Reprinted from

Wi"I~SSWQridJanuary and February 1956

LOUDSPEAKERENCLOSUREDESIGN

by E. J Jordan

I

GOODMANS INDUSTRIES. LTD. AXIOM WORKS, WEMBLEY, MIDDX. rei: WEMbloy 1200

LOUDSPEAKERDESIGN

ENCLOSURE·By E. J. JORDAN*

I.-Alternative Methods Their Advantages and Disadvantages

•

IN the first part of this article the theory under-lying the principal types of loudspeaker enclosure isreviewed, and formula associated with the majordesign factors are given.

This will be followed later by a discussion ofsome recent developments in which an improvedlow-frequency performance has been achieved incabinets of relatively small volume.

THE loudspeaker enclosure .has the task of doingsomething (useful or otherwise) with the low-frequency radiation from the rear of the loudspeakercone, which would otherwise cancel the radiationfrom the front of the cone.

Before examining various methods of overcomingthis, let us establish the principles on which ourfuture arguments will be based.

We shall regard the moving parts of a loudspeakeras a mechanical system which at low frequenciesis analogous to an electrical circuit, as shown in itssimplest form in Fig. 1.

The complete analogy is revealed by an examina-tion of the electrical and mechanical equations viz.

d2S dSForce=M- +R- +SKdt2 di

d2Q dQ QE.rn.f. = Ldt2 + R dt + C·

where M = mass, L = inductance, S = displace-ment, Q = charge, C = capacitance, K = stiffnessand R = resistance.

There are, of course, other analogies,but the abovelends itself more readily to discussions of the pro-posed nature.

Assume for a moment that the loudspeaker ismounted on an infinite baffle. It WIll be seen,that the power developed in Ra (Fig. 1) is afunction of the current through it. Comparing

the above equations it will be seen that i (= d2).is analogousto the cone velocityv (= ~~). Hence

it is the cone velocity,and not the displacement, thatis responsible directly· for the radiated outputpower, v2Ra'

From this it would seemthat, if the radiated poweris to be independent of frequency, the resistive

WIRELESS WORLD, JANUARY 1956

components of the circuit should be high relativeto the reactive components. This is not so inpractice, since at frequencies where the wavelengthis longer than twice the cone diameter the value ofR, falls as the frequency is lowered. The reactanceof M, also falls, however, and the increasingvelocityresulting from this may largely compensate for thefall in R, to the extent that the radiation remainssubstantially constant, down to a frequency where

1wMc - -C -+ O. Here the velocity of the cone

w c

rises sharply, and is limited only by Rd' R, and Ra'This produces an increase in the radiated power andis the resonant frequency of the loudspeaker.

Belowthis frequency, the impedance of the circuitrises as the frequency falls, due to the reactance ofCc, consequently the radiation falls very sharply.The resonant frequency may thus set the limit tothe low-frequency response of the loudspeaker.

The above may be shown by considering theexpression for the radiated power at the frequenciesbeing discussed:

Force2 21TpJ2 •P = v2 Ra =-Z2 . -- . (1Tr2l, where r IS

M Cthe radius of the cone.

.Above resonance if RM< < XM (mass)

P Force2 f2Cl: X2

M•

This is the condition of mass control, and sinceXM2 Cl:J2, P is independent off.

Above, at, or below resonance, if Rill > >XM

P Force2 f2 f2Cl: RM2 . Cl:

This is the condition of constant velocity, and Pfallswithf at the rate of 6dB/octave.

Belowresonance if RM < < X~{(stiffness),

P Force2 f2 f4Cl: X

M2' Cl:

This is the condition of constant amplitude and Pfalls with f at the rate of 12dB/octave.

Above resonance if RM is comparable to XM

P Force2 f2Cl: z..:2 •

and P falls with frequency at a rate determined byJ2

the ratio =--;;--,--==---;:-RM2 + XM2.

In all cases the radiation resistance is small

*Goodmans Industries Ltd.

3

relative to the total. mechanical impedance of thesystem; its effect on the velocity has therefore beenneglected.

So far, it has been assumed that the loudspeaker ismounted in an infinite baffle. The analogous circuitis similar to that of a loudspeaker mounted in freeair, except that the baffle produces a large increasein Ra and a small increase in La'

It is very important to realize that any baffle orenclosure may be represented in the analogy by aseries impedance Z, which will tend to reduce thecone velocity, but, depending upon the nature ofthis additional impedance, partial compensation maybe effected by resonant phenomena over at leastpart of the low-frequency range.

The effective mechanical impedance presented tothe cone, due to any acoustical impedance ZA isgiven by: Zy = ZA (1TT2)2where ZA is the vector sum of Z; and the acousticimpedance due to the mounting.. At low fre-quencies

Z R . L 2'TTpf2 • O.85pwr = r + JW r i=::::! -- +.7 ---

C 'TTr

Impedance Curves.-A very convenient way ofmeasuring the effects of the enclosure on the outputof the loudspeaker, is to plot the impedance/frequencycurve of the loudspeaker, when housed in the en-closure. If a base line is drawn at a value equalto the clamped impedance of the voice coil, then theimpedance curve relative to this line is directlyproportional to the velocity of the cone.

The relationships .between the electrical impedance(ZE) the mechanical impedance (Zy) and thevelocity (v) of a loudspeaker system, are as follows:where B = flux density in the magnet system,I = length of. voice coil conductor enveloped byflux, i = current flowing in coil.

Back e.m.f. due to the motion of the coil:B212i

E IX Blv = --z,Motional impedance of the coil:

E B212Zm = T IX Zy

Total electrical impedance:ZE = Z,," + Zm

where Z,," is the clamped impedance of the voice coil.1

From above v Ct Z Ct Zmy

If the component -parts of Zy are expressed in c.g.s.

Fig. I. Simplified electrical analogue of mechanical propertiesof a moving-coil loudspeaker.

terms then Zm will be in electro-magnetic units.Impedance curves often give a far more accurateassessment of the performance of an enclosure thanpressure response curves, since the latter depend notonly on the cone velocity but, in the case of ventedenclosures, upon the port radiation as well. Pressurecurves are also greatly. affected by diffraction andwhile they are invaluable in demonstrating the overallradiation from a loudspeaker system, they do notshow clearly the action of the various acousticcomponents due to the enclosure on the loud-speaker cone.Wall Mounting.- The nearest practicable approachto the infinite baffle condition is by mounting theloudspeaker in a wall e.g. a partition wall betweentwo rooms.

This method of baffling a loudspeaker ensurescomplete separation between the front and rearradiation of the cone and imposes a relatively lowmechanical impedance to the cone velocity. Theextent of the low-frequency response is limitedby the resonant frequency of the cone.

For wall mounting it is therefore desirable' to usea loudspeaker having a low-frequency, highly-damped cone resonance. The damping in thiscase will be mainly electromagnetic, i.e. a highvalue of Rt:I in the analogy, tending to produceconstant velocity conditions and resulting in a fallinglow-frequency response, as we have shown. Sinceunder these conditions the cone displacement atresonance does not exceed the level required tomaintain the velocity constant, a considerable amountof bass lift may be applied from' the amplifier to .compensate for this loss at low frequencies. Thebass lift required commences at the frequency atwhich the wavelength is equal to twice the conediameter, and has a slope which may be determinedeither aurally, or from the expressions previouslygiven, the latter being possible only when thenecessary loudspeaker parameters are known.

1- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --- - - - - - - - - - - - - - - -- ------ - - - ----- - - - -- --- - - --- - - - -- - -- -- - - - - - _. - ---

l J :

: c velocity of sound in air. SYMBOLS X,b = reactance of air in closed :i C, compliance of air in closed R. R, ('TTr2)2. . cabinet. i: cabinet. Re resistance due to friction in XM = total mechanical reactance. :! C, compliance of cone suspen- cone. ZA total acoustic impedance. !i sion, R, mechanical resistance due to Z, acoustic radiation impedance. :IF force applied to cone. voice coil damping. Z, impedance. due to loud- :! k tillc = wave constant. RM total mechanical resistance. speaker mounting.! L. L, ('1T!'2)2. R, radiation resistance. ZM total mechanical impedance.:L, acoustic radiation mass R. viscous resistance of vent. Z,. motional impedance of coil.i Me = mass of cone system R. total resistance component of!l coefficient of shear viscosity.! M. = mass of air in vent. vent = R, + R.. p density of air.:P radiated acoustic power. v velocity of cone. W 21ff.

! C.g.s. units for mechanical and acoustical quantities, and e.m. units for electrical, have been assumed throughout. I:. ~ ~ ---- J

4 WIRELESS WORLD, JANUARY 1956

A consideration which should be borne in mind,particularly in the case of wall mounting, is that theaperture in which the loudspeaker is mounted willbehave as a tube of length equal to the thicknessof the wall or baffle, and in so doing will exhibita number of harmonically related resonances andanti-resonances, causing irregularities in the trebleresponse. There are, of course, a number of obviousremedies for this, e.g. bevelling the edges of theaperture or mounting the loudspeaker on a sub-baffle.

Finite BafHes.-If the baffle is finite, at somelow frequencies, depending on its size, back-to-front cancellation will occur; and the limitingbaffle size for a given low-frequency extension is:

l=~2f

if the baffle is rectangular and I is the length of -thesmallest -side,

If the bass response is to extend down to a reason-ably low frequency, the necessary baffle size will berelatively large, e.g. a square baffle suitable forreproduction down to 60 c/s will have a side of9.42ft. A loudspeaker acting as a treble unit in acrossover system should be mounted on a baffle largeenough to work down to half the crossover frequency.

For the sake of convenience, baffles often take theform of open-backed cabinets. In such cases, inaddition to the normal baffle action, the cabinet willbehave, more or less, according to its depth, as atuned pipe, and will exhibit a number of harmonicallyrelated resonances, the lowest of which will approxi-..mate to:

cf 2 Cl + 0.85r)

where I is the depth of the cabinet, r = ylA/7Tif Ais the area of the open back.

It is these resonances that contribute' to theunnatural " boomy" quality evident in manycommercial reproducers.

Closed Cabinets.-Alternatively a method ofpreventing back-to-front cancellation, is to completelyenclose the rear of the loudspeaker cone. Underthese conditions, however, the enclosed air willapply a stiffness force to the rear face of the cone.

This may be represented by a mechanical reactanceXcb the value of which is given by:

pc2 (7Tf2) 2

",Vwhere ",2 = piston area of cone and V = volumeof enclosure. .

In the analogy, this reactance appears as a seriescapacitance as shown in Fig. 2.

In order not to raise the cone resonance unduly,the value of C, must be large relative to Cc. Since, fora given loudspeaker system, Cb is the only variable,it must be large.

It has been found that, for a 12-in loudspeakerhaving a fundamental cone resonance at 35 cl»,the volume of an enclosing box would need to be ofthe order of 12 cu ft for its reactance to be sufficientlylow not to impair the low-frequency performance ofthe speaker.

There are a number of factors in the design ofloudspeaker enclosures which should be considered.

WIRELESS WORLD, JANUARY 1956

Fig. 2. Analogue circuit of m.c. loudspeaker in a closedcabinet.

1:::0:Fig 3. Analogue circuit of m.c. loudspeaker in a ventedcabinet.

These are. common to most types of enclosure andare:

Shape of the Enclosure.-As the frequency islowered the radiated wavefront from the loud-speaker cone tends to become spherical, consequentlythe boundary edges of the loudspeaker enclosureconstitute obstacles in the path of the wavefront.This results in (a) bending of the wavefront (diffrac-tion), and (b) secondary radiation from these edges.This secondary radiation will produce interferencepatterns, causing irregularities in the frequencyresponse of the system. ' .

These effects are largely dependent on the shapeof the enclosure, and will be smallest for a sphericalenclosure, and greatest for a cube. Since the cabinethas to be a presentable piece of furniture, there arecertain limitations on its shape. Fortunately, however,the effects of diffraction are not very serious, and itis not difficult to reach a compromise.

Corner Position.-Consider a source of soundthat is small compared to a wavelength and situatedin free space. The radiation from this source willbe of equal intensity at a given distance in all direc-tions, i.e. spherical.

If a large flat wall is placed near the sound source,then the total radiation will be concentrated into ahemisphere, and its intensity will then be doubled.Similarly, if a second wall is placed near the soundsource at right angles to the first, the total radiationwill be concentrated into one-quarter of a sphereand its intensity will be four times greater. A thirdwall at right angles to the other two will increase theintensity eight times.

A loudspeaker standing in the corner or the roommay, at medium low frequencies, be regarded assimilar to the second case, and approaching thethird case as the frequency calls to a point wherethe wavelength is much greeter than the height ot thespeaker above the floor.

Construction.-At frequencies where the wave-length is comparable to the internal dimensions ofthe enclosure, reflections between inside wall faceswill occur, resulting in standing-wave patterns,which in turn will produce irregularities in thefrequency response of the system.

5

These standing waves may be considerably re-duced (a) by lining the enclosure walls with soft 'feltor wool thus providing absorption at points of maxi-mum pressure, (b) by hanging curtains of the samematerial near the centre of the enclosure, therebyintroducing resistance at points of maximum velocity.

A further point to be considered is that thematerial (usually wood) trom which the enclosureis made, possessing both mass and compliance, willbe capable of movement and will resonate at one ormore frequencies and, in so doing, will (a) behaveas a radiating diaphragm and (b) modify the airloading on the cone, both of which will produceunwanted coloration in the reproduction. Theenclosure should, therefore, be made of as thickand dense a material as possible.

Vented Enclosures, Reflex Cabinets.-Onemethod of overcoming the disadvantage of the I

closed cabinet, is to include in the cabinet wall anorifice or vent.

An enclosure, suitably vented, will apply to therear of the loudspeaker cone an impedance whichoffers the cone a maximum degree of damping at, ornear, its resonant frequency and the radiation fromthe vent around this frequency will be more or lessin phase with the frontal radiation from the cone, i.e.,the back radiation is inverted. Before we describethe nature of this impedance, we will describe theHelmholtz resonator, the principle on which thedesign of vented and reflex cabinets is based.

For the benefit of readers not familiar with thisresonator, it consists simply of a partially enclosedair cavity having a communicating duct to- theoutside air;

An enclosed volume of air will have a stiffnessreactance equal to pc2/ wV.

The air in the duct will move as a homogeneousmass, the reactance of which is given by:

pl'w1T1' " 2

where l7r v 2 is the cross-sectional area, and l' is theeffective length ofthe duct.

This system will have a resonant frequency atwhich the mass of air in the duct will"move mostreadily, bouncing, as it were, on the elasticity of theair in the enclosure. This occurs when the sum ofthe reactances, which are opposite in sign, is zero.

Fig. 4. Variation of reactancewith frequency of the circuitelements of Fig. 3. XL = totalmass reactance of series sec-tion; Xc = total sti ffness re-actance of series section; BLand Bc are the mass andcompliance susceptances ofparallel section, and Xs and Xpthe total series and parallelreactances, respectively.

x

(cl.)

6

Equating the two expressionsand transposing for j,wehave

c jm:-2j = 217 V;,

which is the usual expression for the natural fre-quency of a Helmholtz resonator.

In actual fact, this is only an approximation, sincethe" full expression for the mass reactance shouldcontain a Bessel term for the load on the vent, dueto the air outside the cabinet, but in practice this issmall enough to be neglected.

Some of the air adjacent to the end of the ductmoves with the air in the duct, and thus becomesadded to it. The effective length of the duct, there-fore, is greater than its actual length. Rayleigh showsthat the increase at each end is:

8SI = - r

31T "

where r is the radius of the duct.The total effectivelength is, therefore:

16l' = I + -rv = I + 1.7rv

317

If the duct is not circular, rv = vA/1T,where A isthe cross-sectional area of the duct.

Returning now to the subject of loudspeakerenclosures, a vented cabinet containing a loudspeakerwill exhibit a resonance in accordance with theabove description, which will be reasonably inde-pendent of the loudspeaker cone resonance.

When the cabinet resonance is excited by theloudspeaker, the motion of the air in the vent willreach its maximum velocity and will be in phasewith the motion of the cone. At this frequency,therefore, the air in the cabinet will come under thedouble compressive and rarefactive forces of boththe cone and air in the vent; consequently, its effectivestiffness rises, and the resulting impedance applied tothe rear of the cone becomes much higher at thisfrequency than at any other. .

If the resonant frequency of the enclosure is madeto coincide with that of the cone, the latter receivesmaximum damping at its resonance and any peakin the radiated power at this frequency is removed.

In addition to this, the reduction in cone displace-ment results in a considerable increase in thepower-handling capacity of the loudspeaker and

~:

x IXp ,

! Bc /' "

~

..• /' -:

/' "..• "... /... /

"/

-x

(b) (c)

WIRELESS WORLD, JANUARY 1956

I

in a reduction of harmonic and intermodulationdistortion. Although the velocity and therefore thepower radiated from the cone is reduced around thisfrequency, the overallradiated power from the systemis increased considerably, due to the very high airvelocity at the vent. Unlike the cone there is nophysical limitation to the displacement of the air inthe vent..

Below the resonant frequency of the enclosure thestiffness reactance becomes high, and the systembehaves as though the air mass in the vent werecoupled directly to the mass,of the cone. At somefrequency the reactance of this combined mass willbecome equal to the stiffness reactance of the sus-pension system of the cone. A resonance will occurat this frequency, the amplitude of which willbe considerably lower than that of the initial coneresonance, and the radiation from the vent will bein anti-phase with that from the cone.

Above the resonant "frequencyof the enclosure themass reactance of the vent becomes high, and thecabinet behaves as though it were completelyclosed, presenting a purely stiffuess reactance to therear of the cone. At some frequency the combinedstiffness reactance of the cone suspension systemand the enclosure will become equal to the massreactance of the cone. At this frequency a furtherresonance will occur, and again the amplitude will beconsiderably less than the cone resonance.

Now let us consider the nature of the impedancepresented to the rear of the cone by a vented en-closure. Since this impedance rises to a maximumvalue, a parallel tuned circuit is indicated in theanalogy Fig. 3, where R" and M" are the ventcomponents.

By drawing the reactance sketches for the completesystem, we are able to see clearly the derivation of

(\ IINFINITE BAFF LE

- I VENTED

( \/ \1 \ENCLOSURE-

V

/ A Ai/\ 1\

// / '\ \ \I\.... ../ r--J '\ \~

"- t"---r-.

200

180

160

140

120

'"~ 100o

80

60

40

20

FREQUENCY (cfs)

Fig. 5. Impedance/frequency response of a loudspeaker on aninfinite baffle and in a vented enclasure.

WIRELESS WORLD, JANUARY 1956

000'1_0"

the resonant frequencies described above Fig. 4.Figs. 4(a) and 4(b) show the well-knownreactance

sketches for the series and parallel sections of thecircuit respectively. When these are added, we haveFig. 4(c) which exhibits three critical frequenciesf1> t,and f2' It will be noticed that at fl and f2 thereactance falls to zero, and at I; rises to infinity.The corresponging impedance curve, together withthe impedance curve, taken with a loudspeakermounted on an infinite baffle is shown in Fig. 5.

Whilst a reflex (vented) enclosure is much smallerthan a completely closed cabinet for a given bassextension, the reduction in size is limited by themechanical impedance it imposes on the cone, atfrequencies away from its resonance (f.). In thedesign of these enclosures it is important, therefore,to calculate the impedance over a wide range offrequencies, to ensure that this does not becomeexcessive.

To accomplish this, the various components ofthe enclosure are expressed as follows:Referring to Fig. J.

,VCb=~

"pcR,,'= R, + R,

M =L'V TrY v 2

R, is resistance due to air viscosity in vent

= V21'-pw I'17r"3

. '" ' pck2R, IS radiation resistanceof port =--

217Having already met the first two expressions, thenew symbols appearing in the second two expres-sions are: 1'-, the coefficient of shear viscosity andk = w/c, .the so-called wave number or wave con-stant..It is convenient to express all dimensions in c.g.s.terms.

The acousticalimpedance of the enclosure Zab maybe obtained from the usual expression for an L C Rcircuit of this type, i.e.

Z _ Rv -jw[CbR,,2 + M" (w2M"Cb - 1)]ab - w2 Cb2R,,2 + (wMv~ _ 1)2

where all terms are in acousticalunits.Expressing this as the modulus of the mechanical

impedance"we have:-

'IZ I - [ ,Rv2 + w2Mv2' J ( 2)2

: b - w2Cb 2 R,2 + CwMvCb -1)2 17r

At the resonance of the enclosure, the right-handexpression in the denominator becomes zero, the

, MZ approximates to Cb~" (17r2)2.

This is the dynamic impedance of the circuit andis the value of a purely resistive component whichmay replace the parallel circuit at, a resonance inthe analogy.

"The " Q" of the enclosure is given by w:- " andv

is normally much higher than that of the conesystem and is therefore not critical. It has beenfound that an optimum performance is given by thereflex enclosure if the cross-sectionalarea of the vent

7

Fig. 6. Equivalent circuit of m.c. laud-speaker, mounted in a labyrinth, Ml> Rc Mc Cc Ra La M, RI Mo RI

Cl' distributed mass and compliance of E=rr=E--labyrinth. RI. 'R2• viscous resistance . . 'and absorption resIstance respectively.

~~.~~--is made approximately equal to the piston area ofthe cone.

The required enclosure volume for coincidentresonance is then obtained from a derivation of theformula for a Helmholtz resonator, and is givenby:

V =7Tr2[::. z+\.7r + I ]

In this equation 1 is the length of the duct ortunnel, which usually extends into the enclosure,and the volume of the duct has therefore, beenadded to the expression. Broadly speaking, in-creasing the tunnel length decreases the overallvolume until a point is reached where the increasein total volume, due to the increased tunnel length,is exceeding the reduction in the volume requiredto correctly tune the enclosure. The tunnel lengthfor minimum volume is:

1=~-1.7rw

Another limitation on the length of the tunnel isthat it must not exceed 1/12th of a wavelength atthe resonant frequency of the enclosure, otherwisethe contained air would not behave purely as a mass.

We have seen that the reduction in size of a reflexcabinet is limited by the increase in, mechanicalimpedancepresented to the cone.

There are, however, marketed enclosure designswhich are based on the foregoing principle. Theseare extremely small, yet appear to have a substantialbass response,

It ill evident from the expression for the resonantfrequency of a vented enclosure that the enclosedair volume may become as small as we ,like, andthe resonant frequency made low by having a verysmall vent and tunnel area. Such an enclosure hasa very high mechanical impedance, thereby limitingthe cone velocity at very low frequencies. Also,owing to the very resistive nature of the vent, thetwo lower resonances shown for a loudspeaker ina vented enclosure are highly damped, and theupper resonanceis prominent, resulting in an accen-tuated bass radiation around this frequency, hence,the apparent bass efficiency.

The amplitude and frequency of this upperresonance may both be reduced by facing the coneinto a restricted aperture such as a slit, but this in-troduces serious irregularities in the response andwill be discussedin a subsequent article.The Tuned Pipe.- This is based. on the well-known organ pipe principle. In order to excludemodes of resonance other than the air columnresonance, the end of the pipe remote from thespeaker should be either fully open or fully closed.

In the case of the open pipe, resonances willoccur at frequencies corresponding to all evennumbers of quarter wavelengths,and anti-resonanceswilloccur at all odd numbers of quarter wavelengths.For the closedpipe, the reverseis true.

8

One method of applying these properties toloudspeaker mounting, is to use an open pipe withthe loudspeaker mounted at one end, the length ofthe pipe being such that its fundamental anti-resonance coincides with the cone resonance thussecuringsomeof the advantagesof a reflexenclosure.

A closedpipe may alsobe used in the samemanner,in which case the length of the pipe need only beabout half that of the open pipe. However, theimpedance presented to the cone by this methodis high, and a serious reduction in cone velocitymayresult at low frequencies. The radiation fromthe open end of the open pipe increasesthe radiationefficiencyof this system to some extent.

The length of an open pipe for a given frequencyof anti-resonancef is: .

1= ;/- 1.7 J ~where A is the cross-sectionalarea of the pipe.

The length of a closed pipe for a given anti-resonancefrequencyf is:

I = ~/- 0.85J ~Whilst these pipes are a little more simple to con-struct than reflex enclosures, their overriding dis-advantage is the presence of all resonances andanti-resonances occurring at every quarter wavelength, and it is virtually impossible to damp theenclosure and to absorb all the resonances withoutseverely attenuating the required fundamental. Away of partially overcoming this is, described in apatent by Voigt. This is to mount the speakerin the wall of a pipe which is closed at one end andopen at the other, the position of the loudspeakerbeing 1/3rd of the pipe length away from the closedend. By this means, the first resonance above thefundamental (3rd harmonic) will be cancelled.The Labyrinth.-The labyrinth consists ofa very long tube, usually folded and heavily linedwith absorbing material, with the loudspeakermounted at one end. The labyrinth is probablythe cleanest way of disposing of unwanted backradiation, which, having left the rear of the loudspeaker cone at one end of the tube does not re-appear at the other. It does not really mattertherefore, whether this far end is open or closed.

The analogous circuit is that of a transmissionline and is shown in Fig. 6. The sound energy,due to the back radiation from the cone, is largelydissipated in the resistive components ·RI and R2'where RI is due to the viscous losses between theair in motion and the lining on the internal surfacesof the labyrinth, and R2 is due to the absorptionof sound energy at these surfaces.

As the frequency is increased, R, increases andR2 .decreases. Therefore, if the labyrinth is to beeffective at the lower frequencies, the lining mustbe fairly thick. If, however, this begins to take

WIRELESS WORLD, JANUARY 19.56

Fig. 7. Electrical analogue of m.c. loudspeaker with horn load-ing. R'l~and La1 represent the air load an the side of the dia-phragm not coupled to the horn. Ra2 and La2 constitute the airload at the mouth of the horn.

up an appreciable part of the cross-sectionalarea ofthe labyrinth, the air loading on the rear of thecone, which is normally quite high in this type ofenclosure, will become excessive, resulting in asevere reduction in the radiated power at thesefrequencies. The cross-sectional area should,therefore, be at least equal to the piston area of thecone, and, to achieve the necessary dissipation ofsound energy from the rear of the cone, the effectivepath length of the labyrinth should be as great aspossible, the minimum length being set empiricallyat a quarter wavelength equivalent to the lowestfrequency to be reproduced.

Under these conditions, the impedance presentedto the rear of the cone is quite high and mainlyresistive, so that the cone approaches constant-velocity operation. and behaves. in the mannerpreviously described for this condition.The Hom.-Hom loading is the most efficientform of loudspeaker mounting and, if the hornwere large enough, it would give a performancesuperior in every respect to any other form ofloudspeaker mounting.

The action of the horn can be most readilygrasped by consideration of the analogous circuit.

The major problem in all the systems so fardiscussed has been to compensate for the fall inR, at low frequencies. The use of a transformerwould be an obvious answer if this problem werean electrical one, and, applying this to the analogy,we have Fig. 7. Acoustically,such a transformer isanalogousto the horn, which may be used to matchthe relatively high mechanical impedance of theloudspeaker cone to the radiation resistance, and,by makingthe mouth of the horn large, this resistancedoes not become so low at low frequencies.

From the analogy, since the effective radiationresistance reflected back to the primary of thetransformer is very high, the cone operates underconstant velocity conditions and no resonance isevident.

Below a certain frequency the acoustic resistanceof a horn -falls sharply and its reactance (mass)rises. This cut-off frequency is determined by thedimensions of the horn and, since size-for-sizean exponential- horn maintains its efficiency to alower frequency than a conical horn, the former ismore often used. The cross-sectional area (A",)of the exponential horn at any distance x from thethroat is given by:

A., = Ao £ ?>IX

where Aois the throat area and m the flair constant.

The cut-off frequency is given by: le = ~477The diameter of the mouth should not be less

than a quarter wavelength at fe, otherwise the hornwill tend to exhibit the resonancessimilar to a tunedpipe.

Most text books on electro-acoustics deal veryfully with the horn, and there is little point in ourdoing so here, especially since, due to its size, anadequatelylargehorn is rarelyencountered. Althoughmany small folded horn designs are capable ofimpressive (if not accurate) reproduction, let itsuffice to say that a horn capable of presentinga constant radiation resistance down to 30 cls tothe cone of a 12-in loudspeaker would be over12ft long and have a mouth diameter of about 9!ft.

Conclusion.- The different types of loudspeakerenclosures number as many as the possible com-binations of L C R in series with the analogouscone circuit.

Some time ago, the thought arose that an excellentmethod of designing a loudspeakerenclosure wouldbe to state the ideal velocitycharacteristics,and thendetermine an electrical impedance which, whenplaced in series with the analogous cone circuit,would produce these characteristics. It wouldthen remain to transpose this impedance intoacoustical terms and to evolve an enclosurehavingthe required component values.. This line of development has been followed to a

successful conclusion and will be described in thesecondpart of this article.

WIRELESS WORLD, JANUARY 1956 9