THEORY OF PARAMETRIC ACOUSTIC ARRAYS

JOURNAL DE PHYSIQUE

Colloque C8, supplement ecu n°ll,

tome 40, novembre 1979, page C8-101

THEORY OF PARAMETRIC ACOUSTIC ARRAYS H. HOBAEK and S. TJ0TTA University

of Bergen, Bergen, Norway

Résumé. - Ce travail est une contribution à la théorie de l'antenne paramétrique formée par l ' i n t e raction nonlineaire de deux ondes sonores dans un fluide thermovisqueux. La source des faisceaux primaires est un piston c i r c u l a i r e . L'attention est principalement portée sur l'analyse des propriétés â grande distance de la source de celle des composantes du champ engendré qui a une fréquence égale â la différence des fréquences des faisceaux primaires. Les effets dûs à l'interaction qui se produit dans les régions de divergence des ondes primaires sont comparés aux effets produits par l'interaction de deux ondes primaires considérées comme planes et collimées à faible distance de la source, et comme divergentes à grande distance de c e l l e - c i . Différents modèles sont passés en revue. La discussion de leurs domaines de v a l i d i t é f a i t l ' o b j e t une attention particulière. Les résultats théoriques et expérimentaux sont comparés. Abstract. - The paper is a contribution to the theory of the parametric acoustic array formed by the nonlinear interaction of sound waves in a thermoviscous fluid. The source of the primary waves is a circular piston, and emphasis is put on an analysis of the farfield properties of the generated difference frequency sound field from two sound beams. Effects from interaction in the divergence regions are studied and compared with the effects from a model composed of a plane collimated nearfield region of interaction, and a farfield divergence region. Previous models are analysed with emphasis on a discussion of the range of valdity. Theoretical results are compared with experimental observations.

1. Introduction. - These exists today an extensive literature concerned with the problem of mutual nonlinear interaction between two sound beams. The theory of parametric acoustic array has been treated by so many authors since the first paper was published by Westervelt in 1863, that there would seem by now to be little to add on this point. However, there still appear new results of interest.

Reviewing available experimental observations (see table I) we find that these are obtained under various conditions, and direct comparison between the results are therefore difficult. Often, for example, the observations are so near the sources that any comparison with the existing asymptotic theory is not justified. Further, theory valid within the interaction region, very near the source

Table I. - The experimental conditions of parametric arrays with circular primary sound source, reported in the literature. Author

Bennett e t a l . (1974) Bj0rn0 Eller Esipov Hobaek Moffett

e t a l . (1976) (1973) e t a l . (1975) e t a l . (1977) e t a l . (1976)

Muir Smith

e t a l . (1972) (1971)

f

a (kHz)

21.1 910 14«0 107.5 4270 270 482 3500

f. (kHz)

a (cm)

Type of medium

T (o-)

L

A (m)

R

obs (m)

0.61 8.8 5 2.92 Air * * + 18.4 10 40 1.0 Brackish water 15 12.5 3.7 Fresh water 50 1.0 Sea water 130+ 80 5 46 Fresh water 22 1.12 2 100 0.25 Sea water 30 54 85 50 5.1 * Fresh water 12 8 64 3.81 76.5 130 4.5 Fresh water 1.63 325 0.25

•*• estimated quantities. * information obtained from other sources. L = shock formation distance.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979819

N

F

D E^ W >3dB

N

2.25 1.49 0.83 0.25 0.48 4.56 1.96 5.04 1.84 4.21 1.28

4.90 7.1 4.49 1.2 3.2

2.80 1.52 1.28 1.32 0.80 2.07 2.24 1.57

VLs

Domain of f_ (kHz)

0.01 0.2 0.21 25-200 0.04 1- 5 > Ro = -;r , i f a phase e r r o r much l e s s than 1 i n the Green's f u n c t i o n i s required. Also r >> a i s assumed i n t h i s approximation. Here x = k + i a , k = u/cO a i s the p i s t o n r a d i u s and J1 i s the Bessel-function o f order 1. A time v a r i a t i o n e-iut i s assumed. An improved approximation i s sometimes o b t a i ned i f we keep higher order terms i n t h e expansion o f t h e Green's function. For instance, keeping squar e terms, we get the f o l l o w i n g s u f f i c i e n t c o n d i t i o n f o r the phase approximation, instead o f r >> Ro as i n t h e Born approximation : r >>

C8- 105

a l s o behaves n e a r l y as f o r a plane wave, except i n the p a r a x i a l region. Thus, the near f i e l d may roughly be described as a s l o w l y d i f f u s i n g beam o f plane waves o u t t o about Ro/28, followed by a t r a n s i t i o n r e g i o n where the f i e l d changes from plane waves t o s p h e r i c a l l y spreading waves. I n t h i s t r a n s i t i o n region, extending somewhat beyond Ro, the beam s t r u c t u r e d i s solves w h i l e the p a r a x i a l r e g i o n grows i n l a t e r a l extension w i t h t h e distance and dominates t h e f i e l d For c e r t a i n purposes the Bessel f o r r > Ro/2s d i r e c t i v i t y adequately describes the f i e l d down t o a distance a l i t t l e above Ro/2a (see, f o r example, Zemanek (1971)). However, t h i s does n o t a l l o w f o r a general a p p l i c a t i o n o f the f a r f i e l d s o l u t i o n w i t h i n t h e t r a n s i t i o n regions. F i n a l l y , one important p o i n t t o consider i s

.

t h a t t h e r e i s a lr/2 phase s h i f t between t h e plane wave r e g i o n o f t h e beam and t h e s p h e r i c a l l y d i v e r ging wave o f the main l o b e (a reference t o t h e same propagation distance i s t a c i t l y assumed). T h i s phase s h i f t b r i n g s about a fundamental d i f f e r e n c e between t h e r e s u l t i n g difference-frequency f i e l d o f the parametric array, and the f i e l d o f t h e sum frequency as w e l l as t h e znd harmonics o f each o f t h e primaries. T h i s occurs because the source function, Qab i n the quasi l i n e a r d i f f e r e n t i a l equation (see eq.(lO)) g i v i n g t h e d i f f e r e n c e frequency pressure

*

contains pap, ( a s t e r i x denotes complex conjungate) such t h a t constant s h i f t i n the phase i n both o f the primary waves w i l l be cancelled o u t i n the source f u n c t i o n . On t h e o t h e r hand t h e source f u n c t i o n o f t h e sum frequency contains papb, which i m p l i e s t h a t the sources i n t h e spreading r e g i o n w i l l be i n e x a c t l y opposite phase t o t h e sources i n t h e plane wave r e g i o n ( i n a d d i t i o n the sources are phased like t h e generated wave, o f course). This a1 so a p p l i e s f o r the 2" harmonics. Thus, a t a c e r t a i n distance from the primary source t h e r e should e x i s t a domain where t h e c o n t r i b u t i o n from t h e spreading p r i mary waves cancels t h e c o n t r i b u t i o n from t h e plane wave region. Here the generated f i e l d should van i s h provided the phase s h i f t i n t h e primary f i e l d i s abrupt, b u t since t h e phase s h i f t takes place g r a d u a l l y over a t r a n s i t i o n region, complete cancell a t i o n i s n o t t o be expected. Instead, the f i e l d should experience a minimum i n t h i s domain. T h i s i s probably the physical i n t e r p r e t a t i o n o f t h e minimum which was observed i n t h e 2"harmonic f i e l d by Gould, Smith, Williams, Ryan (1966) and obtained

a

$ (ka)3

Due t o t h e cube r o o t t h i s may improve considerably the range o f v a l i d i t y o f t h e phase approximation, when ka>> 1. T h i s i s v a l i d on and near the symmetry axis. Generalized r e s u l t s v a l i d f a r o f f t h e a x i s are obtained by p u t t i n g L=O i n eqs. ( 5 ) and

(6). The near f i e l d i s h i g h l y f l u c t u a t i n g . However, the l a r g e amplitude o f t h e f l u c t u a t i o n on t h e a x i s i s due t o symmetry. Outside a small region surrounding t h e axis, " t h e paraxi a1 region", the f l u c t u a t i o n s are much smaller. I n f a c t , f o r

r 5 Ro/2a the sound source may be considered as a c y l i n d r i c a l beam o f plane waves, w i t h r a d i u s approximately equal t o a The pressure amplitude i s n e a r l y constant across t h e beam, b u t i t has a r i d g e near the beam edge where i t i s about 20% higher than elsewhere i n the beam. The phase angle o f the f i e l d w i t h i n t h e beam

.

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a n a l y t i c a l l y by I n g e n i t o and W i l l i a n s (1971). Likewise t h e r e i s a phase s h i f t of IT between the f i e l d o f two adjacent lobes. Therefore, i f i n

values accordingly, and i s thus represented Nf, NF by a p o i n t i n t h e ~ ~ - ~ ~ The - pgraphical l ~ ~ repre~ . sentation of the Nf-NF-plane w i l l be r e f e r e d t o as

t h e i n t e r a c t i n g wave f i e l d two adjacent lobes o f "the parameter diagram". one f i e l d overlap a common lobe i n the other, the difThe basic parameters may be combined i n source f u n c t i o n of the difference frequency w i l l be o f ferent ways, and two important derived parameters opposite phase i n the two lobes. The e f f e c t o f t h i s i s t o reduce the c o n t r i b u t i o n t o the difference frequency f i e l d from the higher order lobes when i n t e r a c t i o n s i n the f a r f i e l d of the primary waves are important.

=fi)

-/-

N - (Nf)/NF ND gives A a measure.of the (maximum) d e s t r k t i v e phase-shift w i t h i n t h e array, due t o curved wave f r o n t s , w h i l e NA may be i n t e r p r e t e d as a measure o f t h e r a t i o o f the d i f f e r e n c e frequency Rayleigh distance t o t h e are NO

. N ~

and

array l e n g t h :

4. -,Scaling parameters. Returning t o t h e parametri c array problem i t should be evident t h a t t h e nature o f the a r r a y w i l l be s t r o n g l y dependent on the l e n g t h of the i n t e r a c t i o n region ( a r r a y length) compared w i t h the near f i e l d length of the primary waves. Thus, if LA Ra Here Ra r e f e r s t o

.

t h e Rayleigh distance o f the primary wave o f h i ghest frequency. (The Rayleigh distance of the o t h e r primary wave w i l l be denoted Rb). That there i s a s i g n i f i c a n t d i f f e r e n c e between these two p r i mary f i e l d s i s e a s i l y revealed by considering cont r i b u t i o n s t o the generated sound on the axis, from source p o i n t s l y i n g on an equi-phase surface. The phase difference between such c o n t r i b u t i o n s vanishes as R + m f o r t h e c o l l i m a t e d plane primary wave, b u t reaches a f i n i t e value when the primary wavef r o n t s are curved. Thus, d e s t r u c t i v e interference may occur i f the spreading angle o f the d i v e r g i n g primary wave i s l a r g e enough. I t has proved very convenient t o use a s e t o f parameters t o characterize the s t a t e o f the primary f i e l d i n r e l a t i o n t o t h e a r r a y l e n g t h and the d i f ference frequency wave length. Two independent parameters are s u f f i c i e n t f o r t h i s purpose ift h e p r i mary f i e l d i s a x i a l l y symmetric and n o n l i n e a r a t t e nuation i s n e g l i g i b l e . Vestrheim (1973) introduced two basic parameters as f o l l o w s :

N =

where

2

R- = na /A-

. NA

i s also

p r o p o r t i o n a l t o the r a t i o o f the Westervel t beamwidth t o the beamwidth o f d i f f e r e n c e frequency sound r a d i a t e d from a p i s t o n o f diameter

2a.

The parameter diagram can be d i v i d e d i n t o three regions, a1 1 w i t h c h a r a c t e r i s t i c p r o p e r t i e s o f the parametric arrays, by drawing t h e curves where NA = 1 and ND = 1 respectively. We l a b e l the regions according t o Vestrheim (1973). Typical o f region I i s the plane c o l l i m a t e d primary wave array, when NA >> 1. Typical o f region I 1 i s the Westervelt array, and o f region 111, the d i v e r g i n g primary wave array, when ND >> 1

.

5.

- .We s h a l l now consider models where the prima-

ry waves are divergent, i n an attempt t o approximate i n t e r a c t i o n i n t h e f a r f i e l d . Thus, when r e l a t e d t o r e a l parametric arrays, such models are o f i n t e r e s t when L A > Ray i.e. i n t h e domains I 1

and 111 i n t h e parameter diagram. The model i n t r o duced by Lauvstad e t al. (1964) w i l l form t h e basis o f our i n v e s t i g a t i o n . The basic equation governing p- can be w r i t t e n on t h e form

The soluti.on o f t h i s i s expressed by the Green's Any a r r a y s p e c i f i e d by frequencies, primary source radius, sound v e l o c i t y and a t t e n u a t i o n c o e f f i c i e n t o f the medium, w i l l be assigned d e f i n i t e

function, eq. (1). S u b s t i t u t i n g

H. HOBAEK, and a l .

c8-107

i f k-/ka = 0.2 we f i n d

ND = 0.56

.

s ~ ~ ~ /= E0.87; ~ a~t

The asymptotic theory seems t o e x p l a i n the main features o f t h e observations, except t h e ones by Hobaek and Vestrheim (1977) a t ND < 0.6. However, t h e r e i s reason t o b e l i e v e t h a t these measurements are t o o c l o s e t o t h e i n t e r a c t i o n r e g i o n t o j u s t i f y comparison w i t h an asymptotic theory. We have computed t h e pressure on t h e a x i s as a

f o r t h e primary waves, we o b t a i n from eq. (10)

f u n c t i o n o f distance by using an approach suggested by Muir and W i l l e t t e (1972) f o r primary beams w i t h Bessel d i r e c t i v i t y . I f we p l o t R.p- versus R f o r d i f f e r e n t values o f ND we o b t a i n curves s i m i l a r t o the observations presented i n Fig. 5 i n t h e paper by Hobaek and Vestrheim (1977), i f When

ND g 0.6,

nounced maxima a t which i s e q u i v a l e n t t o eq.(5.1) i n Lauvstad e t a l . (1964). Here Da and Db are the Bessel d i r e c t i 2 2 By v i t y o f the two waves, and A = (d p/dp )opoc,2

.

p u t t i n g t h e lower l i m i t L equal t o Ra, o n l y i n t e r a c t i o n s i n the f a r f i e l d s o f t h e primary waves are included. The i n t e g r a t i o n over

r

i s elementary. Further,

by expanding i n a Fourier-Bessel series, we are l e f t w i t h a simple numerical i n t e g r a t i o n t o perform. T h i s a l s o makes a discussion o f c o n t r i b u t i o n s from d i f f e r e n t p a r t s re1a t i v e l y simple. The a n a l y t i c a l r e s u l t s and f u r t h e r discussions wi 11 be pub1 ished elsewhere. I n Fig. 2 we present some numerical r e s u l t s . The n e a r - f i e l d c o n t r i b u t i o n i s obtained from t h e formula o f Naze and T j b t t a (1965). I n Fig. 2, as i n Fig. 1, the normalized h a l f power angle i s p l o t t e d as a f u n c t i o n o f t h e divergence number

ND. The

curves are computed f o r parameters corresponding t o t h e experiment o f Hobaek and Vestrheim (1977), eeping t h e f i e l d number NF approximately constant $3.1, w h i l e changing k,/ka. The d i f f e r e n t curves

I V correspond t o increasing degree o f accuracy, from p u t t i n g Jo = 1, Jn = 0, n # 0 ( I ) , t o Keeping t h e t h r e e f i r s t terms i n t h e s e r i e s as w e l l as i n c l u d i n g t h e c o n t r i b u t i o n from t h e near f i e l d region ( I V ) . The near f i e l d c o n t r i b u t i o n i s almost n e g l i g i b l e a t E3dB, and i t s e f f e c t i s a c t u a l l y n o t v i s i b l e i n curve I V . I to

ND > 0.6.

however, these curves show proR = 1 - 3 LA. T h i s e f f e c t be-

comes more pronounced as ND decreases. Since i t occurs f o r t h e same range o f ND values, as t h e observed d i r e c t i v i t y i s stronger than p r e d i c t e d asymptotically, we b e l i e v e t h a t these two phenomena are connected. I n t h e next s e c t i o n we demonstrate a theoretical prediction o f w i d t h e f f e c t i n t h e near f i e l d o f m e t r i c array. Also i n t h i s case a a maximum o f R.p- i s found. Thus,

t h e minimum beama c o l l i n e a r paraclose r e l a t i o n t o t h e r e i s strong

evidence t h a t a minimum beam-width e f f e c t e x i s t s i n parametric arrays w i t h d i v e r g i n g primary waves, when ND < 0.6.

6.

-

C o l l i n e a r primary sound beams. We w i l l now introduce a d i f f e r e n t method which

i s a p p l i c a b l e when t h e primary waves are approximated by a beam o f plane waves. This model i s s u i t a b l e when considering i n t e r a c t i o n i n t h e near field. L e t x denote t h e distance along t h e symmetry a x i s o f t h e primary wave beam and a t h e l a t e r a l distance from t h e axis. Then

The increase i n d i r e c t i v i t y over t h e Westervelt d i r e c t i v i t y a t values o f

ND about 0.5,

also n o t i -

ced by Berktay and Leahy (1974), i s s i g n i f i c a n t . It becomes l a r g e r when k-/ka increases. For instance,

kx i s the component o f I(-in the l a t e r a l direction.

the x-direction,

ko i n

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of

I f we assume a small ka and a slow v a r i a t i o n p- w i t h x over a wavelength, eq. (10) becomes

' 5

Where

2

v~

il:! 7 a2 +

l a . This

i s t h e equation 30 used by Novikov e t al. (1976). They derived i t from t h e equation o f Zabolotskaya and Khokhlov(l969). It 0

i s o f f i r s t order i n x and can be solved by using a s u i t a b l e i n t e g r a l transform t o e l i m i n a t e the a dependence. I f we assume the primary waves t o have constant non-zero amplitudes f o r a < a and zero amplitudes f o r a > a, which i s t h e model used f o r t h e n e a r f i e l d i n t e r a c t i o n i n F a l t i n s e n and T j d t t a (1971) and Hobaek and Vestrheim (1971), the s o l u t i o n becomes cumbersome and i s h a r d l y t r a c t a b l e a n a l y t i c a l l y . However, f o l l o w i n g Novikov e t a l . by assuming a Gaussian d i s t r i b u t i o n , we o b t a i n an This can be explani n t e g r a l s o l u t i o n f o r p!(x,a). t e d i n a power s e r i e s and solved numerically. Curves computed o f t h e l a t e r a l amplitude d i s t r i b u t i o n a t d i f f e r e n t values o f x d i s p l a y t h e b u i l d i n g up o f t h e d i f f e r e n c e frequency sound f i e l d , as was a l s o shown by Novikov e t a l . I n t h e i r work, however, t h e primary wave absorption was neglected. When absorption i s included t h i s b u i l d i n g up o f a sound f i e l d w i l l reach a maximum a t some distance and then decay. T h i s e f f e c t i s shown i n Fig. 3, by t r a c i n g t h e l a t e r a l distances t o p o i n t s where the pressure i s 3dB below t h e pressure a t t h e axis. The parameter i s Ld/LA , which i s c l o s e l y e l a t e d LA t o NA , and the o r d i n a t e i s (a/d)/(x/LA) = tg< , where 5 as before i s the angle from the x-axis. Ld = k- d2 and d i s a c h a r a c t e r i s t i c radius o f t h e primary source. A remarkable f e a t u r e i s t h e beamw i d t h minimum which becomes more prononced and moves inwards as Ld/LA decreases. T h i s i s consistent w i t h measurements by Vestrheim and Hobaek (1971) and Vestrheim (1973). Further, on the axis, computations agree w i t h e a r l i e r r e s u l t s by F a l t i n s e n and T j e t t a (1971) and Hobaek and Vestrheim (1971). I t i s found t h a t maximum o f x p- i s c l o s e l y r e l a t e d t o t h e beamwidth minimum shown i n Fig. 3. A more exten-

F i g . 3. - V a r i a t i o n o f h a l f power angle w i t h d i s tance, scaled i n a r r a y lengths, as computed numer i c a l l y . The parameter i s Ld/LA.

s i v e discussion o f t h i s t o p i c w i l l be published e l sewhere.

REFERENCES

/1/

Bennett, M.B., and Blackstock, D.T., (1975), "Parametric a r r a y i n a i r " , J. Acoust. Soc. Amer. , 57, 562-568.

/2/

Berktay, H.O., (1969), "Some f i n i t e - a m p l i t u d e e f f e c t s i n u n d e h a t e k acoustics", i n N o n l i near acoustics, Proc. o f the 1969 Symi5SGi h e l p a t Applied Research Laboratories, t h e U n i v e r s i t y o f Texas a t Austin, Texas, USA.

" , (1972), "Some n e a r - f i e l d e f f e c t s i n e n d - f i r e arrays", i n Proc. Symp. Nonlinear Acoustics,held a t t h e U n i v e r s i t y o f Birmingham, '1-2 A o r i l ' 1971, ( B r i t i s h Acoustical Societv. " , (1972), "Near-field e f f e c t s i n param e t r i c e n d f i r e arrays", J. Sound Vib., 3, 135-143.

$

.

/3/

Berktay, H.O., and Leahy, D.J., (1974), " F a r f i e l d performance o f parametric transmit539-546. ters", J. Acoust. Soc., Amer.,

s,

141

Berktay, H.O., and Shooter, J.A., (1973), " N e a r f i e l d e f f e c t s i n e n d - f i r e 1i n e arrays", J. Acoust. Soc. Amer., 53, 550-556.

H. HOBAEK, and a1

.

Bj8rn5, L., Christoffersen, B., and Schreiber, M.P., (1976), "Some experiment a l i n v e s t i g a t i o n s o f the parametric acoustic array", Acoustica, 5, 99-106. B5e, R., (1976), Thesis f o r the cand. Real. Degree, 1976 ( i n Norwegian), Dept. o f Appl. Math. , U n i v e r s i t y o f Bergen, Bergen, Norway. E l l e r , A.I., (1973), "Adaption o f the NRL acoustic research tank f a c i l i t y f o r experiments i n parametric sonar, w i t h p r e l i m i nary r e s u l t s " , NRL Report 7513, Naval Research Laboratory, Washington DC, USA. Esipov, I.B., Zverev, V.A., Kalachev, A.I., and Naugol 'nykh, K.A., (1975), "Experiment a l research o f the parametric sound source", presented a t V I I n t e r n a t i o n a l Symposium on Nonlinear Acoustics, Moscow 191s. Faltinsen, O., and TjBtta, S., (1971), " I n t e r a c t i o n between sound waves propagating i n t h e same d i r e c t i o n " i n Proc. Symp. Nonlinear Acoustics, h e l d a t the U n i v e r s i t y o f Birmingham, - p r f l 1971, ( B r i t i s h Acoustical Society, b :o ;: 1972). a l s o a v a i l a b l e as r e p o r t No.28 (1971), Dept. Appl Math. , U n i v e r s i t y o f Bergen, Bergen, Norway.

.

Fenlon, F.H., (1974), "On the performance o f a dual frequency parametric source v i a m a t c v d asymptotic s o l u t i o n s o f Burgers' equation , J. Acoust. Soc. Amer., 55, 35-46. Gould, R.K., Smith, C.W., Williams Jr. A.O., and Ryan, R.P., (1966), "Measured s t r u c t u r e o f harmonics self-aenerated i n an acoustic beam", J. Acoust. Soc. Amer., 40, 421-427. Hobaek, H., (1967), "Experimental i n v e s t i g a t i o n o f an acoustical e n d - f i r e array", J. Sound Vib., 2, 460-463. 'I , (1970), "On the acoustical near f i e l d o f Results from a numea c i r c u l a r p i s t o n so;rce. r i c a l i n v e s t i g a t i o n , S c i e n t i f ic/Technical Rep o r t No.32, 1970, Dept. o f Physics, U n i v e r s i t y o f Bergen, Bergen, Norway.

" , (1977),

"Parametric acoustic transmitt i n g arrays-a survey o f t h e o r i e s and experiments", S c i e n t i f i c / T e c h n i c a l Report No.99, 1977, Dept. o f Physics , U n i v e r s i t y o f Bergen, Bergen, Norway. Hobaek, H., and Vestrheim, M., (1971), "Axial d i s t r i b u t i o n o f d i f f e r e n c e frequency sound i n a c o l l i m a t e d beam o f c i r c u l a r cross section", i n Proc. Sym Nonlinear Acoustics, h e l d a t t h e Unive* o f %inningham, 1-2. A p r i l 1971, ( B r i t i s h Acoustic a l Society, London, 1972).

.

Hobaek, H., and Vestrheim, M., (1977), " P a r a m e t ~ i c acoustic arrays formed by d i v e r g i n g sound beams , Acoustica, 3, 74-82. Ingenito, F., and Williams, Jr. A.O., (1971), "Calculations o f second-harmonic generation i n a p i s t o n beam", J. Acoust. Soc. Amer., 2, 319-328.

JOURNAL DE PHYSIQUE Lauvstad, V. , Naze, J. and T jGtta, S. (1964), "Nonlinear i n t e r a c t i o n o f two sound waves", Acta U n i v e r s i t a t i s Bergensis s e r i e s Mathematica Rerumque Natural ium No. 12, 1964. M o f f e t t , M.B., and Mellen, R.H., (1976), "On parametric source aperture f a c t o r s " , J. Acoust. Soc. Amer., 60, 581-583. Muir, T.G., and W i l l e t t e , J.G., (1972), "Param e t r i c a c o u s t i c t r a n s m i t t i n g arrays", J. Acoust. Soc. Amer., 52, 1481-1486. Naze, J. , and T j a t t a , S., (1965), "Nonlinear I n t e r a c t i o n o f two sound beams", J. Acoust. Soc. Amer., 2, 174-175. Novikov, B.K., Rudenko, O.V., and Soluyan, S.I. (1976), "Parametric u l t r a s o n i c r a d i a t o r s " , S o v i e t Phys. Acoust., 21, 365-368. Smith, B.V., (1971), "An experimental study o f a parametric e n d - f i r e array", J. Sound. Vib., 14, 7-21. TjGtta, S., (1967), "Some nonlinear e f f e c t s i n sound f i e l d s " , J. Sound Vib., 6, 255-267. Vestrheim, M., (1971), "Thesis f o r the cand. Real. degree, ( i n Norwegian), Dept. o f Physics, U n i v e r s i t y o f Bergen, Bergen, Norway.

" , (19731, "A parameter representation o f the parametric-acoustic array1',' i n Finite-am li tude wave e f f e c t s i n f l u i d s , Proc. Symposium-Copenhagen, ( I P C Sci. Tech. Press. itd.. , Gui 1ford, ~ nand,~ (1974). l

d

Vestrheim, M., and Hobaek, H., (1971), "Angular d i s t r i b u t i o n o f n o n l i n e a r l y ~ e n e r a t e dd i f f e r e n c e frequency sound", i n roc." s$np. Nonl i n e a r Acoust i c s , h e l d a t t h e U n i v e r s i t y o f Birmingham, 1-2. A p r i l 1971, ( B r i t i s h Acoustical Society, London, (1972). Westerwelt, J.P., (1963), "Parametric acoustic array", J. Acoust. Soc. Amer., 35, 535-537. Zabolotskaya, E.A., and Khokhlov, R.V., (1969), "Quasi- p l ane waves i n t h e n o n l i n e a r acoustics of confined beams", Soviet. Phys. Acoust., 35-40.

g,

Zemanek, J., (1971), "Beam behaviour w i t h i n t h e n e a r f i e l d o f a v i b r a t i n g p i s t o n " , J. Acoust. Soc. Amer., 9, 181-191. Zverev, V.A. and Kalachev, A. I., (1968), "Measurement o f the s c a t t e r i n g o f sound by sound i n t h e superposition o f para1l a 1 beams", S o v i e t Phys. Acoust., 173-178.

14,