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Title: The audio spotlight: An application of nonlinear interaction of sound waves to a new type of loudspeaker design

Author: Masahide Yoneyama; Jun-ichiroh Fujimoto; Yu Kawamo; Shoichi Sasabe

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The audio spotlight: An application of nonlinear interaction

of Sound waves to a new type of loudspeaker design

Masahide Yoneyamaand Jun-ichirohFujimoto

Application

Products

Department,

Technology

Division,

RicohCompany

Ltd.,3-6,1-chorne,

Naka-rnagorne,

Ohta-ku, Tokyo143, Japan

Yu Kawamo and Shoichi

Sasabe

Research

Laboratories,

NipponColumbiaCompany

Ltd., 5-1,Minato-cho,Kawasaki-ku,Kawasaki-shi

210,

Japan

{Received

12May 1982;accepted

for publication

17January1983)

Thisworkwasdoneto devise

a newtypeof loudspeaker.

Thetheoryforsoundreproduction

of

this loudspeakeris basedon nonlinearacousticsof soundwaveinteractionin air. A finite

amplitude

ultrasound

wavethatcanbeamplitude

modulated

byanyaudiosignalisradiated

from

a transducer

arrayintoair astheprimarywave.Asa result,anaudiosignalisproduced

in theair

because

oftheself-demodulation

effectoftheAM soundwavedueto thenonlinearity

oftheair.It

ispossible

to geta flatcharacteristic

ofreproduced

soundpressure

byusinganequalizer.

In some

fundamental

experiments

thecharacteristic

ofthereproduced

soundpressure

isnotquiteflatdue

to animperfect

transducer

array.Improvement

of thetransducer

makesit possible

to geta flat

characteristic.

A special

featureofthisloudspeaker

isitsverysharpdirectivity

pattern,which

makesit possible

to realizea soundspotlight.

PACS numbers:43.25.Lj, 43.25.Vt, 43.88.Ja

INTRODUCTION

Thenonlinearinteractionof finiteamplitudeultrasonic

wavesin theair canbeappliedto a loudspeaker.

1Thispaper

describes

thefundamental

conceptof a loudspeaker

basedon

Ps=-•

o[r r'I t?-•

qr,t---

CO

dr', (3)

wherer is theobservation

pointpositionvector,r' is the

thenonlinearinteractionof soundwavesin air. Also,someof

source position vector and o is the nonlinear interaction

the experimental

resultsfromthe operationof a prototype

space.

loudspeakerare presented.

When two finite amplitudesoundwavesIprimary

waves),havingdifferentfrequencies,interactwith one an-

otherin a fluid,newsoundwaves(secondary

waves)

whose

frequencies

correspond

to the sumandthe differenceof the

primaryWavesmaybeproducedastheresult.

This phenomenon

was first analyzedby Westervelt

2

and is well known as "nonlinear interaction of sound

Whenthe primarywaveconsists

of two continuous

sin-

usoidalwavesandbothareplanarandwellcollimated,

the

integralof Eq. {3) is calculatedin the samemanneras in

previous

papers.

2.5Whenthedirectivity

ofa circular

piston

is takenintoconsideration,

however,

Eq. {3)mustbeused

with the expression

of Muir et al.6

A newtypeof loudspeaker

hasbeendeveloped

onthe

basisof the nonlinearinteractionof soundwavesmentioned

waves,"or the "scatteringof soundby sound."3 Basedon

above.

In thistypeof loudspeaker,

ultrasound

isamplitude

Lighthill'sarbitraryfluidmotionequation

4asshownin Eq.

{1), Westerveltderivedan inhomogeneous

waveequation

whichissatisfied

by thesoundpressure

of secondary

waves

producedby the nonlinearinteraction[Eq. {2)].

modulatedby an audiosignaland radiatedfrom a trans-

t?t

2

•xit?x

j,

(1)

ducerarrayasfiniteamplitude

waves.

Whentheamplitudemodulated ultrasoundwave interactsis a nonlinear fashion

in air, themodulated

signal{theaudiosignal)canbe demodulated in the air. 1

In thefollowing

section,

theprinciple

underlying

this

type of loudspeakeris described.

p: density

of fluid,To:stress

tensor,

I. THEORY

A. Acoustic reproduction by nonlinear interaction of

V2ps

Co

23t2= - Po3t'

AM ultrasound

(2)

q_ ]3 t•p•.

in air

When two sinusoidal sound waves are radiated in the

air,twonewwaveswithangularfrequencies

ofcolq-co2

arise

poCo,gt

by nonlinearinteractionof the two original sinusoidal

In Eq.(2),Psisthesecondary

wavesoundpressure,

p• isthe

waves,whoseangularfrequencies

arecolandco2.

primary

wave

sound

pressure,

]3isthenonlinear

fluidparamThereforeonemightexpectthesecondary

wavewhich

eter,andCoisthesmallsignalsoundvelocity.

corresponds

to themodulationsignal,to appearin theair as

ThesolutionforEq.(2)maybeexpressed

bythesuper- a result of the nonlinear interaction between the carder ulpositionintegralof the Green'sfunctionand the virtual sectrasound

andtheloweranduppersideband

waves,provided

ondsource[fightsideof Eq. (2)]asshownin Eq. (3).

that a finiteamplitudeAM ultrasoundwaveis radiatedinto

2

1532

4

J.Acoust.

Soc.Am.73(5),May1983

0001-4966/83/051532-05500.80 @ 1983Acoustical

Society

ofAmerica

1532

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 185.21.216.148 On: Mon, 03 Mar 2014 21:07:32

cessedby an equalizerhaving -- 12 dB/oct frequencycharacteristicsbeforethe audiosignalis introducedinto the AM

modulator.

B. Harmonic

distortion

In the caseof pure-tonemodulation,g(t ) = sintot, the

soundpressures

arisingfrom both the signalsecondarywave

and the secondharmonic distortion signal are calculated

from Eqs. (6)and (7), respectively,

FIG. 1. Frequencyspectraof an AM waveanddemodulatedwave.

the air. That is, the AM ultrasoundis self-demodulatedby

the nonlinear interaction.

ps(t) = -- (/3p•a2mto2/8poc•ar)

sinto(t-- r/Co),

(9)

pa(t) = •p•a2m2ro2/8poc•ar)

cos2ro(t-- r/Co).

(10)

From theseequations,it is possibleto definethe second

harmonic

Figure 1 showsthe spectrafor both an AM waveand a

demodulated wave. In this case, since the modulation wave

is reproducedin the air, a new type of loudspeakercan be

devisedif the modulationsignalis selectedas the program

audiosignal.

If a finiteamplitudeultrasoundbeam,modulatedby an

audio signalg(t ), is radiatedinto the air from a transducer

array, the soundpressurep•of the primary wave (AM wave)

at a distancex from the array on axismay be representedby

Eq. (4)

P• = Po[ 1 + mg(t -- X/Co)]e- '• sinCOo(t

-- X/Co),

(4)

wherepois the initial soundpressureof the ultrasound,m is

the parameterindicatingmodulationindex,and a is the absorptioncoefficientof cartier sound.

A virtual audio signal sourceoccurs in the primary

soundbeambecause

of thenonlinearityof theacousticinteractionin air. This soundsourcemayberepresentedby Eq. (5)

usingEq. (2) and Eq. (4)

distortion

ratio as follows

• = [ •pa(t)l/[Ps(t)l]

X 100= mX 100%.

(11)

Becausethe secondharmonicdistortionratio isproportional

to m, a gooddistortionratio requiresa very smallmodulation depth to preventcrossinteractionbetweepthe lower

and uppersidebandwaves.The signaland distortionsound

wavesarerepresented

by thefirstandthe secondtermon the

fight sideof Eq. (5),respectively.

The soundpressureof the

signalis proportionalto m, while the distortionis proportional to m2. In accordancewith this relation, if m is selected

lessthan 1, the distortionsoundpressurewill be much less

than the signalsoundpressure.

If the equalizerof -- 12dB/oct isused,the modulation

depthm varieswith the frequencyof the modulationsignal,

as expressedin Eq. (12)

m = too/to2, mois constant.

•

q= p•c•e

•o•tmg

t--•Co+-•-m2g

• t--•Co .

In the aboveequation,the secondterm on the fight side

impliesa harmonicdistortioncomponentarisingfrom the

interactionbetweenthe lower and upper sidebandwaves.If

the primary soundbeamcrosssectionisassumedto be circular with radiusa, then the demodulatedaudio soundpressurep• at the point r from the array, on axis,canbe calculated analyticallyusingEqs. (3) and (5) in the form

o•2

Ps

=t•p•a2m

8poCo4

ar•5

g(t---r).

Co

(6)

On the other hand, the soundpressureof a harmonic

distortioncomponentmay be expressed

as

(t----r).

Pa=t•P•a2m2

16pocgar

•o•2

g2

Co

The Fourier transformof Eq. (6) can be expressedas

(7)

•

Ps(cO)

= -- (/3po2

a2m/8poc•ar)to2exp[

--j(r/Co)CO

] Gs(cO),

(8)

wherePs(to}is the Fourier transformofps(t },and Gs(to}isthe

Fourier transformof g(t }. As evidentfrom Eq. (8},Ps(to}is

proportional

to to2andthusthefrequency

characteristics

of

the reproducedsoundshowa 12dB/oct dependence.Consequently, the audio signal(modulationsignal}must be pro1533

J. Acoust. Soc. Am., Vol. 73, No. 5, May 1983

FIG. 2. Front view of the loudspeaker.

Yoneyama

etal."Audio

spotlight

1533

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 185.21.216.148 On: Mon, 03 Mar 2014 21:07:32

120 -

lOO

120

I

'

10 20

I dl,

50

100

f

( kHz )

2•0

FIG. 3. Sound pressure-frequency

responsecharacteristicsof the transdueerarray,for a point4 m from thetransducer.The inputvoltageis0.5 ¾.

(V)

FIG. 5. Soundpressure

versusinputvoltageat 40 kHz, for a point4 m from

the transducer.

In this case, since the secondharmonic distortion ratio e is

proportional

to 1/o•2, distortionin low-frequency

regions

increasesmarkedly.

and the directivityat 40 kHz (theprimary wave)of the array

are shownin Figs.3 and4, respectively.

As canbe seenfrom

the signalfrequencycharacteristics,

is due to the fact that

Fig.

3,

the

frequency

response

characteristics

of thearray are

Pd(t }isproportional

tom: eventhroughpt

(t }isproportional

not

symmetrical

for

40

kHz.

Moreover,

there

are many harto m. If rn is kept smallto make distortionlow, the sound

monic

resonances

and

antiresonances.

The

frequencyrepressurePt (t} alsodecreases.

sponse

characteristics

of

the

secondary

sound

waveare disTherefore, either the initial soundpressurePo of the

tributed

by

the

resonances

and

antiresonances.

carrier waveor the radiusof the primary beamcrosssection

Figure 5 showsthe soundpressureat 40 kHz, at a point

shouldincreaseto maintain the expectedPt (t }.

4 m from the array, plottedagainstinput voltage.

The soundpressurefrequencyresponsecharacteristics

II. EXPERIMENT

of the secondarywaveproducedby the nonlinearself-interA loudspeake•usinga finiteamplitudeAM ultrasound

actionof the finite amplitudeAM ultrasoundradiatedfrom

radiatedfrom a transducerarray was developedand put to

the array, are shownin Fig. 6. The characteristics

weremeapractical use. This array consistedof 547 PZT bimorph

suredwith modulationdepthrn = 0.5 at a point4 m fromthe

transducers.The fundamentalresonantfrequencyof each

array in an anechoicchamber.The 12dB/oct equalizerwas

transducerwasabout40 kHz. A front view of the array apnot used.In the frequencyregionbelow 1.5kHz, the characpearsin Fig. 2.

teristicsalmostfollow the 12 dB/oct curve.The soundpresThe soundpressurefrequencyresponsecharacteristics

surecharacteristicsof the primary wave have a fiat region

within the frequencies

of 40 _+ 1.5 kHz as shownin Fig. 3.

Thate isproportional

to 1/o•2,in spiteoftheflatness

of

Whenthesideband

spectra

of themodulated

ultrasound

deviatesfrom the fiat range,the soundpressureof the secondary wave decreases.The peak of the primary soundpressureCurveat 60 kHz producesthe peak of the secondary

wave at 20 kHz. All of thesephenomenacan be predicted

100 1•1'

270 ø

'"'

go

70

60

11•02•)0 5•0 1• 2'k

f

FIG. 4. Directivityat 40 kHz of the transducerarray,for a point4 m from

the transducer.The input voltageis 10 V.

1534

J. Acoust.Soc.Am.,Vol.73, No.5, May1983

5'k l(•k 2•k

(Hz)

FIG. 6. Soundpressure-frequency

responsecharacteristics

of secondary

wave,for a pointof 4 m, rn = 0.5, andinputvoltageof 10¾.

Yoneyama

et al.' Audiospotlight

1534

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 185.21.216.148 On: Mon, 03 Mar 2014 21:07:32

92•

FIG. 7. Directivity of secondarywave at 1.0 kHz, for a point of 4 m,

rn = 0.5, and input voltageof 10 V.

FIG. 9. Directivity of secondarywaveat .10.0kHz, for a point of 4 m,

m = 0.5, and input voltageof 10 V.

,

--from

Eq. (9)and the characteristics

of the primarywave.

The measureddirectivitiesof the secondarysignal

Wavesat 1.0,5.0, and 10.0kHz areshownin Figs.7, 8, and9,

respectively.

To checkthe relationbetweenthe secondary

signal

surementresultsatfs -- 5.0 kHz. Theseresultsshowthat the

relationof the soundpressurelevelbetweensignaland dis-

tortion

arepredicted

byEqs.(9)and

(10).Forexample,

if the

resultsof m = 1.0andm -- 0.5 arecompared,it is clearthat

the signallevel (i.e., 5 kHz) decreases

6 dB and the second

soundpressureps

(t) andsecond

harmoniccomponentsound

pressurePa(t) of the secondarywave, the secondarywave

pickedupby audiomicrophone

wasanalyzedby a spectrum

analyzerfor variousvaluesof rn. Figure 10 showsthe mea-'

5

10 15kHz

(a)

92

62 ',-i-.52

•-t--

J. Acoust.Soc. Am., Vol. 73, No. 5, May 1983

10

15kHz

(b>

•6

FIG. 8. Directivityof secondarywaveat 5.0 kHz, for a point of 4 m,

rn = 0.5, and input voltageof 10 V.

1535

5'

FIG. 10.Relationsof secondary

signalsoundpressure

Ps and secondharmonicsoundpressurepa,(a)rn = 1.0,(b)rn = 0.7, (c)rn = 0.5, (d)rn = 0.3,

and (e)rn = 0.1.

Yoneyama

•t a/.'Audio

spotlight

1535

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 185.21.216.148 On: Mon, 03 Mar 2014 21:07:32

1

_1

I J Power

'•

Amp.

I

qMød•atør

I

IEclu•

izer

I

Transducer

Sourcesignal

Carrier

sin ½t

g(t)

FIG. 11. Constructionof the loudspeaker.

co2characteristics,

an equalizerisrequiredfor flat response.

Usually,it is quitedifficultto producelow-frequency

sound

because of distortion.

Onespecial

featureof thisloudspeaker

isitsverysharp

directivity

pattern.Thisloudspeaker

canbeusedasa sound

spotlight.

Sinceanacoustic

spotlight

hasneverexisted

in an

audiblesoundregion,varioususesfor thisloudspeaker

may

be anticipated.

For example,•

the sharpdirectivitywould

makeit possible

to speakto onegroupof peoplewithout

disturbanceto neighboringgroups.In a museumor an ex-

hibit,expensive

soundbarriersbetween

exhibitswouldbe

unnecessary.

harmonicdistortionlevel(10kHz) decreases

12dB. Accord-

ingly,thesignalsoundpressure

isproportional

tornandthat

of thedistortion

isproportional

to m2.

III. DISCUSSION

An entirelynew type of loudspeaker

hasbeendeveloped.Thisresearch

isbasedonthephenomenon

of thenonlinear interaction of sound waves. That is, the self-modula-

tion effect of finite amplitude AM ultrasoundby the

nonlinearity

oftheairhasbeenappliedin theconstruction

of

theloudspeaker.

Thisloudspeaker

consists

of anultrasound

transducerarray, a drivingamplifierfor the array, an AM

modulator,a pure-toneoscillatorfor the carrierfrequency

andequalizerasshownin Fig. 11.

The soundpressure

obtainedfrom the loudspeaker

is

proportional

to thedepthrnof themodulation.

However,rn

shouldbe assmallaspossible

because

the secondharmonic

distortionratio e is equalto m. The soundpressure

of the

secondary

waveisalsoproportional

to thesquareof theini-

tial soundpressurepo

of thecarriersoundandthesquareof

the beamradiusa. Thesevaluesmustbe aslargeaspossible

to obtainadequatesoundpressure

for practicaluse.

Sincethefrequency

response

ofthesecondary

wavehas

1536

J. Acoust.Soc. Am., Vol. 73, No. 5, May 1983

ACKNOWLEDGMENTS

The authorswishto express

theirsincereappreciation

to all the membersof the NonlinearAcousticSocietyof Ja-

panfortheirhelpful

comments.

In particular,

special

thanks

are due to Dr. A. Nakamura and Dr. T. Kamakura for their

generous

discussion.

Finally,theauthors

wishto acknowledgeDr. C. Schueler

forhishelpin revising

theEnglish

of

this manuscript.

•M. Yoneyama,

Y. Kawamo,J. Fujimoto,andS.Sasabe,

"An application

ofnonlinear

parametric

interaction

toloudspeaker,"

Meetingof Institute

of Electronics

and Communication

Engineers

of Japan,PaperEA81-65

(1982).

2p.j. Westervelt,"ParametricAcousticArray," J. Acoust.Soc..Am.35,

535-537 (1963).

3R.T. Beyer,"NonlinearAcoustics,"

NavelShipCommand(1974).

4M. J. Lighthill,"On soundgenerated

aerodynamically,

I," Proc.R. Soc.

LondonA211, 564-587 (1952).

•H. O. Berktay,"Possible

exploitation

ofnonlinear

acoustics

in underwater

transmitting

applications,"

J. SoundVib.2, 435-461(1965).

6T. G. Muir andJ. G. Willete, "Parametricacoustictransmittingarrays,"

J. Aeoust.Soc.Am. 52, 1481-1486 (1972).

Yoneyamaeta/.: Audiospotlight

1536

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 185.21.216.148 On: Mon, 03 Mar 2014 21:07:32

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