence the right ͑ ͒ ͑ ͒ ͑ ͒ To fully analyze the dynamics of the system, we could sides of Eqs. 9 and 10 are coupling terms. Equation 11 ͑ ͒ write down the with full time depen- can be solved for Xc ; substituting this result in Eqs. 9 and ͑ ͒ dence, carry out the time derivatives in these equations, with 10 yields x(t) having the form suggested by Eqs. ͑1͒ and ͑2͒,andfind ␻␻ 2 i I X0 t. Pippard7 treats a problem ͑␻2Ϫ␻2͒ϩ ͩ 1Ϫ ͪ ϭϪ␮͑␻2ϩ␻2 ␨ ei␾͒ , ͑12͒ץ/␨ץ t andץ/␾ץ expressions for I Q 2 I II very similar to ours in this manner. However, we are inter- XI ested mostly in locked states, where everything is in the ␻␻ 2 i II X0 steady state. We seek expressions for the state variables ␾ ͑␻2 Ϫ␻2͒ϩ ͩ 1Ϫ ͪ ϭϪ␮͑␻2 ϩ␻2eϪi␾/␨͒ . ͑13͒ II Q 2 II I and ␨ in terms of known or measurable quantities like m, M, XII Q,and⌬␻, and an expression for the ‘‘sensitivity’’ of the ␮Ӷ ͉ Ϫ 2 2 ͉Ӷ ␻ We assume that 1and 1 X0/XI, II 1, and that II ͒ ͑ ͒ ͑ ␻. This last quantity is of particular interest ϳ␻ ϳ␻⌬ץ/␾ץ ,system I . Taking the real part of Eq. 13 minus Eq. 12 , in the vibration-cancellation context, because it tells how fast keeping terms to first order, and simplifying gives a system tends to move away from some ideal phase ͑e.g., ⌬␻ ␮ ␾ϭ␲) as it is detuned; when considering a mode-locked ϭ ͑␨Ϫ1/␨͒cos␾ . ͑14͒ pair of acoustic engines, a movement of phase away from ␻ 2 ␾ϭ␲ means more case vibration. 7 7 To find a equation, we follow Pippard and assume One may readily show, as has Pippard, that a reactively that in the steady state, energy is conserved in the system, coupled pair of oscillators like ours has two possible locked i.e., states, one nominally near ␾ϭ0 and one nominally near ␾ ϭ␲ ␻ 2 ␻ 2 . Which one is actually selected is of some interest; a m I X0 m II X0 ͩ Ϫ1 ͪ ͉x˙ ͉2ϩ ͩ Ϫ1 ͪ ͉x˙ ͉2 ϭ0. ͑15͒ ␾ϭ I II tendency for our coupled engines to mass-lock at 0, Q X2 Q X2 where both ‘‘masses’’ are moving in the same direction, I II could impair vibration cancellation. Fortunately, we may ͓Equation ͑15͒ may also be obtained from the ratio of the take some comfort in the words of Huygens1 describing his imaginary parts of Eqs. ͑12͒ and ͑13͒.͔ After some algebra, mode-locked pendulum clocks: this leads to the identity . . . the motion of the pendula, while they are in con- X2 X2 1 1 0 0 2 sonance, are not in phase, but they approach and fall Ϫ ϭ ͩ ␨ Ϫ ͪ . ͑16͒ X2 X2 2 ␨2 back with contrary motion . . . I II ͑ ͒ Due to external on our moving case, we expect the Using this identity, with the imaginary part of Eq. 12 minus ͑ ͒ mode with the least case motion to be favored. In a passive, Eq. 13 , yields linear system, the modes that experience more tend ͑␨Ϫ1/␨͒ϭ2␮Qsin␾ . ͑17͒ to decay faster, and we expect similar behavior in our sys- Equations ͑14͒ and ͑17͒ can be combined to eliminate ␨,to tem, although it is neither passive nor strictly linear. In any obtain , like Huygens, we nearly always observe that our en- gines mass-lock in antiphase rather than in-phase. ͓If one ⌬␻ ϭ␮2 ␾ ␾ ͑ ͒ extracts the mechanisms from two metronomes and mounts ␻ Qsin cos ; 18 them on a very light balsa-wood or perforated-aluminum ␾→␲ frame, and suspends this frame so it is free to swing, the in the limit ,thisleadsto ͑ ͒ ␾ 1ץ Huygens phenomenon is readily observable. One of us PS ␻ has experimented with such a system, observing both in- ϭ . ͑19͒ ␻ ␲␮2⌬ץ ␲ phase and antiphase locking of the metronomes, the former Q especially if the two metronome pendula are initially started This last equation is particularly useful for comparing the out with in-phase motion.͔ present results with those for acoustically coupled engines.3 To proceed with our analysis, then, we will let d/dt The results in this section are simple and roughly true ϭi␻, and assume amplitudes and phases are constant in for any mass-coupled pair of oscillators. In the next section, time. We obtain to obtain results specific to mass-coupled acoustic engines,

590 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 590 we will derive differential equations isomorphic with Eqs. in accordance with Rayleigh’s principle relating the fre- ͑9͒–͑11͒ for the acoustic pressure in the engines and the quency of a normal mode to the ratio of the maxi- forces on the case. We will derive equations valid for en- mum potential and kinetic .9,10͔ Hence, an appropri- gines with resonators of arbitrary cross section, and then ate equation for is modify these for specific examples of interest. ␻ 2 ͐ ˙ I P0 V swdV p¨ ϩ␻2p ϩ ͩ 1Ϫ ͪ p˙ ϭ␳a2 . ͑27͒ I I I Q 2 I ͐ 2 PI V w dV B. Acoustic model To make this equation specific to mass coupling, con- Consider two nearly identical acoustic engines, called sider how the moving case of an acoustic resonator couples engines I and II, which are rigidly attached side-by-side and to the gas inside. If the motion of the resonator is essentially have a combined case mass M. Each engine consists of a rectilinear, then ͑in the low-amplitude limit͒ changes in cross prime mover section, where heat is applied and the acoustic section act like local volume- sources ͑or sinks͒.Ina power to sustain the oscillations is generated, and a resonator cylindrical resonator of varying cross-sectional area A(x), section, which comprises the bulk of the volume and is one may write mostly responsible for determining the frequency at which 1 dA͑x͒ the engine runs. The engine cross section may vary with s͑x,t͒ϭ u ͑t͒, ͑28͒ position, but the radius is always much smaller than the A͑x͒ dx c length l. The engines are also assumed to be supporting what where uc is the case velocity in the x direction, which is the is more or less a half-wavelength mode. The long axis is same for all x ͑if the case is assumed rigid͒. We can also find labeled x, and is positive in the same direction as positive uc in terms of the forces on the case; considering the forces displacements and positive forces. The prime movers are lo- exerted by both engines, assuming they are identical in ge- ϭ cated near the x 0 end, and we assume the acoustic pres- ometry, and neglecting viscous forces, we can write sure at the xϭl end is known. p ϩp l dA͑x͒ We assume that the acoustic pressure throughout each ˙ ϭϪ I II ͵ ͑ ͒ uc wdx, 29 engine as a function of x and t can be written as a product of M 0 dx the pressure p(t)atthexϭl end and a spatial weighting function w(x), that is, where M is the total case mass. Note that for each abrupt Ϫ␧ ϩ␧ change in cross section, say between x j and x j , pressure ͑x,t͒ϭw͑x͒p ͑t͒ , ͑20͒ I I ϩ␧ x j dA͑x͒ ͵ ϭ͓ ͑ ϩ␧͒Ϫ ͑ Ϫ␧͔͒ ͑ ͒ ͑ ͒ ͑ ͒ϭ ͑ ͒ ͑ ͒ ͑ ͒ wdx A x j A x j w x j . 30 pressureII x,t w x pII t . 21 Ϫ␧ x j dx ͑ ͒ ͑ ͒ Furthermore, in the manner of Eqs. 1 and 2 , we assume For example, Eq. ͑30͒ would apply to the endcaps of a typi- ␻ ϩ␾ ͒ϭR i( t I) ͑ ͒ ͑ ͒ ͑ ͒ pI͑t ͕PIe ͖ , 22 cal resonator. Combining Eqs. 27 – 29 , and using ͐ ͕͖ ϭ͐l ͕͖ ␻ ϩ␾ V dV 0 Adx, we obtain ͒ϭR i( t II) ͑ ͒ pII͑t ͕PIIe ͖. 23 ␻ 2 8 I P0 Following Morse, and guided by the results in Sec. II A, we p¨ ϩ␻2p ϩ ͩ 1Ϫ ͪ p˙ I I I Q 2 I write the inhomogeneous for engine I as PI ␻ 2 ␳ 2 ͓͐l ͑ ͒ ͔2 I P0 a 0 dA/dx wdx p ٌ2wϪwp¨ /a2Ϫ ͩ 1Ϫ ͪ wp˙ /a2ϭϪ␳s˙ , ͑24͒ ϭϪ p ͑1ϩ␨ei␾͒. ͑31͒ I I Q 2 I M ͐l 2 I PI 0w Adx where a is the sound speed, ␳ is the density, and the source We may now make use of the results previously ob- t is the volume velocity per unit tained for the mass–spring system. Starting with Eq. ͑31͒ forץ/Vץ(density s(x,t)ϭ(1/V ϭ ␻ volume imposed by the moving boundaries. Multiplying pI and a similar equation for pII , letting d/dt i ,andthen Ϫ 2 both sides by wa and integrating over the volume leads to dividing the first equation by pI(t) and the second by pII(t) results in a pair of equations exactly like Eqs. ͑12͒ and ͑13͒, ˙ ͐ ␻ 2 2ٌ 2 ͐ V a w wdV I P0 V swdV ␮␻2 p¨ Ϫ p ϩ ͩ 1Ϫ ͪ p˙ ϭ␳a2 . except that the constant has been replaced by a more I ͐ 2 I Q 2 I ͐ 2 V w dV PI V w dV complicated expression. We can therefore identify the effec- ͑25͒ tive mass ratio ␮ for the acoustic system as ͑Note that ␳a2 is a constant, so it appears outside the inte- ␳ 2 ͓͐l ͑ ͒ ͔2 a 0 dA/dx wdx gral.͒ This last equation has exactly the same form as Eq. ͑8͒, ␮ϭ , ͑32͒ ␻2 ͐l 2 if the right-hand side is zero ͑no coupling͒, with the coeffi- M 0w Adx ␻2 ͓ cient of pI identified as I . In the limit that a is constant, and then all the results obtained for the mass–spring system .and ٌw is zero at the walls, the vector identity ٌ•(w ٌw) should apply to the acoustic system as well -ϭ(ٌw)2ϩw ٌ2w and Gauss’ theorem ͑for turning the vol ume integral of a divergence into a surface integral͒ yield 1. Obtaining Q of engines from experiment We measure Q from the ‘‘load response’’ of each en- 2͒ ٌ 2͐ 2ٌ 2 ͐ V a w wdV a V ͑ w dV Ϫ ϭ ϭ␻2, ͑26͒ gine. Suppose that instead of a moving case, the only ͐ 2 ͐ 2 I ϭ ͑ V w dV V w dV volume-velocity source is a small orifice at x l e.g., a

591 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 591 valve͒ leading to a small volume, forming an acoustic RC Ref. 3. The sensitivity to frequency difference of two engines 3 network which serves as a load. In the steady state, the dif- coupled by a narrow half-wavelength tube of area Ac is ferential equation for the loaded engine leads to ␦ ␾ ␲ A␶ץ ␻ ͩ ͪ ͯ ϭF ͩ ͪ , ͑38͒ ␻ H 4 A r⌬ץ ␻ P2 i␻U˜ ␲ ͑␻2Ϫ␻2͒˜ ϩ ␻ ͩ Ϫ 0 ͪ˜ ϭϪ␳ 2 e ͑ ͒ acoust. c c 0 p i 1 p a , 33 Q P2 ͐l w2 Adx ␦ 0 where rc is the radius of the coupler and is the effective ␦ϭ␦ ϩ ␥Ϫ ␦ where ˜p is the complex pressure amplitude at the loaded end thermoviscous boundary layer thickness ␯ ( 1) ␬ ; ␦␯ is the viscous penetration depth, ␦␬ is the thermal pen- of the resonator, P is its magnitude, and U˜ is the volume e etration depth, and ␥ is the ratio of isobaric to isochoric velocity exiting that end. Multiplying both sides by ˜p*/i␻, specific heats. FH is a quantity called the ‘‘Helmholtz fac- ˜ ˜ where p* is the complex conjugate of p, and taking the real tor,’’ which is proportional to 1/A␶ but depends only weakly ␦ part leads to on rc and . A␶ is arbitrary and normally refers to a typical ␻ ␳ 2 cross-sectional area of the engines; if the engines have a 2 2 a ϭ ͑P2ϪP ͒ϭϪͫ ͬR͕˜p*U˜ ͖/2 . ͑34͒ constant cross section A0, then choosing A␶ A0 makes FH Q 0 ͐l 2 e 0w Adx ϭ1. The Helmholtz factor represents the effect on sensitivity of nonuniform cross section, e.g., of using engine geometries Since R͕˜p*U˜ ͖/2ϭW˙ , the time-averaged load on the en- e L that are more like Helmholtz resonators and less like straight gine, it follows that ducts. Combining our present general theory with that of P2 Ref. 3, we may find FH analytically. A full expression for FHץ l ␻ QϭϪͫ ͵ w2 Adxͬ ͩ ͪ . ͑35͒ that is precisely accurate over a wide range of coupler diam- ˙ ץ ␳ 2 0 2 a WL ␦ ␦ eters has a very complicated dependence on rc , ␬ ,and ␯ , ͑In this context, power flowing out of the engine is positive; but reasonable accuracy is obtained with for constant heat input to the engine, pressure amplitude 2 l ͒ ͑ ͒ ͑ ϭ ͵ 2 ͒ ˙ ץ 2 ץ drops as the load increases, so P / WL is negative. FH w A x dx, 39 A␶ l Strictly speaking, the left side of Eq. ͑35͒ should be c 0 ӷ␦ →␭ 2Q/n, but this is moot unless one can find n independently, which is exact in the limit rc , lc /2, where lc is the 2 ˙ ␭ say by measuring the curvature of P vs WL .Wefind that resonant coupler length and is the wavelength. In this ͑ ͒ ϭ for the engines in the present work, P2 vs W˙ is quite lin- limit, Eq. 39 predicts FH 2.83 for the engines in the L ϭ , and the data have sufficient scatter to discourage higher- present work, letting A␶ equal the endcap area at x l and 11 order fitting. Since nϭ2 simplifies some of the math, it is a performing the integral numerically using a DELTAE simu- ͑ useful assumption. The interested reader may verify that car- lation of a single engine with air at 80 kPa as the working ͒ rying out our derivations with arbitrary n leads to the same gas . For comparison, our DELTAE simulation of two acous- ␻͔⌬ץ ␾ץ ϭ͓ ␻ ␲ results as ours, but for the appearance of 2Q/n instead of Q tically coupled engines predicts FH ( / ) / / ͑ Ϸ ͓(␲/4)A␶ /A (␦/r )͔ϭ2.80, a difference of only 1%. We as long as PI,II P0). c c may also compare our present theory with the experimental results in Ref. 3; in that work, two sets of sensitivity data 2. Uniform-diameter engine were obtained, one on the engines used in this work ͑which To enable ready comparison of theory to numerical have a variable cross section—see Fig. 4͒ and another on simulations in a later section, we consider the simple ex- engines that were nearly uniform in cross section. The ge- ample of an engine with a single cross-sectional area A0 ometry of the engines was the only thing that varied between throughout. If we further assume that the pressure distribu- the two experiments, so if A␶ is chosen to be the same for tion in such an engine is simply w(x)ϭϪcos(␻/a)x, then both types of engine, the ratio of the measured sensitivities ͐l ϭϪ ͐l 2 ϭ 0(dA/dx)wdx 2A0 and 0w Adx A0l/2. Our formu- should be las for ␮ and Q become sensitivity ͑var. A͒ F ͑var. A͒ ϭ H ␳ 2 ͑ ͒ ͑ ͒ 8 a A0 8 mg sensitivity const. A FH const. A ␮ϭ ϭ ͑36͒ 2 2 ␻ lM ␲ M ͐l 2 ͑ ͒ ͑ ͒ 0w A x dx var. A ϭ . ͑40͒ and ͐l 2 ͑ ͒ A 0w dx const. A 2 ץ ␻ 2 ץ ␻ A0l P mg P This formula predicts a ratio of 3.09, whereas the data and Qϭ ͩ ͪ ϭ ͩ ͪ , ͑37͒ -simulations in Ref. 3 both indicate a ratio of 3.08, a negli ˙ ץ ␳2 2 2 ˙ ץ ␳ 2 4 a WL 2 a WL gible difference. ϭ␳ ␳ where mg A0l is the mass of gas in the engine and and a are assumed constant. III. COMPARING THEORY WITH NUMERICAL SIMULATION 3. Application to acoustic coupling To compare our theory with numerical simulations, we A useful check on part of our theory is to compare its used DELTAE11 to model a mass-coupled pair of engines with predictions for acoustic coupling with those published in uniform cross section. DELTAE ͑Design Environment

592 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 592 for Low-amplitude ThermoAcoustic Engines͒ integrates the one-dimensional wave equation in the small-amplitude ͑‘‘acoustic’’͒ approximation, including thermal and viscous effects, according to the thermoacoustic theory of Rott.12 To simulate a mass-coupled pair of engines, we force all four endcaps in the model to move with the complex case veloc- ity, and require that the case be the complex sum of all the end pressures the cross-sectional area, divided by the case mass ͑we ignore forces on the heat ex- ͒ ␻ ␻ changers and stack .Wethenvary II relative to I by vary- ing the ambient gas temperature inside simulated engine II, and note the effect on the phase difference ␾ between the engines and on their amplitude ratio ␨. The results of this simulation may be compared to the predictions of Eqs. ͑17͒– ͑19͒, using Eqs. ͑36͒ and ͑37͒ for ␮ and Q. In order to obtain ␻ of a pair of mass-locked thermoacoustic⌬ץ/␾ץ Q, we simulate a single engine with no moving endcaps, and FIG. 2. The sensitivity engines with uniform cross section, as predicted by theory ͑line͒ and a large real impedance instead of a perfectly rigid endcap at DELTAE simulation ͑symbols͒. xϭl. As the end impedance is lowered, the power flowing out of the end increases and the pressure drops; the deriva- of an engine as the natural resonance of a column of gas with ˙ ץ 2 ץ tive P / WL is therefore obtained. the engine’s geometry and temperature distribution, or as the While these simulated engines have a simple shape, we frequency with which an engine with that geometry and tem- strive to make them similar in other respects to the real en- perature distribution will run. When the frequency difference gines we studied. The simulated engines are filled with 3 bar between two engines is largely determined by the tempera- of xenon, with an ambient temperature of 300 K; they run at ture of the gas in the resonators, the distinction between Ϸ Ϸ 56 Hz and have Q 200 and mg 0.06 kg; they are 1.6 m these definitions is unimportant; but when two engines are Ϸ ϫ Ϫ3 2 long and have A0 2.3 10 m . They are also set to run weakly coupled, even a very small change in frequency can with cold-end acoustic pressure of about 16 kPa, since this is mean a relatively large change in phase, which can mean a close to where our experimental engines were operated. significant change in the acoustic power flow between the First, we will compare the Q that best describes the engines. This in turn can cause significant differences in the ͑ ͒ simulated mode-locked states according to Eq. 17 with the engines’ hot-end temperatures, and these differences can ul- Q determined by observing the simulated load-response of timately dominate the frequency difference. In this circum- ͑ ͒ one engine as suggested by Eq. 37 . By combining the two stance, the difference in the running frequencies of the two ͑ ͒ ␮ equations, as well as Eq. 36 for , one may obtain the engines may vary considerably from the difference in their putative identity gas-column resonance frequencies. This distinction is of little practical importance in the big 2 ץ ␨Ϫ ␨ 2 1/ 4A0 P ϭ ͯ ͯ . ͑41͒ picture, since one is generally more interested in how sensi- ͑ ˙ ץ sin␾ ␻M WL tive the locked states are to objective external changes one engine sitting in the sun, the other in the shade; one engine The quantities on the left-hand side define the locked state, driving a bigger load, etc.͒ and not to changes in the ‘‘natural and the quantities on the right are well-defined constants that frequencies,’’ particularly if those frequencies are dominated are independent of the locked state, if we restrict our atten- by the engines’ internal responses to external changes. How- ␾→␲ ˙ → tion to the limit , WL 0. In our simulations, every- ever, the natural frequency difference is a very convenient thing is held constant but Q; for each point, the exact pore analytic tool, and useful for expressing general results about geometry of the stack is altered slightly to change the per- mode-locked systems—e.g., the phase difference ␾ between ␻ ⌬␻⌬ ˙ ץ 2 ץ ͑ formance of the engine i.e., change P / WL). We find that the engines is determined by , however that comes over the range 70рQр230, there is virtually no difference about. between the left and right sides of Eq. ͑41͒.TheseQ values We compare our theory with simulation by calculating ⌬␻ϭ␻ Ϫ␻ are similar to those of our actual devices, and typical of the the frequency difference two ways; the first, II I , passive Q’s of acoustic resonators with geometry and gas is the difference in the gas-column resonances; the second, ⌬␻ϭ⍀ Ϫ⍀ properties similar to our engines. II I , is the difference between the frequencies at ␻ which the engines will run, uncoupled, if they have the same⌬ץ/␾ץ Next, we may inquire whether the sensitivity of our simulated system matches theory. We use Eq. ͑19͒, temperature distribution as when coupled. along with Eqs. ͑36͒ and ͑37͒ for ␮ and Q, and compare that Figure 2 shows the comparison of simulations and ,␻ suggested by the simu- theory; moving along the horizontal axis from left to right⌬ץ/␾ץ(result to the value of (␻/␲ lations in the limit ␾→␲. In order to make a meaningful the coupling is weakened by increasing the mass M of the comparison, we must address one subtlety, that of calculat- combined engine cases, thereby lowering ␮, but leaving Q ⌬␻ ⍀ Ϫ⍀ ing , the difference in natural frequencies of two acoustic unchanged. As the coupling gets weaker, II I overesti- engines. This question has direct bearing on our experiments mates the value of ⌬␻. Therefore we might expect that in an as well. One can imagine defining the ‘‘natural frequency’’ experiment on a weakly coupled system, the values of ␨ and

593 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 593 ␾ for a given locked state might agree well with theory, but ␻ will be smaller than⌬ץ/␾ץ the experimentally determined its ‘‘true’’ value. Acoustically coupling3 two engines so that they lock near ␾ϭ␲, and keeping ͉␾Ϫ␲͉ small enough so that 95% of the case vibration is canceled when their frequency differ- -␻Ϸ2. Typi⌬ץ/␾ץ(ence is as much as 1%, leads to (␻/␲ cally, an engine might have Qϳ100 and ␮ϳ10Ϫ3, implying ␻⌬ץ/␾ץ(that the mass-coupled sensitivity is (␻/␲ ϭ1/␲␮2QϷ104. This implies that the mass coupling is re- ally very weak, and can be ignored in the vibration- cancellation application. Absent heroic efforts, we are con- fined to weakly coupled engines in the laboratory as well. The engines we use for our experiments are charged with 3 bar of xenon ͑as in the simulations͒ and lightly constructed; FIG. 3. Normalized sensitivity of a mass-locked system with external resis- 2 3 ␣ϭ ␻ still, we find that 1/␲␮ QϷ10 . tance Rc as a function of Rc / M, for various strengths of mass cou- pling. The darker line corresponds to ␮Qϭ0.135, the value we estimate for A. Possible complications our experimental system of coupled thermoacoustic engines. In many presentations on mode-locking, few quantita- tive results are given because it is usually so difficult to left side of Eq. ͑11͒, with consequences in Eqs. ͑12͒–͑14͒. estimate the coupling strength. In Huygens’ workshop, his The energy conservation equation, Eq. ͑15͒, becomes clocks could have been coupling by uniform motion of the ␣ wall, elastic in the wall, or even motion of the nearby Q ͑X2ϪX2͒ϩ͑X2ϪX2 ͒ϭ X2 , ͑42͒ air. ͑Huygens himself suggested that it was the ‘‘impercep- 0 I 0 II ␮ c tible agitation of the air caused by the motion of the pen- with consequences in Eq. ͑16͒. Equation ͑17͒ becomes dula’’ that enabled the clocks to ‘‘communicate.’’1͒ In addi- tion, one would have to characterize the ‘‘load response’’ of 2␮Qsin␾ the clock mechanism to obtain the coupling strength. ␨Ϫ1/␨ϭ . ͑43͒ 1ϩ␣2ϩ␣␮Q By contrast, we can successfully characterize the cou- pling of our thermoacoustic engines, given the of the Finally, in the limit ␾→␲, and with the approximation ␨ analytic results and our experience at making acoustic and ϩ1/␨Ϸ2 ͑which is true for our experimental system, but is thermodynamic measurements. To illustrate how difficult the not generally true for all systems far from the center of the problem can become, however, and why it often is not worth ‘‘capture band’’͒,wefind the effort to make quantitative analyses of mode-locked sys- ␾ 1 1ϩ␣2ϩ␣␮Qץ ␻ tems, we will consider two of many possible subtle compli- ϭ ͫ ͬ . ͑44͒ ␻ ␲␮2Q 1ϩ␣/␮Q⌬ץ cations: external dissipation on the case and heat leak be- ␲ tween the engines’ hot ends. Recall from Eq. ͑19͒ that without external resistance, the -␻ϭ1/␲␮2Q, so we may view ev⌬ץ/␾ץ(sensitivity (␻/␲ 1. External dissipation erything in the square brackets on the right side of Eq. ͑44͒ So far we have treated the coupling between the engines as the effect of external resistance on sensitivity. Figure 3 as purely reactive, but we know the actual coupling imped- shows this multiplying factor as a function of ␣, for a num- ance must be richer in detail, and in particular it must contain ber of different values of ␮Q. some resistance. One possible consequence of external resis- The behavior of these is not immediately intui- tance ͑friction in the engine mounts, etc.͒ is that it may dis- tive. For strong coupling ͑large ␮Q) the addition of dissipa- courage the ␾Ӎ0 mode. In an extreme limit, imagine that tion increases the sensitivity, making the system more the case mass becomes negligible and the only impedance is weakly coupled; but for systems that are weakly coupled a dashpot connecting the case to a rigid wall. Obviously the already, the addition of small amounts of external damping mode that encourages maximum motion of the dashpot will actually enhances the coupling, quite dramatically. ͑For our suffer more damping than the mode that encourages mini- acoustic engines, this suggests that adding a small real com- mum motion. This may help explain why the in-phase, ␾ ponent to the coupling impedance results in a phasing, be- Ӎ0 mode was rarely observed by us ͑or Huygens͒. There are tween the pressure at the endcaps of the resonator and the other, more subtle consequences of adding resistance to the case motion, that is more favorable to energy transfer be- coupling, however. tween the engines.͒ Each has a minimum at ␣ϭ1 Consider the mass-and-spring model of Fig. 1. Imagine Ϫ␮Q,if␮Qр1. Only after the resistance becomes fairly connecting one side of a dashpot with mechanical resistance large does the coupling strength begin to diminish again. ␾ ⌬␻ Rc to the case M, and the other side to a rigid wall, so that This could have quite a noticeable effect on any vs ϭ ϩ ␻ ϭ␻ ϩ␣ the coupling impedance Zc Rc i M M(i ), where data. ␣ϭ ␻ ͑ ͒ Rc / M. The additional dissipation alters the analysis In contrast, Eq. 43 is only a small perturbation away ␻2␣ ͑ ͒ presented in Sec. II A. A new term i Xc is added to the from Eq. 17 , for small external resistance. In addition, the

594 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 594 perturbation appears only as a multiplicative constant, so it is virtually impossible to tell from ␨ vs ␾ data whether there is external dissipation, as opposed to whether one’s values for ␮ and/or Q are inaccurate. We have not pursued this much further, or tried to de- vise a reliable method for measuring Rc . In principle, one can determine it from measurements of case motion; that is, the phase between the motion of the case and the pressures inside may indicate whether the pressure forces are driving a reactance or a reactance and a resistance. Based on such measurements, we estimate that for our system, ␣ ϭ ␻ ϳ Rc / M 0.1, a plausible value.

2. Heat leaks between engines We now briefly discuss another : heat flow- ing between neighboring prime movers due to a temperature FIG. 4. Experimental setup for exploring the behavior of coupled thermoa- difference that develops as a result of mass coupling. Imag- coustic engines. The insulation surrounding the prime movers and the ine that in a given locked state, engine I becomes ‘‘loaded,’’ temperature-control coils is not pictured. feeding power into engine II, which is then ‘‘driven.’’ The ˙ ץ 2 ץ amplitude P in engine I will fall and its hot temperature T I H,I have very high P / WL , and hence high Q, implying that will rise; conversely, PII will rise and TH,II will fall. If the they would allow very little phase shift for a given difference ˙ two hot ends have some thermal contact, a heat leak Qcross in natural frequency ͓see Eq. ͑19͔͒; this behavior is indeed ϭ Ϫ UA(TH,I TH,II) will flow between them, where UA is observed in simulations. However, the engines allow large some effective heat-transfer coefficient. This has the effect of differences in pressure amplitude for small frequency differ- draining more power from engine I and feeding more into ences. engine II, making the amplitude imbalance between the en- Having given the reader a taste of the challenges in- ˙ ץ 2 ץ gines that much greater. Hence, it makes P / WL greater volved in making a good mass-locking measurement, we will than it would otherwise be, making Q larger and enhancing proceed to describe our experiments with small, mass- the coupling. This additional coupling will not be apparent coupled thermoacoustic engines. when making a load measurement on a single engine, unless great care is taken to duplicate the conditions of the coupled IV. APPARATUS AND EXPERIMENTS state ͑e.g., pump the gas out of the other engine so it cannot start, and manipulate its heat input so it always has the TH To explore mass-locking of acoustic engines experimen- that it would have in the locked state corresponding to the tally, we use two nearly identical thermoacoustic engines first engine’s P͒. which are rigidly attached side by side ͑these are the same This also points to a larger issue: the effect of different engines that were used for a portion of the acoustic coupling modes of operation on engine coupling. Engines may be run work described in Ref. 3͒. Two neighboring ends of the en- with constant power, constant hot-end temperature, or some- gines are connected by a coupling duct which can be inserted thing in between ͑such as constant pressure amplitude, or or removed from the system by means of valves, enabling constant power delivered to a load, etc.͒ The general prin- the system to be acoustically coupled, or not, as desired. ciples developed so far should still apply; that is, if one can Each engine consists of a section of duct of varying cross find the ‘‘Q’’ that characterizes the engines in their typical section ͑the resonator͒ connected to a thermoacoustic prime mode of operation, along with the effective moving gas mass mover, by which we mean an electric heater, a water-cooled and the case mass, one should be able to estimate the cou- heat exchanger, and a stack of parallel plates between them; pling strength. Care must be taken, of course, to always re- these sustain the oscillations in the resonator when the en- examine the energy balance whenever the mode of operation gine is running. is changed. For instance, consider two mass-locked engines Figure 4 shows the essentials of the experimental setup, whose hot-end and cold-end stack temperatures are held con- and some details of the thermoacoustic prime mover hard- stant. This example is of interest because material properties ware. Note that the engines share a water jacket that cools usually limit how hot an engine may be run, and the need for their cold heat exchangers, and both are welded to a common maximum power density and efficiency may require that the plate at the other end. The engines are also joined by two engine be run at this upper limit. If one such engine experi- narrow, 50-cm-long plates that are welded between the 50- ences an extra load as a result of being mass-coupled to cm-long ‘‘fat’’ sections of the resonators. A section of each another engine, its hot temperature will tend to rise and its resonator is wrapped with copper refrigeration tubing, which oscillating pressure amplitude will fall; the temperature con- circulates water from a temperature-controlled bath. Thus the trol will decrease the power flowing to that engine to main- gas temperature in each engine can be varied independently, tain its temperature at the upper limit, lowering the pressure enabling control of ⌬␻. A pressure sensor is mounted at the still further. Thus one might expect that such engines would end of each resonator where it joins the coupling duct. We

595 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 595 take the signal from the sensor in resonator I ͑pressure sensor ͒ I to represent pI(t), and that from pressure sensor II to rep- resent pII(t). Using pI as a reference to a lock-in amplifier and using pII as the input signal allows a direct measurement of ␾. Independent monitoring of pressure sensors I and II gives PI(t)andPII(t). Accelerometers on the cases measure case vibration in the axial (x) direction, and numerous ther- mocouples ͑not pictured͒ provide temperature data on the gas near the hot end of the stack, the cooling water, the water in the coils wound around the resonators, and the gas inside the resonators. Multimeters monitor the input voltages and currents to the tube heaters that power the engines. The ac- celerometers are used to verify that the two engines, welded together, act like a rigid structure, and that vibrations are indeed canceled when the phase between the engines ap- ␲ FIG. 5. Load responses of the engines used in the mass-locking experiment. proaches —we find that cancellation of case vibration is at The squares are engine I data and the circles, engine II; the lines are linear least 99.5% complete when ␾ϭ␲Ϯ0.002. The engines are fits. Each data set on a given engine was taken with the neighboring engine ϭ hot, but not running, to approximate the heat leak that exists when both filled with xenon, with the mean pressure pm 303 kPa; they run at 50 Hz, in a room temperature environment ͑rejecting engines are running. heat at 300 K͒. The combined case mass of the engines, including water in the heat exchangers and such, is M at the end of the engine, p is the pressure in the tank, and ␪ Ӎ11 kg. T is the phase by which p leads p . To assist us in interpreting our data, we simulate the T In these measurements, we not only had to contend with system using DELTAE. We use a single simulated engine, heat leaks between the engines, as discussed earlier, but also ␮ without moving endcaps, to help us estimate and Q for the ordinary heat leaks from the engines to the ambient. Rather system; we use a pair of simulated engines, whose combined than attempt to subtract the ambient heat leaks from the elec- case mass moves in response to the oscillating pressure, to tric power going into the engines and manipulate the electric compare simulated mode-locked states with measured ones. power to maintain constant heat input, we chose to simply The simulated mode-locked system involves a few approxi- maintain constant electric power, and characterize the load mations; for instance, the forces on the heat exchangers and performance of the engines as they were, leaks and all. Fig- stacks are neglected, and the changes in cross-section are all W˙ is fairly linear inץ/P2ץ ure 5 shows the results; evidently treated as abrupt ͑including the truncated cones shown in L ͒ the region studied, but the data have sufficient scatter that a Fig. 4 . Nevertheless, we expect that the agreement between higher-order fit is probably not warranted. The data for each experiment and simulation should be reasonable. engine were taken over the range of pressures which that The single simulated engine is used to estimate w(x), engine experienced when the data on locked states were the pressure in the engines. The only pressure taken. Of interest is the difference in slopes between I and II; reading taken in each experimental engine is at the end of the apparently the two engines do not perform identically, de- ͑ ͒ resonator the ‘‘cold end’’ ,asshowninFig.4,sowetrust spite being constructed ‘‘identically.’’ One difficulty in pre- DELTAE’s linear acoustics to estimate what the pressure dis- dicting the performance of these particular engines, and in tribution in a given geometry should be. Combined with our modeling them, is the use of tubular heaters in the prime knowledge of A(x), this allows us to numerically carry out movers ͑see detail in Fig. 4͒. These heaters are robust and ͑ ͒ ͑ ͒ the integrals in Eqs. 32 and 35 . easy to bend into a desired shape, but are not as well under- ˙ ץ 2 ץ To obtain Q,wealsoneed P / WL . To measure this stood thermodynamically as standard shell-and-tube or quantity experimentally, we use a precision leak valve in parallel-plate heat exchangers. series with a small cylindrical ‘‘tank’’ of volume Vϭ150 ϭ ϭ ¯ We find that QI 73 and QII 50, with Q, the average, 3 cm to form an acoustic RC load, and attach this load to the being 62; by simulating this measurement in DELTAE, we end of one engine resonator at a time, observing changes in find Qϭ61. Because of the difficulties in precisely modeling the end pressure of that engine as the valve is gradually the hot ends of these engines, and the large difference be- opened. The other engine is pumped out for this measure- tween QI and QII , this striking agreement must be a bit ment, so it cannot start; some heat is still applied to it, how- coincidental. The difference in measured Q between the en- ever, to maintain it near the temperature it would have if it Ϫ ¯ 2 gines adds a slight offset of order ͓(QII QI)/Q ͔ to Eq. were running. The actual power dissipated at each valve set- ͑17͒. We do not treat the question of the offset rigorously, ting is estimated by using a pressure within the but merely measure it and subtract it from our simulated and tank, along with the one at the end of the engine, to measure theoretical results. the pressure difference across the valve. This pressure We may also use an experimental observation to obtain difference and the tank volume V are used to an approximate value for ␮, the effective mass ratio. Observ- 13 ˙ calculate the dissipated power according to WL ing the engines when they are coupled but not mode-locked, ϭ␻ ͉ ͉͉ ͉ ␪ ␥ V p pT sin /2 pm , where p is the oscillating pressure and measuring the frequency and phase of both engines as a

596 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 596 ␨Ϫ ␨ ␨ϭ FIG. 7. Amplitude imbalance 1/ , where PII /PI , versus phase dif- FIG. 6. Frequency as a function of the phase between two mass-coupled ference ␾ of mass-locked state. Circles are data points, crosses are DELTAE engines that are on the verge of locking. simulations, and the broken line is a fitto␨Ϫ1/␨ϭ2␮Qsin␾ϩC , with ␮Q and C free parameters. The solid line is the same equation, but using our estimated ␮ and Q. The difference in their slopes at ␾ϭ␲ is about 15%. function of time, one may obtain a plot of frequency versus The DELTAE points have had the offset C subtracted from them as well. phase for the engines as they beat. Such a plot is shown in Fig. 6; note that the frequencies are their lowest near ␾ already in place. An extra 130 kg of mass is clamped to the ϭ␲,3␲,5␲, . . . and their highest near ␾ϭ0, 2␲,4␲,..., engines during the acoustic measurement, to remove all ef- consistent with the idea that a moving case shifts frequencies fects of mass-coupling. Subsequently, during the mass- up. By comparing the high and low values, we find that the locking measurement, after the engines have established a ͉⌬ ͉ Ϸ 2ϭ 2 ϩ␮ frequency shift f / f 0.002; if we assume f f 0(1 ), stable mass-locked state we momentarily open the acoustic then ␮Ϸ0.004. Since this includes the moving gas in two coupler valves and record the phase, which can then be re- ␮ ␮ ␾ engines, we may guess that for one engine, that is, the lated to a frequency difference. Figure 8 shows a plot of A , ϭm/M to use in Eqs. ͑17͒ and ͑18͒,is␮ (expt.)Ϸ0.002. the phase at which the system is mode-locked due to acoustic Using our theory, together with our DELTAEmodeltofind coupling, versus the normalized frequency difference ϭ ⌬␻ ␻Ϸ ⍀ Ϫ⍀ ␻ ␾ ϭ␲ w(x), we find that the effective mass m 0.024 kg for our / ( II I)/ . Near A , the data are linear, and engines; with Mϭ11 kg, ␮ (theo.)Ϸ0.0022. we use the slope in this region to obtain the relationship ␨Ϫ ␨ ␾ ␾ ⍀ Ϫ⍀ We are now poised to predict 1/ versus for our between and II I . We have chosen a relatively engines; we choose ␮Qϭ0.0022ϫ62ϭ0.135 as our ‘‘theo- narrow-diameter coupler for this part of the work, so that the retical’’ coefficient in Eq. ͑17͒. This equation can be com- acoustic coupling is fairly weak ͑although not as weak as the pared to actual phase differences and amplitude ratios at mass coupling͒. Thus the phase of the acoustically locked various locked states for our engines. We found that the en- state is a sensitive measure of frequency difference. We al- gines would readily mode-lock, and the phase could be var- low the engines to lock through mass coupling ͑i.e., with the ied easily by changing the temperature of one of the cooling acoustic coupler’s valves closed͒, record the steady-state am- coils; however, we found that the states were not particularly plitudes and phases, and then throw the valves open momen- ␾ ⍀ Ϫ⍀ stable, but very sensitive to small changes in temperature. tarily and record the phase A , thus obtaining II I .As Only by very carefully ‘‘teasing’’ the system could it be made to settle at a particular state ͑largely because the hot temperatures need to equilibrate to establish a stable state͒. Figure 7 shows a comparison between theory and experi- ment. Considering all the approximations that have been made to get to this point, the agreement between the three methods is remarkably good. We also try to compare theory and experiment for Eq. ͑19͒, but here we must grapple with the difficulty in measur- ing ⌬␻. The most straightforward way to measure the dif- ference in frequency between two coupled engines is to de- them and measure the beat frequency; however, we know from Sec. III that this frequency difference is not nec- essarily the ‘‘natural frequency difference’’ required by theory. In addition, when the engines are decoupled, the hot- end temperatures immediately begin to shift, making the measurement of a long beat period somewhat difficult. We FIG. 8. The phase of the locked state when the engines are acoustically have chosen an indirect, but simpler, method to find ⌬␻ coupled, achieved when the coupler valves in Fig. 4 are open, versus the Ӎ⍀ Ϫ⍀ ⍀ Ϫ⍀ frequency difference estimated by measuring the beat period when the II I . We record phase versus II I for the engines valves are closed. In this measurement, an extra 130 kg of mass is clamped when they are acoustically coupled, using the coupler that is to the engines, to discourage any mass-coupling.

597 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 597 as we estimate for our system to make the experimental and theoretical curves coincide. We believe inability to measure the ‘‘true’’ value of ⌬␻/␻ is the biggest source of error. Since ⌬␻/␻ is of mostly theoretical interest anyway, we consider what effect objective changes might have on our system. The DELTAE model, which agrees well with theory and experiment as demonstrated by Fig. 7, suggests that a 0.03-K change in the resonator gas temperature of one en- gine, within the section wrapped in copper tubing, would cause a change in phase angle ⌬␾ϭ␲/16. The measurement of gas temperature in our experiment was too coarse to verify this, but if we assume it is correct, it demonstrates again how very sensitive mass-coupled engines are, compared to the acoustic coupling necessary to cancel vibration. For com- parison, consider an acoustically coupled system of two en- FIG. 9. The phase difference of the mass-locked state versus the frequency gines, designed so that 95% of the case vibration is canceled difference estimated from the corresponding acoustically-locked state. The when ⌬␻/␻ϭ0.01. This implies ͉⌬␾͉ϭ͉␾Ϫ␲͉р0.05 when quantity ␣ is defined in Sec. III A 1, and represents the degree of external ⌬ the difference in resonator gas temperatures is T/Tm dissipation in the system, such as that caused by friction in the supports. ϭ ⌬ ϭ 0.02, or T 6 K if the mean temperature Tm is about 300 ␲ ⌬␾ ⌬ Ϸ K. For such a system, (Tm / ) / T 1; for our mass- ␲ ⌬␾ ⌬ Ϸ locked system, (Tm / ) / T 600. ␾ expected, we find that A drifts, since the hot temperatures In conclusion, we have shown that the Huygens effect, ⍀ Ϫ⍀ drift as soon as the valves are closed, and thus II I is a or mass-coupling, is sufficiently weak in thermoacoustic en- moving target. We find that measuring the phase about 5 s gines that it may be ignored in many circumstances, although after the valves are opened gives a repeatable, if not wholly a precise measure of its weakness will often be difficult to ␾ accurate, value for A . obtain. It is also apparent that the natural mode for the en- Figure 9 shows our mass-locked ␾ vs ⌬␻/␻ data, com- gines to mass-lock has the correct phase relationship to mini- pared with two theoretical predictions—one assuming no ex- mize motion of the common mass. This is further encourage- ternal dissipation, and another assuming moderate external ment that mass-locking will not interfere with the canceling dissipation. We were able to achieve mass-locking only be- of engine vibrations by acoustic coupling. tween 3␲/4Ͻ␾Ͻ5␲/4; this is in contrast to the acoustically There are some circumstances, of course, where the coupled engines studied in Ref. 3, which would lock between mass-coupling could be significant. Engines that use a liquid ␲/2Ͻ␾Ͻ3␲/2. We may use our theory to predict the cap- as the working fluid ͑such as a liquid-sodium thermoacoustic ture bandwidth; starting with Eq. ͑18͒, we ask what angle ␾ engine6͒ could have ␮ϳ1. This could also be true of acous- makes ⌬␻ a maximum. This is equivalent to seeking ␾ such tic engines whose resonators include moving parts, such as that d(sin␾cos␾)/d␾ϭ0. This occurs when ␾ϭ(2n magnets for generating electricity. This strong coupling ϩ ␲ ϭ ␾ ␾ϭ 1 1) /4, n 0,1,2,...,andhencewhensin cos 2.For might in fact be an advantage, if the proper suspension or a system whose band center is nominally at ␾ϭ␲, then, we supports for the engines can encourage them to lock in an- predict tiphase, thus lessening or eliminating the need for any acous- tic coupling. ⌬␻ ␮2Q ␲ Ϯͩ ͪ ϭϮ at ␾ϭ␲Ϯ . ͑45͒ ␻ 2 4 max ACKNOWLEDGMENTS The experimental phase limits agree well with this predic- tion, but the predicted frequency bandwidth is much nar- The authors wish to thank Bob Keolian and Bob Ecke rower. for teaching us about mode-locking; Bill Ward, Scott Back- The shapes of the experimental and theoretical curves haus, and Bob Hiller for useful discussions; and David Gard- have some qualitative similarities, but as expected ͑recall ner for assistance with the experiments. We are also grateful Fig. 2͒, the slope of the experimental curve is much less than to Andrew Piacsek ͑of Central Washington University͒ for ␾ providing an English translation of Huygens’ letter. This that predicted by theory. We found that measuring A ‘‘im- mediately’’ ͑within 0.5 s͒ after the valves were opened ap- work has been supported by the Office of Basic Energy Sci- peared to approximately double the slope; however, the ences in the U.S. Department of Energy. valves were operated by hand and took several tenths of a

second to open, so these data are of dubious value. While the 1 ͑ ͒ C. Huygens, in Ouevres Completes de Christian Huyghens,editedbyM. simplest theory no dissipation predicted that the mass- Nijhoff ͑Societe Hollandaise des , , The , coupling in our system should have been about 25 times 1893͒,Vol.5,p.243͑a letter to his father, dated 26 Feb. 1665͒. weaker than the acoustic coupling, the data suggest it is only 2 G. W. Swift, ‘‘Thermoacoustic natural gas liquefier,’’ in Proceedings of about 10 times weaker. External dissipation brings these the DOE Natural Gas Conference, Houston, TX, March 1997. 3 P. S. Spoor and G. W. Swift, ‘‘Mode locking of acoustic resonators and its numbers into closer agreement, as shown by the dashed line application to vibration cancellation in acoustic heat engines,’’ J. Acoust. in Fig. 9, but it would require three times as much dissipation Soc. Am. 106, 1353–1362 ͑1999͒.

598 J. Acoust. Soc. Am., Vol. 108, No. 2, August 2000 P. S. Spoor and G. W. Swift: The Huygens phenomenon 598 4 A. B. Pippard, The Physics of Vibration (Omnibus Edition) ͑Cambridge U. Applications ͑Acoustical Society of America, Woodbury, New York, P., Cambridge, 1989͒,p.392. 1989͒,p.53. 5 B. Van der Pol, ‘‘The nonlinear theory of electric oscillations,’’ Proc. Inst. 11 W. C. Ward and G. W. Swift, ‘‘Design environment for low amplitude Radio Engineers 22,1051͑1934͒. thermoacoustic engines ͑DeltaE͒,’’ J. Acoust. Soc. Am. 95, 3671–3672 6 G. W. Swift, ‘‘Thermoacoustic engines,’’ J. Acoust. Soc. Am. 84, 1145– ͑1994͒. Fully tested software and user’s guide available from Energy Sci- 1180 ͑1988͒. ence and Software Center, US Department of Energy, Oak 7 A. B. Pippard, The Physics of Vibration (Omnibus Edition) ͑Cambridge Ridge, TN. To review DeltaE’s capabilities, visit the Los Alamos ther- U.P., Cambridge, 1989͒, Chap. 12. moacoustics web site at http://www.lanl.gov/ 8 P. M. Morse, Vibration and Sound ͑American Institute of Physics, New . For a beta-test version, contact [email protected] ͑Bill York, 1986͒, pp. 104, 313. Ward͒ via Internet. 9 J. W. Strutt and Lord Rayleigh, Theory of Sound ͑Dover, New York, 12 N. Rott, ‘‘Thermoacoustics,’’ Adv. Appl. Mech. 20, 135–175 ͑1980͒. 1945͒,Vol.1,Sec.88. 13 A. M. Fusco, W. C. Ward, and G. W. Swift, ‘‘Two-sensor power mea- 10 A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and surements in lossy ducts,’’ J. Acoust. Soc. Am. 91, 2229–2235 ͑1992͒.

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