Yoneyama 1983 .pdf

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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

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