ON THE AXIAL FIELD OF A PARAMETRIC ACOUSTIC RADIATOR

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ON THE AXIAL FIELD OF A PARAMETRIC ACOUSTIC RADIATOR

Francis Fenlon

To cite this version:

Francis Fenlon. ON THE AXIAL FIELD OF A PARAMETRIC ACOUSTIC RADIATOR. Journal

de Physique Colloques, 1979, 40 (C8), pp.C8-119-C8-125. �10.1051/jphyscol:1979821�. �jpa-00219526�

JOURNAL DE PHYSIQUE Colloque C8, supplement au n"ll, tome 40, novembre 1979, page C8-119

ON THE A X I A L FIELD OF A PARAMETRIC ACOUSTIC RADIATOR FRANCIS H. FENLON

Department of Engineering Saienoe and Medham.es, and the

Applied Research Laboratory, The Pennsylvania State University P.O. Box 30, State College, PA 16801, U.S.A.

Résumé . - Cet a r t i c l e étudie le champ axial d'un radiateur acoustique paramétrique asymétrique, pro- duit par l ' i n t e r a c t i o n non-linéaire des ondes primaires d'une ouverture rectangulaire dans un f l u i d e . Ce problème est étudié au moyen d'une nouvelle solution de l'équation parabolique non-linéaire du second ordre. La forme fonctionnelle de cette solution est particulièrement intéressante parcequ' e l l e permet d'approcher la structure du champ produit à la fréquence, la différence des fréquences primaires. Pour toute distance moindre que la distance Rayleigh de l'onde primaire d'une ouverture axisymétrique ( t e l , par exemple, un piston projecteur carré ou c i r c u l a i r e ) la solution se rapproche assyitiptotiquement de la forme déjà dérivée par Novikov, Rudenko et Soluyan / 6 / . Dans le champ l o i n - tain du radiateur paramétrique, e l l e approche aussi de façon asymptotique la forme dérivée au préa- lable par cet auteur/10/. A 1'encontre toutefois des approximations généralement f a i t e s , la nouvelle solution évite de combiner a r t i f i c i e l l e m e n t la solution de l'équation d'une onde sphërique et celle d'une onde plane-ondes produites par un radiateur paramétrique. Bien des a r t i c l e s nous permettent d'affirmer que ce procédé a été souvent u t i l i s é dans le but de d é f i n i r le champ établi par la d i f - férence de fréquence produite dans le f l u i d e par les ondes primaires d'un piston radiateur p l a t . Finalement, dû à sa forme relativement simple, cette nouvelle solution devrait f a c i l i t e r l ' é t a l o n - nage de radiateurs paramétriques au moyen de mesures faites dans un champ a proximité de ces r a - diateurs.

Abstract. - In this paper the axial f i e l d of an asymmetric parametric acoustic array generated by non-linear interaction of the primary waves of a rectangular aperture in a f l u i d is investigated via a new solution of the second-order non-linear parabolic 'wave' equation. The functional form of t h i s solution is p a r t i c u l a r i t y intriguing because of the insight i t provides into the composition of the difference-frequency f i e l d . Within the primary wave Rayleigh distance of an axisymmetric aperture ( i . e . square or circular piston projector) the solution asymptotically approaches the form derived by Novikov, Rudenko and Soluyan / 6 / . Likewise, in the f a r - f i e l d of the parametric array i t . asymp- t o t i c a l l y approaches the form previously derived by the author / 1 0 / . Unlike previous approximations however, the new solution completely obviates the necessity of a r t i f i c i a l l y matching plane and sphe- r i c a l parametric array solutions in order to define the difference-frequency f i e l d of a plane piston radiator. F i n a l l y , on account of i t s comparative simplicity the new solutions should enchance the f e a s i b i l i t y of calibrating parametric arrays from nearfield measurements.

1. Introduction. - During the past seven years the f e a s i b i l i t y of calibrating parametric acous- t i c arrays via measurements made in the v i - c i n i t y of the primary wave projector has been greatly f a c i l i t a t e d by refinements in the ana l y t i c a l formulation of nearfield models. Other than mentioning the principal advances of t h i s period however, these models w i l l not be considered here. For further details the i n - terested reader is referred to Hobaek's / l / comprehensive review of parametric array r e - search prior to 1978. Before continuing, we note as this point that the following discus- sion is confined to the case of 'unsaturated' parametric arrays where the time waveform is distorted and quasi-linear analysis is thus permissible.

One of the f i r s t comprehensive i n v e s t i - gations of the difference-frequency f i e l d

formed by the primary waves of a finite-amplitude projector was made by Willette / 2 / via numerical evaluation of the f i e l d integrals. In order to gain deeper insight into the physical form of the near- f i e l d solution other investigators including Berktay / 3 / , Bartram and Payson Fugitt / 4 / , and Rolleigh / 5 / made use of geometrical and physical approximations to simplify the volume source dis- t r i b u t i o n function, thus permitting them to derive analytical approximations of the difference-fre- quency f i e l d . These models were of course limited by t h e i r inherent assumptions. For example, Rolleigh's / 5 / model is based upon the assumptions that the difference-frequency signal undergoes sphe- r i c a l spreading at the face of the projector, and that viscous absorption has no effect upon the non- linear interaction within the near-field of the parametric array. Although these assumptions are quite reasonable at frequencies where the primary

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979821

~ 8 - 1 2 0 JOURNAL OE PHYSIQUE wave absorption loss within the Rayleigh distance of

t h e projector is very small, they become increasin- gly inadequate a t higher frequencies. Moreover, s i n - ce s p e c i f i c a l l y nearfield model s do not asymptotical- l y approach the well established f a r - f i e l d solutions /1/ a t large ranges, i t i s d i f f i c u l t t o dgtermine the ranges a t which they cease t o provide a r e l i a - ble approximation of the difference-frequency f i e l d . The same limitation applies t o a subsequent near- f i e l d model derived by Novikov, Rudenko and

Soluyan /6/. In t h i s model the f i e l d i n t e g r a l s trans- verse t o the direction of wave propagation a r e eva- luated a n a l y t i c a l l y by using Gaussian beams t o appro- ximate the s p a t i a l d i r e c t i v i t y of the primary waves.

On account of the l a t t e r advantage Hobaek /1/ correc- t l y notes t h a t t h i s model i s of greater u t i l i t y than i t s predecessors /3/

-

^/5/, because apart from neglec- ting primary wave side-lobe s t r u c t u r e , i t does not de- pend on other more r e s t r i c t e d physical assumptions.

I t should be noted however, t h a t the advantages of Gaussian beams, although somewhat novel in Non- l i n e a r Acoustics, have previously been exploited t o g r e a t advantage i n Nonlinear Optics, as f o r example, Asby

'

s /7/ analysis of second-harmonic generation i n optical resonators c l e a r l y demonstrates.

More recently a generalization of Novi kov, Rudenko and Soluyan's /6/ nearfield analysis of axi symmetri c parametric radiators by Novi kov, Rybachek and Timoshenko /8/ has overcome the limi- t a t i o n s of the previous model /6/ by including the e f f e c t of diffraction-induced spreading losses on t h e primary waves. This solution

,

which represents the fundamental mode of a more general analysis undertaken independently by the author /9/ via Gauss-Laguerre modes of the homogeneous progressive wave equation i n the frequency domain, asymptotically assumes the form of a f a r - f i e l d solution previously derived by the author /lo/. In addition t o these models /8/, / 9 / , another generalized model has recently been developed by Mellen /11/ f o r both axisymmetric and asymmetric f i e l d s which attempts t o s t r i k e a balan- ce between analytical approximation and numerical analysis. Despite i t s comprehensive features howe- ver, t h i s model s t i l l r e l i e s too heavily on numerical evaluation, which a1 though of s l i g h t importance i n system design s t u d i e s , does not lead t o simple clo- sed-forms approximations from which physical i n s i g h t s can be gained.

2. Analysis.

-

The propagation of progressive f i n i t e - amplitude waves i n a dispersionless thermo-viscous

f l u i d can be described by means of a paraxial f i e l d equation derived by Zabolotskaya and Khokhlov /12/

which assumes the form of a nonlinear parabolic equation in the frequency domain. I f wave propa- gation occurs i n the direction of the positive z a x i s , t h i s equation can be expressed i n r e c t i l i

-

near Cartesian coordinates as

where pu = Fb {p) i s the Fourier transform of the acoustic pressure p ,

ot,

i s the ther-viscous a t t e - nuation c o e f f i c i e n t , po i s the density of the f l u i d , co i s the small-signal speed-of-sound, 0 i s the c o e f f i c i e n t of nonlinearity 1131 and k i s the acous- t i c wavenumber.

As previously shown by the author / l o / the above equation can be reexpressed in t h e form of an integral equation given by

*

exp {-cxW(z-2')) dx' dy' dz'

For the case of a plane rectangular aperture radia- t i n g simultaneously a t angular frequencies wl and u2 i t will now be assumed, as shown i n figure 1, t h a t the pressure f i e l d s a t the face of the pro- j e c t o r can be represented by the Gaussian p r o f i l e

where xo and yo a r e 'spot s i z e s ' of the function i n the

x

and y d i r e c t i o n s , respectively.

FRANCIS H. FENLON

Po

4

BEAM R A D l l -

_

Fig. 1.

-

Source Configuration.

Assuming f u r t h e r t h a t t h e primary waves a r e n o t s i - g n i f i c a n t l y perturbed by 'secondary' waves genera- t e d v i a n o n l i n e a r i n t e r a c t i o n i n the f l u i d

-

^{a con-}

d i t i o n r e s t r i c t e d t o weak f i n i te-ampli tudes-they are thus given c o r r e c t t o f i r s t - o r d e r by

G 2

p (x,y,z)=(i/2a)p expt-a - ~ K ~ [ ( z ~ ~ / z ) (x/xo) +

Wa OR '"11

2 2 1/2

where zox=koxo12

,

zoy=koy0/2

,

z ~ = ( z ~ ~ z ~ ~ )

,

and K~ = ka/ko

Provided t h e p r o j e c t o r does n o t r a d i a t e d i r e c t l y a t t h e difference-frequency (i.e. W- = wl

-

w2), t h e f i e l d o f t h e l a t t e r which r e s u l t s from n o n l i n e a r primary wave i n t e r a c t i o n i n t h e f l u i d i s g i v e n along t h e beam a x i s as,

where A ( z l ) = f (F~+z/zox)-(z'/zox) ( l - & ~ z / z o x ) l i

and a T = a + a - a w *

'9

^W-

w i t h PG = 8 lp: pEZ ko zO/po cOZ

,

t h e assumption

K~ K~

*

1 having been made on t h e grounds t h a t f o r most a p p l i c a t i o n s o f i n t e r e s t w- << wl. w2.

I n order t o g a i n f u r t h e r p h y s i c a l i n s i g h t w i t h a minimum o f a n a l y s i s t h e above i n t e g r a l w i l l now be evaluated approximately f o r small and l a r g e va-

-9

r l o w i - l u e s o f t h e viscous absorption l o s s a '

-

t h i n t h e 'matching d i s t a n c e ' r'o=ro(wo/w-). P r i o r t o t h i s however, i t i s expedient t o r e p l a c e t h e Gaussian beam parameters by those o f a u n i f o r m l y e x c i t e d r e c t a n g u l a r a p e r t u r e i n t h e h o r i z o n t a l and v e r t i c a l planes, by matching b o t h beams a t t h e i r r e s p e c t i v e half-power p o i n t s . This g i v e s zoX=roX/2,

'/2

Zoy = roy/2

.

^{Zo =}

^Pox

r o z ) 12 = (zox zoy)

%

where rox = k0ax/2 2

.

^{roy = k011:/2}

^,

ro = (r,, roy)

'4

2 G

and

k=B

pO1po2 ko r0/p, co

,

s i n c e Pot zohY

= POI1 rox,y

,

( a = 1, 2).

Thus, w i t h z = r, and aT =

9

ro, t h e d i f f e r e n c e - frequency f i e l d f o r a s T = aT/K- << 1 becomes

c8-122 JOURNAL DE PHYSIQUE ( ~ - % / Z ) e x p ( - a r ) 2

*-

(aTSxy/SxSy

-

In e x ~ ( - &

*-

r )

,

f o r a+ << 1

1-Sx -(r/ro)SyCSy

[ I -

I)

exp ( - a T r / r o ) ] e x p ( - a * r )

,

f o r a + y.1

-

In order t o exemplify the good agreement of t h i s where the l a t t e r approximation is only valid f o r model with experimental r e s u l t s , i t s predictions a r e

aTr/ro < 1, but may be used t o estimate the f a r - compared in f i g u r e 2 w i t h data measured by Muir and f i e l d presence by l e t t i n g r / r o ⁺ l/aT i n S,, S and ^{W i} 1 l e t t e /14/. Similarly good agreement between the-

Y ory and experiment has a l s o been obtained f o r other Sxy.

where

PEAK PRESSURE (dB re 1 p Pa)

140

130

0.4 1.0 10.0

r (METERS)

Fig. 2.

-

Comparison of theory and experiment /14/

a s r + m f o r a 7.62 cm diameter c i r c u l a r piston projector radiating simultaneously a t 482 kHz and 418 kHz i n fresh water with a; = ol-yro(wo/u-) = 0.116 np.

values of af thus lending a high degree of credibi- a s r + m 1 i t y t o the model.

I t should be noted t h a t the axisymmetric form of p,,, ( r ) , which i s identical t o the fundamental mode-zf the a u t h o r ' s /9/ more general solutions, and

sxy=/{l-(r/rox)S:l { I - ( r / r OY

)s2}.

Y

reduces t o Novi kov, Rudenko

,

and Sol uyan' s /6/

nearfield solution f o r r/ro << 1.

J

P ( r ) e i ( ~ - % / 2 ) In ( 1

-

^ir/Ld)

,

Ld ' ~ - r ~ / 4 -

*-

On the other hand, in the f a r - f i e l d of the para- metric array, where

In the case of an axisymmetric r a d i a t o r ( i . e .

a square or c i r c u l a r p i s t o n ) , rOX = r = r0 giving K- r / r o >> 1, p,-

OY ( r ) becomes

2 1+4r / r o 2 2 p ( r )

-.

i ( ~ - P s / 2 ) ( r o / r ) I n (l+i/a+) exp (-a r )

.-p,

/2

* -

^a-

P,

-

(rf

-.

l - i ~ - r / r ~ ^[aTr/ro

^-

^In 11

1-i 4r/~-r,, 1 2

= V K - ~ 1 2 ) ( r 0 / r ) + l n ( l + l / a + )+i tam-l[l/p+!~

FRANCIS H. FENLON c8-12.3

exp ( - a r)

"'-

-+ - i ( ~ - P , / 2 ) ( r ~ / r ) I n ( a ' ) exp (-a r )

,

T w-

f o r a ' << 1 T

+ i ( ~ - 2 g / 2 a T ) ( r o / r ) exp ( - a r ) 9

w- f o r a ' >> 1

T

T h i s f a r f i e l d s o l u t i o n which was f i r s t deduced by Berktay /15/ i n an e n t i r e l y d i f f e r e n t manner, and subsequently r e d e r i v e d by Novikov, Rybacheck, and Timoshenko /8/ i s an approximate form o f a more general a x i symmetric f a r - f i e l d s o l u t i o n p r e v i o u s l y d e r i v e d by t h e author

/lo/.

The l a t t e r , which r e - s u l t s from a more p r e c i s e e v a l u a t i o n of t h e f i e l d i n t e g r a l along t h e a x i s

,,

can be expressed as.

ca

where El(-iaf) =

1 ^cX

^dx i s t h e exponential i n t e g r a l f u n c i o n .

i a ' T

The r e s p e c t i v e magnitudes o f El(-ia' ) and l n ( 1

+

^i/a;) are d e p i c t e d i n f i g u r e 3 as f u n c t i o n s T

PRESENT THEORY EXPERIMENTAL DATA

SL;,*(~B r e l p P a - m - kHz) Fig. 3.

-

Magnitude c h a r a c t e r i s t i c s .

Fig. 4.

-

Phase c h a r a c t e r i s t i c s .

Fig. 5.

-

Conversion e f f i c i e n c y c h a r a c t e r i s t i c s . I n f i g u r e 5 t h e conversion e f f i c i e n c y

I

rpU- ( r ) au-r

TI

- = l i m

- I

c a l c u l a t e d from t h e f a r f i e l d s o l u t i o n w i t h t h e a i d o f f i g u r e 3 i s compared w i t h experimental r e s u l t s obtained by M o f f e t t and Mellen /16/ as a f u n c t i o n o f t h e scaled primary wave source l e v e l SLlS2 = 20 loglO r o f o ) f o r equal unsa t u r a t e d primary wave amplitudes ₌ Pol = PO2

,

where f o = (wl

+

w2)/4a

,

and f- = w-/2n.

Returning now t o t h e more general expression f o r p ( r ) which was d e r i v e d f o r t h e case o f a r e c -

W-

t a n g u l a r aperture, t h e f a r - f i e l d ' g a i n ' o f t h e para- m e t r i c a r r a y r e l a t i v e t o t h a t o f an i d e a l v i r t u a l - e n d - f i r e a r r a y which would be formed i n t h e absence o f d i t t r a c t i o n - i n d u c e d losses, Go, i s given by

o f a; = L /La, where L = r;

,

and t h e i r phase cha- r a c t e r i s t i c s a r e d e p i c t e d i n f i g u r e 4.

/Go\ = l i m r + w p

P, ( r )

c8-124 JOURNAL DE PHYSIQUE

3. Conclusions

-

Closed-form a p p r o x i m a t i o m f o r t h e

w i t h experimentnal data, a new expression has been obtained f o r t h e ' g a i n ' o f a d i f f r a c t i o n - l i m i t e d l + i a f

+/im

a x i a l f i e l d o f a parametric a r r a y formed by t h e p r i -

W 2

-

where p ( r ) = i ( K p 123 )(r /r) exp (-a r), t h e su- parametric a r r a y r e l a t i v e t o t h a t of an i d e a l v i r -

W- - w T 0 W-

t u a l -end-fire-array. T h i s expression g i v e s r e s u l t s p e r s c r i p t W i n d i c a t i n g t h a t t h i s s o l u t i o n corresponds

which c o i n c i d e w i t h Berktay and Leahy's /18/ nu- t o t h a t o r i g i n a l l y d e r i v e d by Westervelt /17/ i n h i s

m e r i c a l e v a l u a t i o n o f t h e f i e l d i n t e g r a l s . I n gene- pioneering paper on the parametric a c o u s t i c array.

r a l , t h e comparatively simple forms o f t h e appro- T h i s new closed-form approximation assumes t h e

ximations derived h e r e i n enables them t o be evalu- form o f Berktay ' s /15/ approximation f o r axisymme-

t r i c parametric arrays, ated v i a desk c a l c u l a t i o n s , thus s i m p l i f y i n g t h e l G o l = t a s k of p r e d i c t i n g t h e near and f a r f i e l d performan-

= af

a+ i l n ( l + i / a + ) l when a+,=aiy ( i . e . rox =

ray).

ce o f parametric acoustic arrays under unsaturated

I

^{I I I I I} c o n d i t i o n s .

REFERENCES l n [ l

+

ⁱ { 11

a + [ l G ( a i X / a i y ) 1/2+-$ j x ) 'I2]

/1/ Hobaek. H., U n i v e r s i t y o f Bergen S c i e n t i f i c / Technical Report No. 99, Berqen, Norwav 1977.

mary waves o f a r e c t a n g u l a r a p e r t u r e i n a f l u i d have been d e r i v e d v i a quasi l i n e a r i z a t i o n . From these approximations, which a r e i n good agreement

I

^-

^-

^~

-

m

-15 /2/ Willette.J.G., J. Acoust. Soc. h e r . , 1972,

-

52, 123.

u0

-

-

8' /3/ Berktay, H.O., J. Sound and v i b r a t i o n , 1972,

-20 1.25 -

-

20, 135.

2.5 /4/ Bartram, J.F., and Payson F u g i t t , R., J.

3.75 Acoust. Soc. h e r . , 1974, 55, S23.

5 -

-

-25 6.25 /5/ R o l l e i g h , R.L., J. Acoust, Soc. h e r . , 1975,

7.5 58, 964.

10

-

/6/ Nsvi kov, B. K., Rudenko, O.V., and Soluyan ,S. I.,

.

^{.* ? 3 , .} S o v i e t Physics-Acoustics, 1976,

-

^{21, 365.}

/7/ Asby, R., J. Opto-Electronics, 1969, 1, 165.

m ^-

Fig. 6.

-

R e l a t i v e gain c h a r a c ~ ~ ~ i s t i c s . /8/ Novikov, B.K., Rybachek, M.S., and

Timoshenko, V.I., Soviet. Physics-Acoustics, 1977, 23, 354. -

-

-

I n f i g u r e 6 t h e axisymmetric g a i n c h a r a c t e r i s t i c compu-

t e d from the above expression i s depicted as a /9/ Fenlon, F.H., J. Acoust. Soc. h e r . , 1978, 63, S10.

f u n c t i o n o f La/Lx = l / a f x and L /L _{a Y} = l/a;y

, _{/ l o /} -

Fenlon, F.H., J. Acoust. Soc. h e r . , 1974, where Lx =

rok

^{and L} = r '

.

These c h a r a c t e r i s -

Y 'JY

-

^{55, 35.}

t i c s a r e i n complete agreement w i t h those obtained

/11/ Mellen, R.H., J. Acoust. Soc. Amer., 1976, by Berktay and Leahy /18/ v i a numerical e v a l u a t i o n

-

59. S28. -.

.

^{- - -}

o f the f i e l d i n t e g r a l s , where i t should be noted

/12/ Zabolotskaya, E.A., and Khokhlov, R.V., t h a t t h e l a t t e r r e s u l t s are p l o t t e d as f u n c t i o n s o f

3 .

S o v i e t Physics-Acoustics, 1969,

15,

35.

= (2/n)

* /

^{(L~/L, ) and}

3s,

^{= (2/n)14}

JW .

/13/ Beyer, R.T., American I n s t i t u t e o f Physics Handbook, McGraw H i l l , New York, 1972.

/14/ Muir, T.G., and W i l l e t t e , J.G., J. Acoust.

Soc. h e r . , 1972,

-

52, 1481.

FRANCIS H. FENLON /15/ Berktay,H.O. ,J. Sound and vibration, 1965, .- 2 ,

462.

/16/ Moffett, M.B., and Mellen, R.H., J. Acoust.

Soc. Amer., 1977,

61,

325.

/17/ Westervelt, P.J., J. Acoust. Soc. h e r . , 1963, 35, 535.

1181 Berktay, H.O., and Leahy, D.J., J . Acoust.

Soc. h e r . , 1974,

55,

539.