Virtual arrays for underwater reception

J. Sound Vib. (1969) 9 (2) 295-307

V I R T U A L ARRAYS F O R U N D E R W A T E R

RECEPTIONt

H . O. BERKTAY AND C. A . AL-TEMIMI

Department of Electronic and Electrical Engineering, The University of Birmingham, Birmingham 15, England (Received 15 March 1968)

A simple device is described which utilizes interaction between an incoming signal wave of low frequency and a locally-generated pump column to provide highly-directional reception. The device consists of a pump transducer and a receiving transducer placed along the acoustic axis of the former. Predictions made as a result of theoretical considerations (based upon simplified models for studying non-linear interaction between the two waves) are in good agreement with experimental results. Considerations affecting the design of a practical device are discussed and suggestions are made about forming arrays of such devices for beam shaping, side-lobe reduction or beam deflection.

1. INTRODUCTION Possible underwater transmitting applications of virtual end-fire arrays (formed by the non-linear interaction of two acoustic waves of relatively high frequencies) have been studied elsewhere. This phenomenon has already found some practical applications and further developments in this field look very promising. The present paper discusses a device which also exploits non-linear interaction between two acoustic waves for wide-band, highly-directional reception of underwater acoustic signals. The device consists of a transducer transmitting a pump wave of high frequency (wl) and a receiving transducer mounted along its acoustic axis. An incoming plane wave (of frequency w2) will interact with the locally-generated pump wave, producing interactionfrequency components. In this paper only the sum and the difference-frequency terms will be considered; in other words, the received signal will be at a frequency Q = w~ ~ w2, where COl >~ £02.

A study of interaction between an incoming acoustic plane wave and a locally-generated pump column was reported in reference 1. The analysis was restricted at that time to the pressure field in the far field of the virtual array. When the pump frequency is much higher than the frequency of the signal wave, the effective length of the virtual array (which is of the order of 1/IA [ where A is the absorption parameter, "1 + -2 - ~, and [A I ~ 1 when £01 >> £02) is large [2, 3]. Hence, to be in the far field of the array, the receiving transducer would have to be placed at a great distance from the pump transducer, with the resulting large attenuation of the interaction-frequency component. This would reduce the attractiveness of the device in practical applications. The analysis is now extended to the near field of the virtual array using the techniques discussed in detail in the literature [1-5]. Briefly, the method used assumes virtual sources 1 Presented at the British Acoustical Society meeting on "Acoustic Arrays", Birmingham University, 2 January 1968. 295

296

H. O. BERKTAY AND C. A. AL-TEMIMI

(at the frequency g2) to be created as a result of the non-linear interaction between the signal and the pump waves and it is further assumed that the waves radiated from these sources propagate linearly. This quasi-linear approach (used in conjunction with a very simplified picture of the pump wave in the vicinity of the transducer) has produced results which agree with the measured characteristics of such receiving devices very well. Some practical problems which would arise in the use of such a device are considered briefly in section 5. 2. THEORETICAL CONSIDERATIONS The basic configuration to be considered is shown in Figure 1. The pump transducer is a square piston of sides 2b, placed in a vertical plane with one pair of the sides (and the direction of propagation of the signal wave) on a horizontal plane. The signal wave is assumed to propagate horizontally, at an angle 0 to the axis of the pump transducer which is taken as the x axis. The receiving probe is on the x axis.

Pump column

o- I~S__ Pumpaxis

_ _

~ving probe

Figure 1. Geometry of virtual array. The field in the vicinity of a square transducer shows a Fresnel diffraction pattern [6]. A complete analysis of the source function at the interaction frequencies and then of the scattering behaviour of the interaction components would be very complicated and laborious. In the present paper, two simplified cases will be studied. In the first case, the pump column will be considered to be of large cross-sectional dimensions compared with the wavelength at the pump frequency, thus causing the interaction-frequency component to be radiated as a plane wave. (This may be approximated to in practice by placing the receiving transducer in the near field of the pump transducer.) In the second case, the cross-sectional dimensions of the pump column is considered to be negligible, so that the interaction components arising from a small length can be assumed to radiate as a spherical wave. This is a concept which is more difficult to justify, but can be shown to represent the case where the receiving transducer is in the far field of the pump transducer. In either case, then, the pump and the signal waves can be represented as plane waves travelling in different directions, Pl = Pl exp (--~1 x). cos (col t -- kl x),

(1)

P2 = P2 exp (-~z. x cos 0 + y sin 0). cos (w2 t -- k2 x cos 0 + y sin 0).

(2)

In practical cases, the dimensions of the pump transducer would be such that 2b~2 < 1 ; then the y term in the exponent in equation (2) can be neglected.

VIRTUAL ARRAYS FOR UNDERWATER RECEPTION

297

2.1. THE PLANEWAVE CASE The source function at the interaction frequencies on a surface x = constant can be written as

q(x,y,z,t)=-

QPI P2(cos 0 + 2 4 Y ) e x p ( - ~ l + ~2cos0.x).

poCo

.sin (~2t - x . k s + k2 cos 0 q: k2ysinO)

(3)

where the upper signs are used for the sum frequency and the lower signs for the difference frequency. The radiation of the interaction-frequency component can be studied in two stages. The surfaces of constant - x form apertures which are insonified with a phase taper of ±k2 sin # along the y direction. As the aperture is radiating at a frequency ~ , the preferred direction of radiation will be along an angle 4 = sin-l(oJ2sin0/~'2). In the near field of the wafer (calculated for frequency ~2) the radiation can be assumed to form a well-defined column in the direction 4, while in the far field, a diffraction pattern of the form sin M / M , directed along 4, will be obtained. In the present case the receiving transducer is assumed to be in the near field of all such apertures. Hence, the pressure within the column will be

p(R, 8, t) = - e x p ( - ~ R ) [I2P I P2(cos 0 + 7)/2p0 co3].exp (-~1 + ~2 cos 0 - ~. x). . sin ($2t - K R - x. ks 4- k2 cos 0 - k). dx,

(4)

as K = kl 4- k2, ks 4- k2 cos 0 - K = qzk2(l - cos 0) ~ q:fl, say. I f oil >> oJ2, tAI ~ 1, then,

p(R, 8, t) - -[I2Ps P2(cos 0 + 7). exp (-~R)/2p0 C3o].R sin (I2t - K R + fiR/2). . sin (Rfl/2)/(Rfl/2).

(5)

The directivity function (for variations in 0) can be calculated directly as

D(O) = ]p(R, 8, t )/p(R, O, t)l = sin (Rfl/2)/(R3/2).

(6)

Here we assume that cos 0 + ~, - 1 + V as 0 < 1 and 7 is of the order of 3. This directivity function is identical to that of a continuous end-fire array of length R, receiving signals of frequency co2.

D(O) has zeros at sin (8/2) = (n)t2/4R) 1:2, 3 dB beamwidth of the main lobe is 0~ = 2V'A2/R;

(7)

the first side lobes occur for sin (0/2) = ~3~2/4R and are about 13.5 dB below the main lobe. 2.2. THE WIDTH OF THE PUMP COLUMN IS SMALL

I f the distance R is greater than R0, the Fresnel distance~f of the transducer at frequency ~2, an approximate calculation can be made using the source function calculated in the previous section. In the present case the sources on surfaces of x = constant will be assumed to contribute to the pressure at the receiving transducer as a near-field effect for R - Ro ~< x ~< R and as a far-field effect for 0 ~
298

H.O. BERKTAY AND C. A. AL-TEMIMI

The near-field component, p., can be calculated from the work of the previous section, and the far-field component, Ps, using the methods discussed in reference 2: sin (fiR0/2)

p,(R, 0, t) = -[£2P 1P2(1 + 7')-Ro/2po c3]. exp (-~l + a2. R). ~

•sin (g2t - K R + ft. R - Ro/2). Similarly, using equation (3) for ps(R, O, t) -

.

(8)

IAI < 1,

g'22Pl P2(1 +

4~rpoc4 y)" (2b)2 exp (-~1 + ~2. R). R-Ro

sin (k2 b sin 0) ( k2 b sin 0 J

cos ($2t - K R + fix). dx. R - x

(9)

x=0

The integral in equation (9) can be shown to reduce to (Ci 2 +

Si2) 1/2 .cos (Ot - K R

where

+ fiR - ~ )

tan • = Sio/Cio, ~a

f

Cio-jSio=

and

exp(-JV)

v~Ro

Therefore, pAR, O, t) =

~2/~?

)') (2b)2 exp (_~q + ~2.R)(Ci 2 + S,:2"d/2 oj •

sin (k2 b sin O) cos (I-2t (k2 b sin 0)

K R + f i R - - @).

I0 0-9 0'8 0"7 0-6 0"5 0'4 0"3 0'2 01

\

/

0

,SR

Figure 2. Directivity functions. - - - - , A plot of sin BRI2 BRI2

A plot of

R f ° "J"~

~L-xdX

"

1

with

R/Ro = 1o R / R o = 8. R/Ro = 5

(10)

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It can be shown [7] that for R ~ Ro the pressure component p, predominates, while for

R >>Ro, p: is the main component of the resultant pressure. Hence, for R ~ R0, D(O) - sin (flRo/2)/(flRo/2)

(11)

where Ro "=. 1.5b2/Al, while for R >> R0, D(0)=

sin (k2 b sin 0) ,,~.2

-(k~

"tt~° + Si2)l/2/l°gE(R/R°)"

(12)

The first term in equation (12) is an aperture factor arising from the finite dimensions of the column, and is nearly unity in most practical cases. Then, equation (12) becomes

h(O) - (Ci~ + Si2)l/2]logE (R/Ro).

(13)

D(O) calculated from equation (13) is shown in Figure 2 as a function of fiR for various values of R/Ro; that calculated from equation (7) is also shown in the same figure. These normalized curves are very similar in many respects. The basic difference between them is the presence of zeros in the directivity function calculated from equation (7), while the other curves only exhibit finite minima. The 3-dB beamwidths calculated on the bases of equations (7) or (13) differ by a few per cent only. 2.3. PARAMETRICAMPLIFICATIONAND THE BEAMWIDTHOF THE RECEIVER From equation (10), for 0 = 0,

p.r(R,O, t )

-

Q2PI P2(1 + ~') (2b)2. exp (-=1 + =2" R). loge (R/Ro). cos (Qt - KR). (14) - 4,rrpo co4

Further, as pointed out in section 2.2, this is the larger component of the pressure at frequency Q when R >> R0. Similarly, for 0 = 0, the signal frequency pressure is given by

p2(R, 0, t) = P2 exp (-=2 R). cos (to2 t - k2 R). (15) The ratio of the amplitudes of p:(R,O,t) and p2(R,O,t) at the receiver represents upconverter type of parametric amplification [2] arising from the non-linear interaction process. 4 2 o -2 -_---g ~ - - --4

7=5.5 ~--~---I-4

....

~---L--~._~__.lt__4_..,K

-12 -14 -16 -18 -20

7=0

I

I

50

I

I

I00

I..I

I

150

Figure 3. Parametric gain versus signal frequency. Po r a 2 p ]

frequencyfl = 5.9 MHz. ), = 2C'---~o' L-----3-' e p j p=p"

I

I

I

200 250 fzkHz~-

I

I

300

I

I

:550

, Theoretical; - - - , experimental. Pump

300

B E R K T A Y A N D C. A. A L - T E M I M I

H.O.

A detailed discussion of such parametric amplification is being prepared separately. For the purposes of the present paper it is perhaps sufficient to note that the parametric amplification can be written as ~r22Pl (1 + y ) S , G(R) -~p~ . exp (-cq R).log,(R/Ro). (16) In this expression the only terms dependent upon the signal frequency are £2 and R0. In a practical device where eo~ >> to2, the dependence of g2 and R0 on the signal frequency is small. This means that parametric amplification is nearly constant over a wide range of signal frequencies. The values of G(R) calculated for a particular receiving device are shown in Figure 3 together with measured values. It can be seen that the measured values of gain are in agreement with the predicted values and vary very little in the signal frequency range 50 to 350 kHz. 3. EXPERIMENTAL WORK Two different experimental models have been tested. The details of the devices are given in Table 1. TABLE 1

Pump transducer

Pump frequency (MHz)

Transducer dimensions

Transducer casing dimensions

Length of array

Location of experiments

(1)

2.85

3 cm x 3 cm

5 cm strip

1"75 m

Belvide revervoir

(2)

5.9

1 cm x 1 cm

1 cm x I cm (approx)

Variable 0.1-1-0 m

A tank of 4 x 9 x 18 ft

3op ~, 2o E ,n

I0 8 6 5 4

g

I

I

5

I0

1

I

I

2

3

I

I

I

4 56

IIII

8 I00

I

I

I

2

:5 4 5 6

I

I

I

I I I

8 I000

f2 kHz

Figure 4. Beamwidth versus signal frequency f2. Pump frequency f~ = 2.85. R = 1.75 m. Theoretical; - - - , experimental. The 2.85 MHz device was tested at a reservoir using a pulsed low-frequency source (placed at a distance of about 10 m) and a continuous pump wave. The measured values of the 3-dB beamwidth at the difference frequency as a function of the signal frequency are shown in

Plate 1. Signals obtained from the receiving transducer. Upper trace--at the incoming signal frequency showing the effects due to multiple path. Lower trace--at the interaction-frequency illustrating the improvement due to the directivity of the receiving device.

(facing p. 301).

VIRTUAL ARRAYS FOR UNDERWATER RECEPTION

301

Figure 4. The straight line gives the values calculated from equation (7). Parametric amplification for 0 = 0 was measured (for varying p u m p intensity) at different signal frequencies. These results are plotted in Figure 5. The 5.9 M H z device was tested in a tank, the signal source (again pulsed) being at a distance of about 5 m from the p u m p transducer. Some of the measured directivity patterns are I

101 Pump Frequency f I =2"85...~o Ir~ _HZ

R : I .8 m

8

,j

+T

+Z

25

30

35

----°-

I

i

I

I

40 4.5 50 55 Pump transducer voltage (dBrel. to IV)

Figure 5. Parametric gain versus pump pressure, x - - x, f2 =3 3 kHz; Zx--A, f2 = 47 kHz; e - - e , f2 = I00 kHz; o - - o , f2 = 121 kHz;n--,E3f2 = 190 kHz; + - - - t - , f 2 =240 kHz; o--©,f2 = 300 kHz.

I-0 °.

•

///..~/ ?:\

°"i/ 0.7

/;/ ~

l.".

0.z

]'~1

;

0'2

.".1~,

Lj, -50

i i,~:li,:

i

-20

-I0

~

~t'V:i'i\ ,~ \ L I : I ~'}t~ I ~ I

0'1 I

0 1 0 0

X

I0

20

30

,-

F i g u r e 6. Directivity patterns of the virtual array, fz = 5-9 MHz; R = 50 cm. x - - x ,f~ = 56 kHz; . . . . . , ~ = 140 kHz; . . . . ,f2 = 300 kHz.

shown in Figure 6 and the 3-dB beamwidths obtained for various array lengths are plotted in Figures 7 and 8. The pulsing of the signal wave was necessary (particularly in the experiments made in the tank) to avoid erroneous results due to multipath effects. By using short enough pulses, signals arriving via various paths could be identified. The upper trace of Plate 1 indicates

302

H. O. BERKTAY AND C. A. AL-TEMIMI

the signal frequency component obtained at the receiving transducer after passing through a communication receiver tuned to oJ2. The multipath effects are clearly illustrated. The lower trace of Plate 1 shows the interaction-frequency wave received (on a different receiver) at the same time. Clearly, the array discriminates against the pulses coming via different angles by virtue of its directionality.

_R=5Ocm

Ig

, ...'~,~

50 40

~

R=llOcm

5O

~ 20-

"

" ~ ~'o... ~'~"f't. °%.

8 6

m

t"t~

4 I

I

5

IO

I

I

I

I I I Illl

t

I

2 3 4 5 6 8 I00 Signol frequency (f2kHz)

I I lilt 4 56 81000

I

2 3 • ~,

Figure 7. Variations of beamwidth with the signal frequency. - - - , experimental. Pump frequencyf~ = 5.9 MHz.

Theoretical; - - - ,

T40$0 Q)

.... ; O o o o

~ 20 "o

N8 6

si 4 IIII I0

I

I

i

2

3 4 5678 R (cm)

[ i illtl

I

I00

I

I

2

25

Figure 8. Variation of beamwidth with the length of the virtual array (R). , experimental. Pump froquencyf~ = 5.9 MHz. 4. D I S C U S S I O N

, Theoretical;

OF THE RESULTS

4.1. DIRECTIVITY PATTERNS

The features of the directivity patterns shown in Figure 6 are very similar to those of an end-fire array of the same length (with real elements) receiving the signal directly. Some of the "zeros" were found to be as much as 40 dB below the value along the preferred direction. The relative level of the first side lobes varies between - 1 0 and - 1 4 dB. The presence of zeros suggests that the interaction mechanism is as discussed in section 2.1. However, the pressure component p, was neglected in calculating the directivity function given in equation (12). Some calculations made showed that if p, is also considered, the

VIRTUAL

ARRAYS FOR UNDERWATER

303

RECEPTION

curves of the directivity function of equation (12) (see Figure 2) are generally pulled down and, due to phase cancellations, near-zero values are obtained for fiR~2 = rr. Thus it becomes very difficult to differentiate between the directivity functions obtained from sections 2.1 and 2.2. Also, it must be remembered that the configurations considered (for the behaviour of waves in the near field of a transducer) in sections 2.1 and 2.2 were both of a very simplified nature and give approximate answers. The real picture is, of course, much more complex. 4.2. BEAMWIDTI-IMEASUREMENTS Figures 4, 7 and 8 show that there is very good agreement between the measured values of the 3-dB beamwidth of the virtual receiving arrays and the values calculated on the basis of the very simplified theory outlined in sections 2.1 and 2.2. We have been unable to explain

f2 = 20 kHz

0.4 02

10

Io, l\ 08- \

:gmHz

fl

0.4 (32

-45

-30

-15

0

>

15

30

45

Angle (Degrees)

Figure 9. Sample directivity patterns for an array of 42 cm, with signal source about 15 m away. 100

5

2

°iI

~o

i

I

t tt~l

I

I0

2

I

I

I L tttl

5

I

I00

2

I

i

I

5

III

lOOO

Signal frequency (kHz)

Figure 10. Beamwidth against signal frequency for the same conditions as in Figure 9. Theoretical; • , experimental, f~ = 6 MHz; R = 42 cm. quantitatively the deviation of the measured values at low frequencies for a fixed array length or at short lengths for a fixed frequency. One possibility is that the sphericity of the signal wavefront (as the source is at a finite distance) has an increasing effect at the larger values

304

H . O . BERKTAY AND C. A. AL-TEMIMI

of 8. This could cause increased directivity. An array of 42 cm, with a p u m p frequency of about 6 M H z was tested at the transducer testing facility of the Admiralty Underwater Weapons Establishment at Portland. The signal wave was transmitted from a distance of about 15 m, approximating to a plane wave more closely than in the previous experiments. Two of the directivity patterns obtained are reproduced in Figure 9 and the 3-dB beamwidth at various signal frequencies is shown in Figure 10. As can be seen, the agreement between the theoretical and experimental results was much better than previously. This suggests that the anomalies noted above were probably due to the sphericity of the signal wave.

~ _

R = 50

crn

"Ox ,

I,

_o~ !\

x/

o. i!

/! ,,,x%÷

(x, x /-

.\

p'iil~

t

~, \ x \

_o,

I ,I

o.ot

-30-25-20-15-I0-5

0

5

t ~ I0

"V

15

I ~t

20 25 3 0

Figure 11. Single and double beam directivity patterns, f~ = 5.4 MHz, f2 -- 50 kHz. R -- 50 cm. double beam separation d = 6 cm.

x ~ x , Single beam; o - - o ,

1.0

~x\

ii,,"/

/

/,.,.,,/,,,/ -V, I -20

I -15

I

L -I0

0'7

x :~

0.6

0.5

~

"f"\

o4

V,

,.-

0.3

!j

02 0.1

I"

I -5

o.01

I o

i 5

l

e \ ...-x.."~~, i Ikvl ~ t~\r I0 15 20

o

Figure 12. Doublepump beam directivitypatterns, dis the separation ofthebeams.fl = 5.4 MHz; f2 = 50 kHz. R = 50 cm. x - - x, d = 6 cm; - - - , d = 8 cm; - - . - - , d = 10 cm. 4.3. PARAMETRICAMPLIFICATION The parametric amplification obtained with the 5.9 M H z array at various signal frequencies is shown in Figure 3. The absolute values of the amplification m a y have an error of up to 2 dB because o f the limited calibration accuracy of the high-frequency probes used. However,

VIRTUAL ARRAYS FOR UNDERWATER RECEPTION

305

the agreement between the calculated and the measured values is very encouraging. Also it is worth noting (from the Figures 3 and 5) that it is possible to obtain an up-converter parametric amplification of the order of 0 dB in such a device even at low pump intensities. This has a bearing on the signal-to-noise performance of such a device. The parametric-amplification curves of Figure 5 have some additional features of interest. Even for low-intensity pump waves the bandwidth of the 2.85 MHz device appears to be smaller than the one working at 5.9 MHz. This was found to be due to shadowing by the transducer mount. As the signal frequency is increased, the length of the shadow is also increased, reducing the effective length of the virtual array and thus its gain. As the pump intensity is increased, the gain of the array shows a saturation effect due to the extra attenuation of the high-intensity pump wave. At low signal frequencies (where shadowing could be neglected) this effect is less noticeable. However, at higher frequencies the increased attenuation of the pump wave within the shadow region causes the gain reduction to be more drastic. In this way, the effective bandwidth of the device deteriorates at the high intensities. 5. CONSIDERATIONS REGARDING A PRACTICAL DEVICE 5.1. ON DEVICEDESIGN The experimental results so far available support the use of the formula for the beamwidth,

OB= 2"V'A2/R,

(7)

as a guide-line. For water, equation (7) reduces to 0B = 139/v'F-~ degrees,

(17)

where F2 is the signal frequency in kHz and R in metres. Thus to obtain beamwidths of the order of 10°, say, at 30 kHz, a device of about R = 6.5 m would be required. To avoid excessive absorption of the interaction-frequency waves, the pump frequency could be chosen within the band 1 to 1.5 MHz. Considerations regarding the choice of the dimensions of the pump transducer are less clearly defined. Provided that interaction between the two waves is confined to within a region comparable to the Rayleigh distance of the pump transducer (i.e. (2b)2/Al) the sphericity of the pump-wave wavefronts can be ignored and the simplified analysis of the sections 2.1 and 2.2 can be justified. Yet, some of the experimental results reported here were obtained for values of R extending up to about 4 (2b)2/)q, showing that the device could work even when the receiving probe is well in the far field of the pump transducer. The pump-transducer size which would give a Rayleigh distance of about 6.5 m at 1.5 MHz is about 8 cm x 8 cm. But, the experimental results would appear to justify the halving of these dimensions. The pump intensity required for parametric amplification of about 0 dB can then be predicted from equation (16). For R = 6.5 m, R/Ro ~ 4, 1 + y - 3.5 and S = 64 x 10-4 m 2, G is of the order of unity when the pump intensity is of the order of 0.017 w/cm 2. Hence, the pump power (acoustic) required is of the order of 1.1 w. 5.2. NOISECONSIDERATIONS The sensitivity of low-frequency acoustic underwater systems is generally limited either by ambient noise or by reverberation. Because of the parametric-amplification effect, the pressure at the receiving transducer at the interaction frequency can be maintained at about the same level as that at the signal frequency. Sea noise at the same frequency band as the 21

306

H . O . BERKTAY AND C. A. AL-TEMIMI

signal will also be amplified if within the beam of the device. In addition, ambient noise in the frequency bands, 2oJi - oJ2,oJ~ + oJ2andat 3oJ1 - oJ2willinteract with the pump frequency or with its second harmonic (which must be present together with the pump wave due to its distortion) and will produce noise components at a frequency of o~t - oJ2. However, at these high frequencies the directivity of the virtual array would be much higher, thus reducing the magnitude of these noise components. Yet another noise component is that present in the water at the frequency oJ~ - oJ2. At these frequencies the ambient noise is restricted to thermal noise. To discriminate against this noise component, the gain of the receiving transducer can be made large. For example, in the 5.9 MHz device tested, the dimensions of the receiving transducer used was 1 cm x 1 cm, giving a receiving gain of about 43 dB. An advantage of the virtual receiving array is that a resonant transducer can be used for reception still providing a bandwidth wide enough for practical applications. For example, if a 1.5 MHz system is used, a receiving transducer Q of 30 still provides a bandwidth of about 50 kHz. In conclusion, one can say that the use of a virtual array instead of a physical one need not cause any deterioration of the signal-to-noise performance for reception of low-frequency acoustic signals. 5.3. EFFECTS OF TURBULENCE

If there is turbulence within the interaction volume of the two waves, the coherence of the difference frequency sources along the virtual array is impaired. In transmission applications of virtual end-fire arrays, the two frequencies used are close to one another, and phase variations caused by turbulence therefore may be expected to be coherent. The non-linear interaction process then removes these phase differences.Therefore, the source function at the difference frequency is less susceptible to turbulence effects. However, in the present application oJl >>oJ2. The effects of turbulence on the phase of the low-frequency signal wave can be neglected as compared with those imposed on the pump wave. The interaction process then produces a source function, the phase variations of which are substantially equal to those in the pump wave. As a result, the array gain is very susceptible to turbulence. This was observed in a qualitative manner during experiments with the devices discussed in the previous sections. A way of avoiding deleterious effects due to turbulence is to place the receiving device in an oil-filled (or water-filled) container which is "transparent" at the low-signal frequencies. Such a device could be towed by a ship (if made neutrally buoyan0 or could be placed under water, for example for propagation studies. If the two transducers placed at the two ends of such a container (say a long tube) are used for transmitting pump waves at two different frequencies (as well as acting as a receiver for the signal waves coming in the opposite direction) the device could be made to look in the fore and in the aft directions simultaneously. 5.4. ARRAY OF VIRTUAL ARRAYS

In a physical array, the relative level of the side lobes can be reduced, for example, by "shading" the transducer suitably. The simple virtual array discussed above does not provide any means of reducing the side lobes. If, however, a broad-side array of virtual arrays is formed, the side-lobe level (as well as the width of the main beam) can be reduced. For example, consider two such devices placed parallel to each other with a spacing d, working at the same pump frequency oJ1. If the received outputs are added, the overall directivity function becomes the product of that for a single array with the interferometer term cos ({k2 dsin 8).

VIRTUALARRAYSFOR UNDERWATERRECEPTION

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The results of some double-beam experiments are shown in Figures 11 and 12. These results exhibit good agreement with predictions. 6. CONCLUSIONS The agreement between the estimated and the measured performance of the virtual arrays is very encouraging, particularly in view of the simple models used in the theoretical work. The evidence presented in this paper is thought to be sufficient for initial design of such devices for particular receiving applications. REFERENCES l. H. O. BERKTAY1965 J. Sound Vib. 2, 462. Parametric amplification by the use of acoustic nonlinearities and some possible applications. 2. H. O. BERKTAY1967 J. Sound Vib. 5, 155. A study of travelling-wave parametric amplification mechanism on non-linear acoustics. 3. S. TJgTrA 1967 J. Sound Vib. 6, 255. Some non-linear effects in sound fields. 4. P. J. WESTERVELT1957 J. acoust. Soc. Am. 29, 199. Scattering of sound by sound. 5. H. O. BERKTAY1965 J. Sound Vib. 2, 435. Possible exploitation of non-linear acoustics in underwater transmitting applications. 6. A. FREEDMAN1960 J. acoust. Soc. Am. 32, 197. Sound field of a rectangular piston. 7. H. O. BERKTAY1967J. Sound Vib. 6, 100. Comments on "Some non-linear effects in sound fields". APPENDIX: NOTATION w~, to2, kl and k:

12, K ~1, 0~2~ O~

A ~0, CO

p y

pump and the signal frequency, and the corresponding wave number. frequency and the wave number of the interaction-frequency terms, i.e. .Q = to1 ~ to2 and K = kl ± k2. absorption coefficient of the medium at the frequencies to1, w2 and O, respectively. absorption parameter ~q + ~ 2 - ~, positive for the difference frequency and negative for the sum frequency; A = + 2 ~ oJ2/w~. density of and the velocity of propagation in the medium when undisturbed. excess pressure, as defined in the text. half the so-called parameter of non-linearity of the medium; y = (po/2C~). (d2pldp~)p=po.