1950-012 molloy - response peaks in finite horns

8/3/2019 1950-012 Molloy - Response Peaks in Finite Horns

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TflE JOURNAL OI Ti'll '\COUSTIC;\I. SOCIDTY otr AMtrRICA \/OLLIvI 22, NUMBER SI'PTIMBER, 19.50,..,;.i i.j i" ir1];l :.::;{_ii,!l_l{ilj;'.:Lj ' i, l1 i:.; j ll{i'tFlJ,:t, : 1s:j' :rl{# $,

Response Peaks in Finite HornsC. l. Mor.lovell I'elehon,e Laboralor,ies, tr[ulray I{ill, Nau Jersey(Receivecl May 20, 1950)

In this paper the term hyperbolic holn is usecl to desigr-rate those horns u'hose alea lar,r'is given l;)':S(:t) : ooz.o shax I T sinha:uf 2.Where S(*) is the cross-sectional area of the horn at distance (:u) from the throat, (o) is the flare consranr;(1) is the shape parameter anc (6) is the throat ladius.The pressure on the axis due to a circular mouth, unbaflled horn loudspeaker is derived. In this calculationnse is rnade of the recent results of Levine and Schrvinger [Phys. Rev. 73, 383 (1948)] and some useful adcli-tional functions are comiruted fron their data and presentecl here in glaphical forrn.It is shown how to calculate the frequencies at which peaks occur in hyperbolic horn type loudspeaker fre-quency response curves.It is shown hovv to calcnlate the parameters of hyperbolic and exponential horn tvpe loudspeakers havingpre-determined peaks in their frequency response curves.Sorne experirnental coufrrrnation of the theory is presented.

List of Symbol llEunction defineclby,Eq. (6) Flare constant cm-l-defrnec1 bv trq. (1)c: Horn mouth ladius (crl),40 Horn throat ar:ea (crn2).4 Effective area of clriving unit diaphragrn (crn2) Wave number insicle hor-n: (h2-a2)+; wave-length ol sounclinsicle hom:2r/ Speecl of sound in free air (crn/sec.)D Rec\rrocal of clirectivity factor lor horn urouthDo Horn throat cliameter (inches)E Parmeter clefinecl by trq. (17)F Pararneter clefined by Eq. (17)G Parameter defined by Bq. (17)H Parameter definecl by Eq. (35)i =l-1)+I Function dened by Eqs. (a) and (9)"I Function clefined by Eqs. (4) ancl (9) Wave numlrer in free space a/c:2r/1, (cn't t),( Force factor for r.ecciver. (clynes/ampere)K :alr', -Florll lensth (cr)- :AAM =1./n c l,-i*


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IVTOLLOYto derive an expression for the pressure on the axisproducecl by the horn loudspeaker and then to discussthis function mathematically either to obtain the fre-quencies of the peaks or to obtain combinations ofparameters which yied desired peaks. The ,,pressureon the axis" function is computed by combining thedriving unit characteristics with the horn equationsand with the equations describing the u.orrii. fr",Ioutside the horn. For a sketch of the systern analyzed.in this paper see Fig. 1.

2. BASIC FORMULAS(A) Horn FunctionsArea function:

S():7.ozcoshrf Z sinhe;]2. (1)The ratio of pressure at the mouth to pressure atthe throat and the ratio of velocity at the n1outf, iovelocity at the throat ancl finally the ipedance aithe throat are given by:7!::

Ptt cosblf 'ikT -a2,,, sinhl coshl-l.Z sinhl

;:;{l(+),,.nr+1( .-u't!,,.).o.nr]si"r+[ (r-#2,,.)sinh/*1 .",r,].o,r]-' (3)lF!) . (-t), u,'n, ],i,, b t + 2,,,u + r . tanhttc o sb tI

These are general functions valid for any ,,Hyperbolic',Horn. If (Z) is put equal to zero they apply to a caten-oidal horn while if (1) is unity trey are valid for anexponentiai horn.(B) Driving Unit CharacteristicsThe driving unit may be characterizecl by two func-tions namely:a. The acoustical impedance iooking into the unitt the place where it couples to the horn throat.b. Its "open circuit" pressure. This is the pressurewhich the unit could develop at the place where itcoupes to the horn throat if the horn were replacecl bvan infinite impedance at that place.

. Frc., 2. lacliation inrpedanc_e ol -an unflangecl tube computecltrom clata of H. Levine ancl J. Schs,inger [h1.s. Rev. 7j, 3g3(ie48)1.

The "open circuit" pressure may be cornputed fromthe usual closed coupler response cuLves.sThe impedance looking into the driving unit mustinclude the contribution which arises from electro-mechanical coupling.(C) Acoustic Field outside the Horn

The radiation field produced by the horn is assumedto be the same as that produced by a circular tubehaving the same radius as the horn mouth. It is furtherassumed that the pressure is constant over the hornmouth. Under these assumptions the theory for un-flanged tubes recently developed by Levine andSchwingera is applicable. For the case wliere the hornis baffled the pressure on the axis may be calculatedsimply by putting I equal to unity in the formula forthe unbaffled case.The freld theory provides two essential pieces of in-formation frrst, it supplies a formula for the impedanat the horn mouth and this is necessary for the calcula:tion of the particle velocity in the horn mouth andsecondly, it supplies a relation between particle ve-locity at the hrn mouth and axial pr.rrui., which isthe final quantity to be calculated, The radiation m-pedance fiom an unbaffled tube r,vas caculated fromth. dutu given in reference 4 and is piotted in Fig' 2'The absolute value of the free fleld pressure measureoon the axis may be obtaineci by equating the to.tarpower radiated by the horn mouth to the pov/er passulg

3 F. F. Romanorv, J. Acous. Soc. Arr. !3,294 (1942)., .^,a H. Levine and j. Schwinger, Phys. Re. 73, 383 (1948)'

m=rm+lxm=RADIATI.N-" (pc)(LTNEAR vELoqly) = TuA RADU5


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RESPONSE PEAKSrough a sphere of large radius (r). It is given by :ll:ip'@/r)(ka).f lf,l, (s)where rr : r",/ D)i . (I/ha). ()Tbe quantiLy (f) y",t.computed from the data of refer-ence4and is plotted in Fig.3. It is of interest to notethat (l) is the ratio of the pressure produced by anunbaffed tube to that produced by a baffled one havingthe same particle velocity at the mouth.(D) Axial Pressure Function

IN FINITE HOINS

may now bewrite for the

,.,-._o'og,oa /-----------r rs rlr: F,iro of lcE PRESSuQE oN P at5Pr?ooucfo 8v ar urJFLAtJGEo lugE lo rHAt

PRODUCED BV A qIGIO PsION iiJ BALE\ = z/(u I aREOUENCY fi._.P5,c : SPEEO OF 5OUN'O (CMISEC) - ruaE RDtus (cra)The results of the preceding sectionscombined. BY Thevenin's theorem wevelocity at the horn throat.

pc(2"!zs)1

and zs: Iff, (g)where 1 and J are merely the numerator and clenom-inator respectively of Eq. (a). Hence

kFrc. 3. I is the ratio of the pressure on the axis produced by ar-runflanged tube to that produced by a rigid pistn in a baie.Bqs. (1) and (11)) among the parameters are specifiedand for which it is desired to calulate the (17-n)unspecified parameters. For example if it is desired tocalculate the usual frequency response curve for aa- --- -J \w/ r errvgiven, several desired values being selected. Of thehorn parameters any four may be specifred and thefifth calculated by use of Eq. (1). The nine driving unitparameters are given. Finally lp I is computed by useof trq. (11). Thus we rnight say that fifteen parametershave been specified numerically and two equations(namely (1) and (11)) have been given and the seven-teen parameters of Table f determined. It should bepointed out that the above formulation does not implythat physically rneaningful solutions can be obtainedfor all possible choices of parameters and relations. ftis, however true that when solutions are obtainable.they wilt be found by the process indicated in the aboveformulation. If we desire to have the quantity (


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554 C' T'3. This is identical with problem 2 except that inthis case an exponential horn is specified, instead of the

more genera.l hyperbolic horn of problem 2'Certain assumptions will now be made. These are theones which were applicable to the particular practicalnroblems which the author had been engaged in solving.-Obviously other assumptions must be made to flt otherCASCS.

Assumptions

MOLLOYtion Number 3 and Eqs' (a) and (9) we haveis real and l is a pure imaginary. Hence under therestrictions the denominator of Bq' (11) is aimaginary. Further, examination of the numerati

frequeqcy. The basic Problem ito find the frequencl.es tor whtch:Jz,*I:0'Equation (1) can be transformed into:

lan\t Fy yz+G

Eq. (11) shows that, regarded as a function ofquency, it has neither poles nor- zeros in the frequiange-O(z(.o. Hence the poles of lll mustspotrd, to the zeros of the denominator of Eq. (11

ali dissipation from the system. As noted before,makes profound change in peak heights bui.r,analysis have resulted in giving l2 | a set of polestead of finite peaks. Physicaliy this has happene{obvious that the simplifying assumptions made inicause the assumptions have postulated the removr. It is assumed that the "open-circuit" pressureis constant over the frequency range investigated.2. It is assuined that the source impedance a"pue stifiness.

pom

where

impedance terms must be included in the calculationof the completed response curve as these terms controlthe heights of the peaks and the depths of the dips.They do not, however' appear to exert much influenceon the calculation of the lower peak frequencies'In virtue of the assumption just made the sourceimpedance is a pure teactance and may be written

lplisa

(13) where(14)

z": (g/hi),to:

(A pc')/ (A oS + (V) / (A o)3. It is assumed that the mouth impedance of thehorn is given by z*: iNko,, (15)where 1/ is the end correction function given in refer-ence 4 Fig. 4. It is a slowly varying function ol ha.Equation (15) gives quite a good approximation to thereactive portion of the mouth impedance for values of less than 1.5.

4. CALCIILATION OF PEAK T"REQUENCIEST'OR A GIVEN SYSTEMThis is the frrst problem mentioned in Section 3'Now by Bq. (13) z" is a pure imaginary and by assump-

FREQUENCY IN CYCLES PER SECONDFrc. 4. Comparison between calculated horn response,nd mesured horn response.

- 1+[P- (T{tanha./ 1f 7 tanh)a]F :plI-BE-a(a-Pr)E'lG:a(a-Pr)\-PE)

7d.: abp:Qt:N&/l'y :bl : (h, - a2) .l

When the parameters ofF and G are determinedfor (y) and this is easilYby the formula v: c/2'rla2l/l)'f.It should be noted that Eq' (17) is a tra'n

known methods for soiving transcendental eqequation and has infinitety many roots' Howeverthose roots yielding p"ul f..qo.ncies for whicbbasic assumptions of lhis analysis are valid shoullutitized. A1l others should be discarded' Any of the'

i


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RESPONSE PEAKS(. CALCULATION OF PARAMETERS FORHYPERBOLIC HOR.NS

This is the second problem mentioned in Section 3'fnr^rf..t of the assumptions macle above is to reduceiti. rytt.* to one wlrere there are seven physica vari-Ji.r'un two' give' relations which must be satisfiecl'These are:a=Flare constanl:1=Shape constant/:Horn lengthi=Radius of horn mouthfto=Raclius of horn throat0=Stiffness coefircient looking into clriving unit:Frequency of first horn PeahX2nEquation (1)Equation (17)Suppose that l, a, and (, deflned by Eq. (21)) aresi;; and that it is desired to calculate a, T and R'hysicaily this is the problem where one has a given

IN FINITE HORNS 555Tnnr,a I

Number Symbolart ofsystem Definitionlpl@r&oI&TA'lLdSRvI(LERzE

I234567891011l2I

1FieklPalametels

IIornParameters

Driving unitPrameters

Fee freld axial pressureFrequencyX 2zDistnce-along axis fron-r horn tnouthto field pointHorn mouth radiusI{orn throat radiusHorn lengthI{orn flare constantHorn shape constanLDiaphragm areaDiaphragn massDiaphragm stifinessDiaphragm resistanceVolunle of air betrveen diaphragm antlhorn throatElectron-rechanical coupling constantBlocked electrical inductanceBloched electrical resistanceApplied voltage

74r.)1,6t7

iVing unif, all of whose constants are fixed excthroat diameter and it is desired to couple it to a hornsuch that the frrst peak frequency of the response curvewill have a predetermined value. It is desired to designa horn which will produce this result. Ilowever, thespace rvhich the horn can occupy is limited so tht thereis an upper limit on the horn length and horn mouthradius.Before giving a detaiLed solution to the problem auseful relatiorl, employed in the subsequent anaiysis,and its mathematical consequences will now be stated.A:QT. (1e)The rnajor reason for using Eq. (19) is that it proclucessubstantial simpliflcation of the characteristic Eq. (17).A secondarv conseouence of this relation is that hornsfor which ii is trrre Lave all their neaks above cut-off.Norv bv Eqs. (2) and (i4) we rn;y wrilewhere

Hn-_- - - ,trcoshalf I srral1/-\4rr\A )'

l(A,f pcr/Sa)*V)combining (19) ancl (20) ancl solving for (1) rve

tion of the problem posed at the beginntng ot thtssection.Procedure for the Calculation of HyperbolicHorn Parameters1. List the given parameters:

Horn length (cm) (l)Horn mouth radius (cm) dFrequency of first peak in () unitsReceiver diaphragm efiective area (A) cmzReceiver diaphragm stiffness (dynes/cm) (,Sa)Volume between diaphragm and horn throat (cm3 (Tz))'2, Compute (H) bV means of Eq. (21).3. Compute (Hl) and determine the maximum per-missible value of (). Call this quantity ('."*)' This iscalculated as a root of the equation(2al)sinh(2al): Hl.

The root may be determined analytically if higir pre-cision is necessary (as it may be if (a) lies near (^*))or it may simply be read from a graph of the function!:x sinhr when an approximate value is suffrcient.4. Assume a value of () such that 0( uSan',*.5. Calculate both values of l from E'q. (22)'Carry out the following steps for each value of 1.6. Calcuiate the following quantities:a: alB: at/Td (values of and d given in steP 1)/, this function is the ordinate of Levine andSchwinger's encl correction curve':(\r'd)/L

1l- (l/) - (o'- '/*o tanha)7. Solve the characteristic equation for y where!:bt. For the situation assumed here (a:Bl) the

(20)

(21)andhat'e

aTt r' =;@;'h?oi\rG - @/H)sinh2at)+(I -2(a/H)sirrh2at)+l . (22)Thus there are two values of ? which are permissible.:lther Tay be chosen. Since (1) must be ,.o1, th. fut-r.-:::i Tq., the radica of Eq. (22) cn never be nega-rtve' This imposes the folloring restriction on (at)(.2at)sinh(2al)


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556characteristic equation is

c. T. MO:LLOYequation. a

tany _ _E+PG-PE) .yyt8. Calculate the value of () corresponcling to (y)'

Do= (inches).2.54(coshl* T sinhal')11. It is to be noted that for a given value of (iionly one of the correspondi|8 tyo "?Iy91 of (?-) wiyied a solution. The second value^of (?) -willgive-

6. CALCULATION OF'PARAMETERS FOREXPONENTIAL HORNSThis is the third problem mentioned in SectionfAs has been previously pointed out, the exponenhorn is u ,p".l case of the general hyperbolic.hornwhich the- parameter (T) has the valu-e unity'physical ,ruiiubi.t of the llob-t9* are then the1 I -l:^ l^^..- ^-l-- -^'I.rr"n u. given for the hyprbolic horn only now 1:1There is irerefore, less freedom of choice of parameter

olution but for a different value of (). Thus for t[n:)+(y/l)'). given set of parameters listed in step 1 there are twrf this varue of (,) is crose to the cresired :")": ll !f i",*i"*"r;JJt'i"3dlTJi;:;i,iiff$;:nd tqgiven in step 1 then a more. accurate value of Jl),:i""ld "'i ,n" course of this investigation a series of sixteiliven in step 1 then a more accurate value oI11,,,:1out" In the .otr. of this investigation a series of sixteeie determined by the method given below .lt^ t]l::,:: n'pr-Uoli. frorns wee designed and points in the neighiillrrtru uJ Lrfu urvLrrvs b.--': "--' ' . - a hvperboirc nolns wele oesrgllcu ir,f tu PUTTIL lrl u-rtr llclgll-:curate value of () is sufflcientlv close to th9 .d:tti:1 ;ih;;J ;f tt. p"uf.t cicuhtecl. Thtt" sttow. if-ii

then the corresponding-values "f (o) u"d i:ti:g r.rrril -- -=(?') are solutions to the problem' If a.better,approx.ll1?- (1) The solution corresponding to the smaller valuiotr is desired or if the fust rough value of () is widely of ifl had the larger throt diameter. ,.,

1"1 9

o.4 0.6 0.8 IFrc. 5. Chart for exponential horn parameters'

in the design of this class of horns than in the ger;;;'b;iLiype. This is iliustrated.bv the tvpical"-rpf. ,rs"d oi the general hyperboli-c case' Here it w1priir" to frx the"parame"ts k, {, .'' lld {^ti*pendently and to solve for the remaining three parani.tt o, o u.rd. l. In the correspondirg case of an e;p"nti to.r, it is possible--t o fr\ P,- 1, and d and t!i"uirring three a, R6, and -H variable are determin-Physica$ this means that for the hyperbolic horn.u .hoot" the peak frequency, horn length, m0radius and driving unit constant H and still be a; b.i"g abie to find a horn which fulfills allconditions. In the case of the exponential horn.we upchoose horn length, peak frequency and mouth raurlbut must u.."p the value () wttictt the solutionlfr" .q.tutions ^produces' rf hs is- n9t a satisfactt;;i";,';;;" altration must be made in the variablesk. d, and,a new .rut r. ot .omputed' It is often possib

@o.^9$ 9 ru ocid c; c; c;citoo8040

Io,8u.oo.4


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by this means to design a suitable exponential horn butii;" a matter of compromise and the choices are morei*iar However, the possibiiity of determining a suit-,bl, ."pon.ntial horn may be explored fairly rapidlyiy ur. of ttt. charts,of ,Fig. 5 and in a practical probleminight be well to do this frrst before embarking on then orr elaborate ,procedure involved in the calculationof a hYPeroollc norn'

Procedure for the Calculation of ExponentialHorn ParametersChoose the parameters in steps I,2 and 3 ancl regarclthem as fixed'1, Frequency of frrst horn peak ()2. Horn length (l).3. Horn mouth radius ().4. Compute K:hd,, M:l/(a) ancl read L:a.d, fromchart of Fig. 5. Also comPute (o).

f! - stzt.If this value of (II) is attainable from the permissible

values of receiver coustants the design task is essentiallyaccomplished and it remains only to obtain more ac-curate values of the parameters a, I:I andRe as explainedbelow. Il however, the value of () is not sa,tisf actory,aiterations must be made in some or all of the na-rameters h, l, a and steps (4) and (5) repeatect. Ifvalues of h, I and R6 inside the domains of these vari-ables, prescribed by the problem, will not yield a satis-factory value of () then it is not possible to design anexponentiai horn for the set of specified conditions andrecourse must be had to a hyperbolic horn.6. Determine a better pproximation of (o) bysolving the transcendental equation below for (o).(rlrol: This is merely the characteristic Bq. (17) wiihI set equal to unity.)1I:_ tan_1[(a_Nak)/bl.b

This may.be most easily done by plotting (l) as a func-:lon o1 () on a large scae plot usilg values of () inthe neighborhood oithe () f step (a).., 7. Using the improved value -of @) compute thethroat radiuso- d'e-" 'l' Y*l.e the improved value of (a.) compure ().-^r/'. usrng this value of (H) make final adjustment ofrec^eiver constants so that Eq. (19) is satisfid.rr1 ctrc.ular mouth exponential horn was designed byme method given abve. The system constants areusted below.

lt nil Anl /\/\t\l /vI

\FIRSl PEAK FREQUENCY466 C.P.5,

I 500 rooo 3000 5000FREQUENCY N CYCLES PER SECONDFrc. . 4[easured axial restonse curve Io circular rnoulh horn.

Flare constant (a) 0,A7945Volume in front of diaphram (Z)Frequency of rst peakDriving unit diaphragm area (,4 )

2.416 cmB480 c.p.s. (design value)8.77 cmzDriving unit cliaphragm stiffness (,S) 45.X106 clynes/cmThis system was assembled and its response curvemeasured. This is reproduced in Fig. . It is to be notedthat the frrst peak came at 4 c.p.s. instead of at thedesign flgure of 480 c.p.s. It is likely that this smallcliscrepancy is due mainly to minor differences betweenthe actual parameter of the system and those assumedfor the purpose of calculation. Despite the small dis-crepancy, it is felt that the result obtained showed goodagreement with the theory and taken together with thecalculated and measured pressure response curves shownin Fig. 4 may be regarded as a verification of the theorypresented in this paper.

7. SUMMARY1. Formulas from which the axial sound field of horntype loudspeakers can be computeci, subject to the re-strictions set forth in the text, are given.2. A comparison between a measured and a computedaxial response curve is given (Fig. a). The agreement issufciently good to be regarded as a verification of thetheory presented.3. A method has been given for calculating the pa-rameters of "Hyperbolic Horns" which will have oneof their response curve peaks at a predetermined fre-quency.4, An exponential horn, designed according to theabove principles was constructed and its measured re-sponse curve peak was found to occur very nearly atthe calculated frequency.5. A method has been given which enables one tocletermine the frequency at which the first responsepeak rvill occur for any hyperbolic horn system whose

parameters are given.

RESPONSE PEAKS IN FINITE HORNSt5

o ozo>so@oGI rn

G _t5-20L300

() from the xelatioa ;;J;;rt,f:+q.,'- - - tia'roj?5-)

(tt.t")(cliam. 2.01")ilii'i"il j,',1,,i.",

1950-012 molloy - response peaks in finite horns

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