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A theoretical and experimental study of the behaviour of a parametric array in a random medium
Journal of Sound and Vibration (1981) 74(3), 39.5410
A THEORETICAL BEHAVIOUR
AND EXPERIMENTAL
OF A PARAMETRIC
ARRAY
STUDY
OF THE
IN A RANDOM
MEDIUM? N. P. CHOTIROS
AND
B. V. SMITH
Department of Electronic and Electrical Engineering, University of Birmingham, Birmingham B 15 2 TT, England (Received 17 October 1979, and in revised form 2 August 1980)
The fluctuations of a parametric signal in a random medium have been investigated. A theoretical expression for the mean square amplitude and phase fluctuations has been
derived, by using the small perturbation method. A simplified version of the theoretical model, which predicts the axial amplitude fluctuations of the difference frequency signal. was computerized. Laboratory experiments were carried out in a heated tank in which the fluctuations of a parametric signal and a linearly generated signal were compared. A similar experiment was also carried out in an estuary. The experimental results obtained were consistent
with the theoretical
estimates.
1. INTRODUCTION The parametric array technique of generating narrow beam sound underwater, from relatively small radiating apertures, has received considerable interest since the scheme [2]. The sound is generated by transmitting two was originally proposed by Westervelt higher frequency signals and employing the non-linearity of the medium to generate sum and difference frequency signals within the medium. By this method, a narrow beam signal at the difference frequency is obtained. The behaviour of such a parametric array in water which contains random and time-varying refractive index inhomogeneities is of importance in practice. Refractive index variations arise because of factors such as local turbulence and the thermal microstructure. There is a considerable amount of published literature on the scattering of acoustic waves from random refractive index inhomogeneities, the fundamentals being excellently covered in the books by Chernov [3], Tatarski [4], and more recently Uscinski [5]. In a parametric array the two higher frequency interacting waves, hereafter called the primary waves, are both subject to amplitude and phase fluctuations through the mechanisms of scattering. Consequently, these fluctuations are passed on to the difference frequency volumetric source density. In addition as the difference frequency wave propagates from the parametric array it has further fluctuations imposed upon it. Therefore, the total effect of a random medium upon the difference frequency wave may be treated by combining and adapting the existing theoretical results for the scattering of single frequency acoustic waves. Some theoretical and experimental investigations of these random effects on the parametric array have been reported previously [l, 6,7]. In this present paper the t Some of the material in this paper was presented
at an Institute
of Acoustics, Underwater
Group, Meetlne
Ill. 395
0022-460X/81/030395
+ 16 $02.00/O
@ 1981 Academic Press Inc. (London~ Limited
396
N. P. CHOTIROS
AND
B. V.
SMITH
theoretical results are extended and generalized to include parametric arrays whose primary waves have arbitrary spreading geometries. Also, some results of experiments carried out on a high frequency modelled parametric array operating in a laboratory tank and on a parametric array in a tidal estuary are presented, and comparisons are made with the theoretical predictions. 2. THEORY Theoretical results for the axial amplitude fluctuations of a difference frequency acoustic wave produced by the non-linear interactions between planar primary waves in a random medium have been presented by Smith [6]. The approach used in reference [6] is followed here but the results are generalized for primary waves with arbitrary spreading characteristics. The scattering geometry under consideration is shown in Figure 1. The transducer at 0 transmits the primary waves and the receiver is assumed to be positioned at Q. The region in which the primary signals interact to produce significant source densities at the difference frequency is called the interaction volume. The extent of the interaction volume largely depends upon the spreading and absorption losses of the primary waves, as well as on the directivity functions of the primary waves.
\
Interaction volume
\
I
\
/’ \ \
/ .
--
/
Figure 1. Scattering geometry of a parametric signal in an inhomogeneous medium. 0, Scatterer; 0, elemental interaction volume; =k, ray path of primary signals; +, ray path of different frequency signal.
Consider the source density in the elemental volume dui. This is given by [2] 4i = (pl&)a(p’)l%
(1)
where p is a non-linearity coefficient, p. is the density of the unperturbed medium, co is the sound speed in the unperturbed medium and pi is the total acoustic pressure at dui. Let the primary waves at the elemental volume dvi be given by the general expressions Pli=Ali
exp[-@1jsil+j(kl.
si-wlt)+Xli+jSlil,
PZ~
exp
Si--2t)+X2i+jS2il,
=A2i
[-~2lSil+j(k2.
where (x1(, xzi) and (Sit, Szi) represent, respectively, the log-amplitude and phase fluctuations of the primary waves at dui; Ali and A2i are generally complex functions of position, but in the ideal spherically spreading wave case their values are real and inversely proportional to range; (Y~and 13~are absorption coefficients.
PARAMETRIC
SIGNAL
IN A RANDOM
397
MEDIUM
The total pressure pi, when second order effects are ignored, is given by pi = pli +p2i. After substitution of pi into equation (l), with retention of only the difference frequency wave component and neglect of frequency spreading, it follows that the source density at the difference frequency, q3iFis given by q3i z (-jW?AriATiIPL?cZ)
exp [-(a1+az)]si]+j(k3.
~--3f)+~1;+~2i
+j(S,i-S2,)1,
where w3 = w1 - w2 and k3 = kl - kz. The elemental difference frequency pressure dpNi at the receiver Q, produced by q3i at dv, is given by dp, = (-jw3po/4rlril)q3i
{exp [-a3lri/
+ $3 . ri +x3i + j&i]) dui.
The total difference frequency pressure, pN, at 0 is then given by integration of dp, over the interaction volume Di: PN
=
exp (-jW)
IHi exp [Xii
+XZ +x3i + j(s,i +SZ +SX)] duiy
where Hi is given by Hi =(~:PAliA2*ii/4~~oc~Iril) exp [-((yl+~~~)Isil-(~glri/+jk3.
(si+ri)+je],
8 being an arbitrary phase angle to be defined. The quantity Hi dui is the unperturbed differential pressure at point Q arising from the source density in the volume dui. In a perfectly homogeneous medium completely free of scatterers the difference frequency pressure would be simply given by PNO= exp (-jw3t)
Hi dvi.
I
This will be referred to as the unperturbed difference frequency pressure. The value of Hi has been discussed in many publications (see e.g., references [2,8-lo]. Therefore, for the present purposes, its value will be assumed to be known. The log amplitude fluctuation, xN, and phase fluctuation, SN, of the difference frequency wave are defined by pN It
therefore
=pNO
exp
(xN
+jsN).
follows that
exp[XN+jSN]=(l
Hidv,)lf Hi exp lili
+x2i +x3< +
j(s,i -s2i +S30] du;.
(2)
Equation (2) relates the overall difference frequency wave amplitude and phase fluctuations to the fluctuations of the individual component waves which constitute the parametric array. By using the method of small perturbations it is possible to simplify equation (2) and obtain more useful expressions for the difference frequency amplitude, xN, and phase, S,, fluctuations. For example, if all the fluctuations are assumed to be small such that ]xPlCC1 and IS,\ K 1, where p = IV, li, 2i, 3i, then the following approximations for the exponential terms in equation (2) may be used: exp
hN
+jsNl=
1 +xN
+jsN,
398
N. P. CHOTIROS
AND
B. V. SMITH
where for convenience xl;+x2;+~3; and Si;-S2;+S3; have been written as x; and S; respectively. When these approximations are used equation (2) becomes
xN+jSN=([ H; du;)-l[
H;(X;+jS;)db;.
In the definition of H; an arbitrary phase angle f3has been so far left undefined. It may now be defined such that Im (5 H; du;) = 0. In practice this automatically happens when the signal is envelope detected. With this definition of 0 the real (Re) and imaginary (Im) parts of equation (3) may be simply separated, giving xN =
J[Re (Hi)xi -
Im {Hidz+
(Hi)Si]doi
sN
=
I Be (Hi)Si+ ImW;)X;ldvi IHidv;
’
.
(4
5)
’
Hence the mean squares of the difference frequency wave amplitude and phase fluctuations become
&=(I Hi duij2 [I [Ej
Re (H;) Re (Hi) + %$ Im (H;) Im (Hj)
--
- Re (H;) Im (Hi)(xiSi+ XjSi)] dvi duj, z=([
H;dv;)-211
[S;Sj - Re (Hi) Re (Hi) +E
-+ Re (H;) Im (Hi)(Sixi+
S~XJI dvidvp
(6) Im (Hi) Im (4) (7)
We have been able to show that these expressions for zNand zN, equations (6) and (7), are also applicable when the method of smooth perturbations (also known as the Rytov method) is used. However, under these circumstances the terms xP and S,, where p = N, i, j, are strictly logarithmic terms. Equations (6) and (7) give the mean square difference frequency amplitude and phase fluctuations in terms of various spatial correlation functions: e.g., &J$ A brief review of these is given in the next section. 2.1.
SPATIAL
CORRELATIONS
OF COMPONENT
WAVES
-The correlation functions in question are x;xj, S;Sj and s. NOWxixj, and %&represent spatial correlation functions for amplitude and phase fluctuations respectively, where the subscripts i, j refer to elemental volumes dv;, dvj. Similarly x;Sj represents spatial cross-correlation functions between amplitude and phase fluctuations at the elemental volumes do;, duj. First one can expand x;,j and S;,j into their components: e.g., ,Y;= xl; +x2; +,y~; and S; = Sl; - S2; + S3;. The fluctuation terms denoted by the subscripts 1 and 2 belong to the waves, at the primary frequencies, propagating from 0 to S; (see Figure 1) while the terms denoted by the subscript 3 are for the difference frequency wave propagating from s; to R. Since both the paths and the frequencies are quite different, it is to be expected that there is negligible correlation between these two groups of fluctuation terms. Therefore, the correlation terms in equations (6) and (7) may be approximated by XiXj’(Xli+X2i)(Xlj+X2j)+X3iX3jv
S;Sj- (Sl;-S2;)(Slj-S2j)+S3;S3j, &$=
(Xl;
+XZi)(Slj-S2j)
+X3iS3j-
Data on the cross-correlation between amplitude and phase fluctuations, i.e., 3, may be obtained from reference [3] for plane waves and from reference [4] for spherical waves.
PARAMETRIC
-
SIGNAL
IN
A
RANDOM
MEDIUM
-
399
The terms K, G, SiiSzi and S’ziSzj are spatial correlation functions for signals transmitted from the same point and received at two different locations. Many aspects of the theory pertaining to such have been explored [3,4,11]. - a geometry The terms G, G, SiiSZjand SZiStjare spatial frequency correlation terms, i.e., the spatial cross-correlation of either the amplitude or the phase fluctuations between the two primary waves. This problem has been studied by Eliseevnin [ 121 for the longitudinal case. In most parametric array applications the frequency difference between the primary waves is small and hence the following approximations are reasonable: (xii + xzi)(xrj +xzj) = ~~~~~~~~ where xc is the amplitude fluctuation of a wave at the average of the two primary frequencies, and (Sli - Szi)(Sij - Szj) = Sd+Sdi,where Sd is the phase fluctuation of a wave at the difference frequency propagating from the source to the elemental volume. Finally, the terms x3ix3j and S3iSsj are correlation functions of amplitude and phase fluctuations, respectively, for two waves both received at Q (see Figure 1) but transmitted from two different sources, the one at dvi and the other at dvi. A study of such a correlation function has not been found in the available publications; it is also very difficult to measure experimentally because of the obvious difficulties in distinguishing two signals at the same frequency arriving at the same receiver simultaneously. However, for an isotropic medium it is possible to demonstrate that reciprocity holds. Therefore, x3ix3j and S3S3, are numerically equal to the corresponding correlation functions of a single wave transmitted from P and received at dui and dvh In the case of the far field, on-axis fluctuations of a narrow beam parametric signal, equation (6) may be considerably simplified to give a very simple and plausible result, because the imaginary part of Hi is negligibly small. Therefore,
-Since the range is long compared to the array, xixj’x3ix3j. The fluctuations of a linear signal at the same range is given by xL, and therefore x~=~==~. A correlation coefficient may be defined as N3ij = X3iX3jI(XZiXZj)1’2,
IN3ijl~ 1.
(9)
Hence, the ratio between the fluctuations of the non-linear and linear signals is given by
ZIIXZ=
HiHjN3ij dvi dujl
HiHj dvi dC(.
(10)
Except for certain rare situations, the value of Nij is positive and varies monotonically as a function of the separation vector between dvi and dvj. Also HI and Hj are positive quantities for the most part with the exception of small regions within the near field of the transducer. In the idea1 case of a perfect end-fire array, Hi and Hi would always be positive. Therefore, it can be deduced that
z/z=
HiHjN3ij dvi dvjl
HiHj dvi dvj 4 1.
(11)
This is an interesting result because it shows that in the far field of a narrow beam parametric array the fluctuations of the difference frequency would be no worse than those of a directly transmitted linear signal. 3. IMPLEMENTATION OF THE THEORY Theoretical estimates of the parametric signal fluctuations have been calculated by using equation (6). Unfortunately, all of the terms in the equation are very difficult to evaluate.
400
N. P. CHOTIROS
AND
B. V. SMITH
Exact numerical answers were not possible. However, since they are integrated over a large volume, some approximations are possible. These are described below. By limiting the problem to on-axis signal fluctuations only, azimuthal asymmetry can be ignored without causing gross errors. Therefore, although the transmitting transducer in the experiment to be described was square, it was assumed that the primary signal profile was Gaussian and symmetrical about the axis of the primary beam. This gave an analytic expression for Hi. The primary beam was also assumed to be perfectly collimated in the near field, and beyond the Rayleigh distance the beam was assumed to be spherically spreading. The justification for adopting this simple, single lobe approximation of the directivity function is that the difference frequency signal is a function of an integral of the primary signals. Therefore, the overall average shape of the primary beam profile rather than its fine structure is what matters. Estimation of the various spatial correlation terms, xixi, etc., presented another problem. For infinite plane or spherical waves at long ranges, they depend only on the separation vector. For infinite plane or spherical waves at short ranges, they depend on both separation and range. In the case of the experimental model, the primary beam is narrow; therefore they depend on both position and separation vectors, and this is the case for which a theoretical estimate is required, However, since these terms are integrated over relatively large volumes, some approximations are possible, and theoretical results for infinite plane and spherical waves may be used with modifications to obtain approximate results. The correlation terms were considered as two separable components, an amplitude factor 7 and a correlation coefficient Nii, defined as follows:
The amplitude factors were approximated by the existing theoretical results for infinite waves. These results in turn were approximated by the simpler expressions applicable to the asymptotic range conditions [4] m >>LO and LO >>m, where L is the range, A the acoustic wavelength and Lo the upper limit of the homogeneous turbulence characteristic length [3,4,13]. Again, since the terms are integrated over a wide range of values of L, using these asymptotic expressions instead of the more cumbersome exact expressions introduced only marginal errors. As an example, asymptotic values of 2 as a function of range L3i are shown in Figure 2.
Figure 2. Linearized
amplitude
fluctuation
coefficient
as a function of range.
The correlation coefficients, Nij, were more difficult to evaluate since there are very few theoretical solutions available even for ideal infinite waves, and a review is given in reference [14]. Longitudinal and transverse components of Nij were assumed to be separable and independent, and the transverse component was assumed to be a range
PARAMETRIC
SIGNAL
IN
A
RANDOM
MEDIUM
401
dependent Gaussian function. Theoretical results based on the works of Chernov [3 j and Karavainikov [13] show that this is a reasonable approximation for most values of range and separation vector, except where the longitudinal separation is small compared to the dominant component of the refractive index spectrum. Again, since the term is integrated over a large volume, this discrepancy may be ignored. Lastly, finite beam width effects had to be taken into account, since the theoretical expressions used were derived primarly for infinite plane or spherical waves. As a consequence of the law of conservation of energy, the transverse component of correlation in a narrow beam is correspondingly narrower than that of an infinitely broad beam. Moreover, in the narrow beam there is a higher degree of negative correlation, and when integrated over the whole beam the positive and negative components tend to cancel. This problem has been discussed by Tatarski for the case of an infinite plane wave in reference [4]. By using equation (6), the above approximations, and a linearized refractive index spectrum of the form proposed by Medwin [15], a computer program was obtained which gave theoretical estimates of the on-axis amplitude fluctuations FN of the difference frequency signal. Further details are given in the Appendix. 4. LABORATORY TANK EXPERIMENTS
A random refractive index field was generated in a water tank, dimensions 1.8 x O-9 x description of the experimental arrangement has been presented elsewhere [l, 7, 141. A temperature microstructure was produced by convective currents, rising through a horizontal perforated metal sheet, generated by a heater array along the tank bottom. Investigations [l, 141 have shown that the temperature microstructure behaved as a passive additive of turbulence, and furthermore that it was the primary cause of the refractive index variations. Amplitude fluctuations of linear acoustic signals in this medium have been observed [l, 141 and were found to agree with theoretical estimates, to within the limits of experimental error. The parametric transmitter used had primary frequencies centred on 9 MHz and a 1 MHz difference frequency signal. The transducer itself was a 1 cm square piezoceramic plate. For comparison, a 1 MHz linear signal was also available from a 2 cm square piezoceramic plate. Small transducers with effective apertures of about 1 mm in diameter [ 1,141 were used for reception. By using measurement techniques outlined previously [l, 141, the normalized average sample variance, V$, of the axial amplitude fluctuations of the difference frequency signal were measured as a function of range. The result=e shomin Figure 3(a). The coefficient of variation, VN, was defined by V’, = {m(/~~/)~}/(Ip~j)~, and for small changes this approximates to V& -3. The subscript N indicates that the signal was non-linearly generated in the medium. A corresponding statistic VL may be defined for the linear signal. Theoretical estimates were computed from the results of section 2, with use of the known refractive index spectral function [l, 141. For comparison, V,, was also measured by using the linear 1 MHz signal, and it is shown in Figure 3(b). The spread in the experimental results, unfortunately, is large enough to cover up any small differences which may exist between VN and VI_,since the two sets of data have very similar values. The cause of the spread may be attributed mostly to the lack of stationarity of the temperature microstructure. Therefore, an alternative technique was required whereby the two types of signal could be observed under identical conditions: i.e., in the same part of the medium and at the same time. This was achieved by making use of the reciprocal behaviour of the refractive index field, as follows. O-9 m3. A detailed
402
N. P. CHOTIROS
AND B. V. SMITH
meowed
values
L(m)
Figure 3. Coefficient of amplitude variation of (a) a 1 MHz parametric signal as a function of range and (b) a 1 MHz linear signal as a function of range.
Both the parametric signal and the linear signal were transmitted simultaneously in opposite directions from the two respective transducers facing each other. Each transducer also functioned as a receiver when a signal arrived from the opposite transmitter. A block diagram of the experimental system is shown in Figure 4. The amplitudes of the two signals were compared, and the result is shown in Figure 5, in terms of the ratio of their coefficients of variation. The spread in the experimental results was small enough to show clearly differences between the two types of signal. Reciprocity was checked by transmitting identical linear signals in both directions.
-IF-
TRANSDUCER
TRANSDUCER
LINEAR TRANSMITTER
PARAMETRIC TRANSMITTER
PRF UNIT
ENVELOPE
ENVELOPE
DETECTOR
DETECTOR
t
t DATA
LOGGER
Figure 4. Block diagram of experiment to compare parametric and linear signals using reciprocity.
PARAMETRIC
1 Standard
SIGNAL
IN A RANDOM
403
MEDIUM
deviation 2
Average
of measured
values (reciprocity
r
check)
i
I
1
00
0.9
I
I
1 0
1
5
L(m)
Figure 5. Comparison of the amplitude fluctuations V,, V, and V,,, of a parametricsignal, a linear signal, and a linear signal transmitted in the reverse direction, respectively, as functions of range. At the longer ranges, the signal was transmitted along a diagonal of the tank in order to avoid multiple path propagation; consequently, the effective value of k, was significantly greater than the calculated value at the centre of the tank.
4.1.
DISCUSSION
OF LABORATORY
RESULTS
Under the given experimental conditions and at the ranges indicated, the experimental results and theoretical estimates agree in indicating that the parametric signal amplitude is subject to smaller fluctuations than the linear signal. From theoretical considerations, it
I’
1
f
-I
12-
1 l-
(#
:
1 o-
Asymptote
0
9-
0
8-
I
I IO
I
1
1
I00
L(m)
Figure
6. Theoretical
comparison
of parametric
and linear signal fluctuations
for long ranges,
K, = 179 rn--‘.
404
N. P. CHOTIROS
AND
B.
V. SMITH
can be shown that this is a consequence of the narrow primary beam. Theoretical estimates of the VN : VL ratio for other ranges show that at long ranges the ratio is less than but approximately equal to unity. At very short ranges, the validity of the theoretical estimates is doubtful, and the individual values of V, and VL would be extremely small and insignificant for most practical purposes. These estimates are shown in Figure 6. 5. EXPERIMENTAL 5.1.
DESCRIPTION
OF
THE
MEASUREMENTS
IN AN ESTUARY
SITE
The tidal estuary experiment took place near Weston-super-Mare, in the Bristol Channel. The main attraction of the site was the very high tides. This enabled the apparatus to be installed and optically aligned on land during low tide, and the underwater experiment to be carried out during the following high tide. The difference between high and low tide levels varied between 7 m during neap tides to 10 m during spring tides. The beach area contained a seaweed belt centred on the mid-water line, growing on the underlying rock and loose stones. Beyond the seaweed the bed rock was overlaid by sand and mud. The transducers were mounted on structures erected on the bed rock. A cross-section of the site is shown in Figure 7, with indications of the tidal conditions during the experiment. The maximum acoustic path length between transmitters and receivers was limited by signal-to-noise considerations, as well as multiple ray path and other logistical considerations. Hence, the transmitter-receiver separation used was limited to 43 m. 5.2. MEDIUM MEASUREMENTS In order to be able to make a theoretical estimate of the signal fluctuations, it was necessary to measure certain medium characteristics. Unlike in the laboratory tank where
Vertical Scale :
-5
?? nr
Tl.T3
I T2
metres
-3
Figure
7. Test site geometry
and water level at high tide.
PARAMETRIC
SIGNAL
IN A RANDOM
MEDIUM
40.5
the medium could be controlled, in this case the medium was very variable. An instrument package consisting of two thermistor probes and an acoustic velocimeter was mounted on a cantilever on the same structure as hydrophone Hl. The thermistor probes measured temperature fluctuations while the velocimeter measured a component of the flow velocity over a path length of approximately O-2 m. Measurements were taken simultaneously with the acoustic experiments, as far as possible. The temperature measurements show that over an interval of approximately 100 s the average sample variance of temperature was in the region of (O.O3”C)*; it was depth dependent and non-stationary, falling to a minimum at high tide. With just two thermistors, it was not possible to measure the structure function directly but from the knowledge of the flow velocity it was possible to make an estimate by deduction. It was also deduced from the flow velocity measurements that turbulent eddies would make a significant contribution to the acoustic signal fluctuations, unlike in the laboratory experiment. The effects of other scatterers were not measured (e.g., air bubbles), although, by using the velocimeter in a different mode, it was possible to measure the total refractive index fluctuations averaged over the 0.2 m path. The equipment and expense required to carry out a comprehensive measurement of the medium was beyond the scope of this experiment. However, by using the available measurements, and using the Medwin spectral model [14,15], it was possible to make linearized estimates of the effective refractive index spectral function caused by temperature microstructure and turbulent eddies. These are shown in Figure 8. These results represent upper and lower bound estimates of the average isotropic refractive index variations, and they can be regarded only as rough estimates since the refractive index fluctuations were in fact neither stationary nor isotropic. 10-‘0,
,
I
Upper bound
Lower bound
1
10-Z kn
1 hb
102
4,
f
f
kob’
kob
IO4
Wove number (rn-‘1
Figure 8. Estimated temperature variations
linearized one-dimensional and turbulence.
refractive
index
spectral
density
function,
caused
by
5.3. ACOUSTIC MEASUREMENTS The non-linear and linear signal fluctuations were measured and compared in a similar way as for the laboratory experiments previously described. The sequence of events was as
406
N. I’. CHOTIROS AND B. V. SMITH
follows, with reference to Figure 7. As high tide approached, preliminary medium measurements were taken; this occurred during the interval labelled A. Then, the acoustic measurements were taken. During intervals B and D, a non-linear pulsed signal was transmitted from Tl to H4, simultaneously with a pulsed linear signal from T2 to Hl. During intervals C and E, a pulsed linear signal was transmitted from T3 to H4 and an identical one from T2 to Hl, simultaneously, to test the reciprocity of the medium. As the tide receded, final medium measurements were taken in interval F. The coefficients of variation, V, and VL, are shown in Table 1. The results show that the medium was not reciprocal. The fluctuations for signals going from Tl or T3 to H4 were very much higher than for those from T2 to Hl. The cross-correlation coefficient was approximately zero. TABLE 1 Acoustic results from the estuary experiment
Period
Hydrophone
Signal type
B B B-C C C
H4 Hl
Non-linear Linear
H4 Hl
Linear Linear
D D D-E E E
H4 Hl
Non-linear Linear
H4 Hl
Linear Linear
0*0588 0.0210
2.80 1.45
0.0472 0.0245
1.93
0.0584 0.0191
3.06 1.59
0.0662 0.0344
1.92
With allowance for the lack of reciprocity, which was assumed to increase the signal fluctuations in one direction by a constant factor relative to those in the opposite direction, an estimate of the ratio V, : V, was made (see Table 1). This represents the ratio between the non-linear and linear signal fluctuations when the signals are propagating under identical conditions. 5.4. DISCUSSIONOF ESTUARY RESULTS
Due to a number of practical factors, the amount of data obtained in the estuary experiment was not as extensive as initially planned, but a number of interesting results were obtained. The apparent lack of reciprocity was very striking. This may be caused in two ways. Firstly, the medium was horizontally stratified to some extent [16]. From Figure 7, it can be seen that the ray path Tl or T3 to H4 is closer to the horizontal than T2 to Hl, although the difference is less than l-5”. Depending on the severity of the horizontal stratification, this may account for the difference in the fluctuation levels. Secondly, turbulence can also account for lack of reciprocity [17], but it is difficult to see how it can cause consistent differences on both sides of the tide peak, since the flow patterns before and after the tide peak must be quite different. By using the spectral estimates in Figure 8, a theoretical estimate of the VL : VN ratio was also obtained and is plotted as a function of range in Figure 9, along with the measured data. Since the medium cannot be described as isotropic by any stretch of the imagination, the theoretical results can be regarded only as order of magnitude estimates. The obvious conclusion, however, is that the difference between the fluctuations of parametric and
PARAMETRIC
I
SIGNAL
IN A RANDOM
I
IO
407
MEDIUM
I
1000
J 10000
Range Cm)
Figure 9. Comparison of the amplitude fluctuations of parametric and linear signals as a function of range, in k, = 40.0 m-‘, - - -. Empirical points: 0. the estuary experiment. Theoretical curves: k, = 21.7 m-‘, -;
linear signals is small beyond the parametric array length, for both upper and lower bound refractive index spectra. 6. CONCLUSIONS A thoretical expression for describing the fluctuations of a parametric signal has been derived based on the small perturbation method, and it has been implemented in the form of a computer program for predicting axial signal fluctuations. The theory was supported by laboratory experimental results. Experimental results outside the laboratory were also consistent with the theory, although these results were very limited. It is quite clear, however, that the fluctuations of a parametric signal would be of the same order of magnitude as that of a directly transmitted linear signal. It can also be deduced from the theory that, although there may be significant differences in the near field, their fluctuations in the far field would be approximately equal. Therefore, there should be no significant disadvantage in using a parametric signal instead of a linear signal, as far as signal fluctuations are concerned. ACKNOWLEDGMENT This work has been carried out with the support of the Procurement of Defence.
Executive, Ministry
REFERENCES N. P. CHOTIROS and B. V. SMITH 1976 in Proceedings of Institute of Acoustics Underwater Group Conference on Recent Developments in Underwater Acoustics, Portland Paper 4.1. The parametric end-fire array in a random medium. P. J. WESTERVELT 1963 Journal of the AcousticalSociety ofAmerica 35535-537. Parametric acoustic array. L. A. CHERNOV 1967 Wave Propagation in a Random Medium, (translated by R. A. Silverman). New York: Dover Publications.
N. P. CHOTIROS
408
AND
B. V. SMITH
4. V. I. TATARSKI 1961 Wave Propagation in a Turbulent Medium, (translated by R. A. Silverman). New York: McGraw-Hill. 5. B. J. USCINSKI 1977 The Elements of Wave Propagation in Random Media. New York:
McGraw-Hill. 6. B. V. SMITH 1971 Journal of Sound and Vibration 17,129-138. A theoretical study of the effect of an inhomogeneous medium upon a transmitter which exploits the non-linear properties of acoustic propagation. 7. B. V. SMITH, W. WESTON-BARTHOLOMEW and N. NAKLI 1973 in Proceedings of the Conferences, Finite Amplitude Wave Effects in Fluids, Copenhagen, 151-155. London: IPC. An experimental study of the parametric end-fire array in a random medium. Difference 8. R. L. ROLLEIGH 1975 Journal of the Acoustical Society of America 58,964-971. frequency pressure within the interaction region of a parametric array. 9. H. 0. BERKTAY 1965 Journal of Sound and Vibration 2, 435-461. Possible exploitation of non-linear acoustics in underwater transmitting applications. 10. H. 0. BERKTAY and D. J. LEAHY 1974 Journal of the Acoustical Society of America 55, 539-546. Far field performance of parametric transmitters. 11. R. T. AIKEN 1969 The Bell System Technical Journal 1129-1165. Propagation from a point source in a randomly refracting medium. 12. V. A. ELISEEVNIN 1974 Soviet Physics-Acoustics 19, 533-538. Longitudinal frequency correlation of fluctuations of the parameters of plane waves propagating in a turbulent medium. 13. V. N. KARAVAINIKOV 1956 Iaroslavl K.D. Ushkinskii State Pedagogical Institute. Fluctuations of amplitude and phase in a spherical wave. 14. N. P. CHOTIROS and B. V. SMITH 1979 Journal of Sound and Vibration 64,349-369. Sound and amplitude fluctuations due to a temperature microstructure. 15. H. MEDWIN 1974 Journal of the Acoustical Society of America 56, 1105-l 110. Sound phase and amplitude fluctuations due to temperature microstructure in the upper ocean. 16. V. M. KOMISSAROV 1964 Soviet Physics-Acoustics 10,143-152. Amplitude
and phase fluctuations and their correlation in the propagation of waves in a medium with random, statistically anisotropic inhomogeneities. 17. Yu. I. BELOUSOV and A. V. RIMSKII-KORSAKOV 1975 Soviet Physics-Acoustics 21, 103-109. The reciprocity principle in acoustics and its application to the calculation of sound fields of bodies. 18. N. P. CHOTIROS 1980 M.ScQ. Thesis, University of Birmingham. The parametric array in an inhomogeneous medium. APPENDIX Computation of the difference frequency mean square fluctuations was carried out by numerically evaluating the two-volume integral given by equation (6). Chernov’s results for a random medium with a Gaussian spatial correlation function were used because of their relative simplicity. These were reconciled with the Medwin refractive index spatial wave number spectrum by employing an equivalent Gaussian correlation coefficient N(r) = exp (-r2/U2,i), where the scale range (Li). This the fluctuations numbers [4]. By using the phase fluctuation
n = 1,2 or 3,
parameter a,i was dependent on both acoustic wave number (k,) and is a valid approximation since Tatarski has shown that at any given range of the sound signal are dominated by a narrow band of spatial wave same method as Chernov, the following expressions correlation functions were obtained [18]:
for amplitude
and
641, A2) Here z is the equivalent determined as follows.
mean
square
refractive
index
fluctuation,
and I1 and I2 are
PARAMETRIC
SIGNAL
IN A RANDOM
MEDIUM
409
When the longitudinal separation AL is greater than ani and the transverse separation component 1 is in the region I < Q,i {(2AL/ k,a ‘,i) + (k,a Ei/2AL)}p and these assumptions are satisfied for the greater part of the two-volume integral, then 11 is approximately given by 11 ={J?rk~a,iLi/(l+4AL*/a~ik~)}
exp (-/‘/Lz:~),
iA3)
where AL = Li - Li (it being assumed that Li < Lj), Li =x; if n = 1, 2 (primary signals), Li = R -xi if it = 3 (parametric signal), and 1’= (yi - yj)‘+ (zi - Zj)*. When the range L; is large compared with the Rayleigh distance, Li >>k,aEi, then II ~12, and 12 may be neglected. When the range is small, Li cc k,aEip then II= 12, and in the case of amplitude correlation, which is given by the difference between Ii and 12, it was necessary to consider second order terms; the resulting expression was approximately [lS] XniXnj-2Jiz
nrani -3 Li exp
(-Z*/U”,~)
{2L;AL +iLf}.
(A4j
These asymptotic solutions were used to form composite linearized approximations of the correlation functions required. Thus, the two-volume integrals were written as a summation of terms, each of the form
X
645, ‘46)
dxi dxj dyi dyj dti drj,
where Xi = Re (H;) or Im (Hi) =FH(Xi) exp {-2(y? + ~:)/bf}, and
-~YniYnj
=
XniXnj
or
SniSnj
=
Fn
(Xi9
Xj)
exp
I-[(Yi
-
Yj)*
+
(zi
-
zj)‘lla ti>,
FH(xi) and Fn(xi) being functions of the longitudinal position vector Xi+It can be seen that the assumption of Gaussian shaded primary beam profiles, which lead to corresponding Gaussian shaded source densities, allowed the integrals in yip yj, Zi and zj to be solved analytically. The remaining integrals in xi and xj were computed numerically, as integrals of the form R
II0
R
0
F(xo,
Xi, ani,
bi, 2)
dxi
dxp
647)
However, the expressions used to obtain the functions F(. . .) were strictly applicable only to infinite plane waves or spherical waves. The following corrective measures were used to take into account the effects of the narrow primary beams. Whenever the beam width bi was smaller than the transverse correlation width U,~,the transverse correlation components in equations (A3) and (A4) were changed to exp (-1*/b?), since correlation width can never exceed beam width. In the case of amplitude correlations, a consequence of the law of conservation of energy requires that the total amplitude fluctuation over a beam cross-section be zero. When the beam width is broad the effect would not be noticeable, but for narrow beams the correlation function F(. . .>in equation (A7) would be reduced by a factor Ki equal to [18] 77X interaction volume cross-section =1n-b? k3 Rayleigh aperture 2 2(R -xi)’
410
N. P. CHOTIROS
AND
B. V. SMITH
It is very difficult to assess the overall error introduced by the various approximations and assumptions made in the above evaluation of equation (6). However, in all cases they have been justified upon physical grounds and the final predictions do agree with experimental observations.
A THEORETICAL BEHAVIOUR
AND EXPERIMENTAL
OF A PARAMETRIC
ARRAY
STUDY
OF THE
IN A RANDOM
MEDIUM? N. P. CHOTIROS
AND
B. V. SMITH
Department of Electronic and Electrical Engineering, University of Birmingham, Birmingham B 15 2 TT, England (Received 17 October 1979, and in revised form 2 August 1980)
The fluctuations of a parametric signal in a random medium have been investigated. A theoretical expression for the mean square amplitude and phase fluctuations has been
derived, by using the small perturbation method. A simplified version of the theoretical model, which predicts the axial amplitude fluctuations of the difference frequency signal. was computerized. Laboratory experiments were carried out in a heated tank in which the fluctuations of a parametric signal and a linearly generated signal were compared. A similar experiment was also carried out in an estuary. The experimental results obtained were consistent
with the theoretical
estimates.
1. INTRODUCTION The parametric array technique of generating narrow beam sound underwater, from relatively small radiating apertures, has received considerable interest since the scheme [2]. The sound is generated by transmitting two was originally proposed by Westervelt higher frequency signals and employing the non-linearity of the medium to generate sum and difference frequency signals within the medium. By this method, a narrow beam signal at the difference frequency is obtained. The behaviour of such a parametric array in water which contains random and time-varying refractive index inhomogeneities is of importance in practice. Refractive index variations arise because of factors such as local turbulence and the thermal microstructure. There is a considerable amount of published literature on the scattering of acoustic waves from random refractive index inhomogeneities, the fundamentals being excellently covered in the books by Chernov [3], Tatarski [4], and more recently Uscinski [5]. In a parametric array the two higher frequency interacting waves, hereafter called the primary waves, are both subject to amplitude and phase fluctuations through the mechanisms of scattering. Consequently, these fluctuations are passed on to the difference frequency volumetric source density. In addition as the difference frequency wave propagates from the parametric array it has further fluctuations imposed upon it. Therefore, the total effect of a random medium upon the difference frequency wave may be treated by combining and adapting the existing theoretical results for the scattering of single frequency acoustic waves. Some theoretical and experimental investigations of these random effects on the parametric array have been reported previously [l, 6,7]. In this present paper the t Some of the material in this paper was presented
at an Institute
of Acoustics, Underwater
Group, Meetlne
Ill. 395
0022-460X/81/030395
+ 16 $02.00/O
@ 1981 Academic Press Inc. (London~ Limited
396
N. P. CHOTIROS
AND
B. V.
SMITH
theoretical results are extended and generalized to include parametric arrays whose primary waves have arbitrary spreading geometries. Also, some results of experiments carried out on a high frequency modelled parametric array operating in a laboratory tank and on a parametric array in a tidal estuary are presented, and comparisons are made with the theoretical predictions. 2. THEORY Theoretical results for the axial amplitude fluctuations of a difference frequency acoustic wave produced by the non-linear interactions between planar primary waves in a random medium have been presented by Smith [6]. The approach used in reference [6] is followed here but the results are generalized for primary waves with arbitrary spreading characteristics. The scattering geometry under consideration is shown in Figure 1. The transducer at 0 transmits the primary waves and the receiver is assumed to be positioned at Q. The region in which the primary signals interact to produce significant source densities at the difference frequency is called the interaction volume. The extent of the interaction volume largely depends upon the spreading and absorption losses of the primary waves, as well as on the directivity functions of the primary waves.
\
Interaction volume
\
I
\
/’ \ \
/ .
--
/
Figure 1. Scattering geometry of a parametric signal in an inhomogeneous medium. 0, Scatterer; 0, elemental interaction volume; =k, ray path of primary signals; +, ray path of different frequency signal.
Consider the source density in the elemental volume dui. This is given by [2] 4i = (pl&)a(p’)l%
(1)
where p is a non-linearity coefficient, p. is the density of the unperturbed medium, co is the sound speed in the unperturbed medium and pi is the total acoustic pressure at dui. Let the primary waves at the elemental volume dvi be given by the general expressions Pli=Ali
exp[-@1jsil+j(kl.
si-wlt)+Xli+jSlil,
PZ~
exp
Si--2t)+X2i+jS2il,
=A2i
[-~2lSil+j(k2.
where (x1(, xzi) and (Sit, Szi) represent, respectively, the log-amplitude and phase fluctuations of the primary waves at dui; Ali and A2i are generally complex functions of position, but in the ideal spherically spreading wave case their values are real and inversely proportional to range; (Y~and 13~are absorption coefficients.
PARAMETRIC
SIGNAL
IN A RANDOM
397
MEDIUM
The total pressure pi, when second order effects are ignored, is given by pi = pli +p2i. After substitution of pi into equation (l), with retention of only the difference frequency wave component and neglect of frequency spreading, it follows that the source density at the difference frequency, q3iFis given by q3i z (-jW?AriATiIPL?cZ)
exp [-(a1+az)]si]+j(k3.
~--3f)+~1;+~2i
+j(S,i-S2,)1,
where w3 = w1 - w2 and k3 = kl - kz. The elemental difference frequency pressure dpNi at the receiver Q, produced by q3i at dv, is given by dp, = (-jw3po/4rlril)q3i
{exp [-a3lri/
+ $3 . ri +x3i + j&i]) dui.
The total difference frequency pressure, pN, at 0 is then given by integration of dp, over the interaction volume Di: PN
=
exp (-jW)
IHi exp [Xii
+XZ +x3i + j(s,i +SZ +SX)] duiy
where Hi is given by Hi =(~:PAliA2*ii/4~~oc~Iril) exp [-((yl+~~~)Isil-(~glri/+jk3.
(si+ri)+je],
8 being an arbitrary phase angle to be defined. The quantity Hi dui is the unperturbed differential pressure at point Q arising from the source density in the volume dui. In a perfectly homogeneous medium completely free of scatterers the difference frequency pressure would be simply given by PNO= exp (-jw3t)
Hi dvi.
I
This will be referred to as the unperturbed difference frequency pressure. The value of Hi has been discussed in many publications (see e.g., references [2,8-lo]. Therefore, for the present purposes, its value will be assumed to be known. The log amplitude fluctuation, xN, and phase fluctuation, SN, of the difference frequency wave are defined by pN It
therefore
=pNO
exp
(xN
+jsN).
follows that
exp[XN+jSN]=(l
Hidv,)lf Hi exp lili
+x2i +x3< +
j(s,i -s2i +S30] du;.
(2)
Equation (2) relates the overall difference frequency wave amplitude and phase fluctuations to the fluctuations of the individual component waves which constitute the parametric array. By using the method of small perturbations it is possible to simplify equation (2) and obtain more useful expressions for the difference frequency amplitude, xN, and phase, S,, fluctuations. For example, if all the fluctuations are assumed to be small such that ]xPlCC1 and IS,\ K 1, where p = IV, li, 2i, 3i, then the following approximations for the exponential terms in equation (2) may be used: exp
hN
+jsNl=
1 +xN
+jsN,
398
N. P. CHOTIROS
AND
B. V. SMITH
where for convenience xl;+x2;+~3; and Si;-S2;+S3; have been written as x; and S; respectively. When these approximations are used equation (2) becomes
xN+jSN=([ H; du;)-l[
H;(X;+jS;)db;.
In the definition of H; an arbitrary phase angle f3has been so far left undefined. It may now be defined such that Im (5 H; du;) = 0. In practice this automatically happens when the signal is envelope detected. With this definition of 0 the real (Re) and imaginary (Im) parts of equation (3) may be simply separated, giving xN =
J[Re (Hi)xi -
Im {Hidz+
(Hi)Si]doi
sN
=
I Be (Hi)Si+ ImW;)X;ldvi IHidv;
’
.
(4
5)
’
Hence the mean squares of the difference frequency wave amplitude and phase fluctuations become
&=(I Hi duij2 [I [Ej
Re (H;) Re (Hi) + %$ Im (H;) Im (Hj)
--
- Re (H;) Im (Hi)(xiSi+ XjSi)] dvi duj, z=([
H;dv;)-211
[S;Sj - Re (Hi) Re (Hi) +E
-+ Re (H;) Im (Hi)(Sixi+
S~XJI dvidvp
(6) Im (Hi) Im (4) (7)
We have been able to show that these expressions for zNand zN, equations (6) and (7), are also applicable when the method of smooth perturbations (also known as the Rytov method) is used. However, under these circumstances the terms xP and S,, where p = N, i, j, are strictly logarithmic terms. Equations (6) and (7) give the mean square difference frequency amplitude and phase fluctuations in terms of various spatial correlation functions: e.g., &J$ A brief review of these is given in the next section. 2.1.
SPATIAL
CORRELATIONS
OF COMPONENT
WAVES
-The correlation functions in question are x;xj, S;Sj and s. NOWxixj, and %&represent spatial correlation functions for amplitude and phase fluctuations respectively, where the subscripts i, j refer to elemental volumes dv;, dvj. Similarly x;Sj represents spatial cross-correlation functions between amplitude and phase fluctuations at the elemental volumes do;, duj. First one can expand x;,j and S;,j into their components: e.g., ,Y;= xl; +x2; +,y~; and S; = Sl; - S2; + S3;. The fluctuation terms denoted by the subscripts 1 and 2 belong to the waves, at the primary frequencies, propagating from 0 to S; (see Figure 1) while the terms denoted by the subscript 3 are for the difference frequency wave propagating from s; to R. Since both the paths and the frequencies are quite different, it is to be expected that there is negligible correlation between these two groups of fluctuation terms. Therefore, the correlation terms in equations (6) and (7) may be approximated by XiXj’(Xli+X2i)(Xlj+X2j)+X3iX3jv
S;Sj- (Sl;-S2;)(Slj-S2j)+S3;S3j, &$=
(Xl;
+XZi)(Slj-S2j)
+X3iS3j-
Data on the cross-correlation between amplitude and phase fluctuations, i.e., 3, may be obtained from reference [3] for plane waves and from reference [4] for spherical waves.
PARAMETRIC
-
SIGNAL
IN
A
RANDOM
MEDIUM
-
399
The terms K, G, SiiSzi and S’ziSzj are spatial correlation functions for signals transmitted from the same point and received at two different locations. Many aspects of the theory pertaining to such have been explored [3,4,11]. - a geometry The terms G, G, SiiSZjand SZiStjare spatial frequency correlation terms, i.e., the spatial cross-correlation of either the amplitude or the phase fluctuations between the two primary waves. This problem has been studied by Eliseevnin [ 121 for the longitudinal case. In most parametric array applications the frequency difference between the primary waves is small and hence the following approximations are reasonable: (xii + xzi)(xrj +xzj) = ~~~~~~~~ where xc is the amplitude fluctuation of a wave at the average of the two primary frequencies, and (Sli - Szi)(Sij - Szj) = Sd+Sdi,where Sd is the phase fluctuation of a wave at the difference frequency propagating from the source to the elemental volume. Finally, the terms x3ix3j and S3iSsj are correlation functions of amplitude and phase fluctuations, respectively, for two waves both received at Q (see Figure 1) but transmitted from two different sources, the one at dvi and the other at dvi. A study of such a correlation function has not been found in the available publications; it is also very difficult to measure experimentally because of the obvious difficulties in distinguishing two signals at the same frequency arriving at the same receiver simultaneously. However, for an isotropic medium it is possible to demonstrate that reciprocity holds. Therefore, x3ix3j and S3S3, are numerically equal to the corresponding correlation functions of a single wave transmitted from P and received at dui and dvh In the case of the far field, on-axis fluctuations of a narrow beam parametric signal, equation (6) may be considerably simplified to give a very simple and plausible result, because the imaginary part of Hi is negligibly small. Therefore,
-Since the range is long compared to the array, xixj’x3ix3j. The fluctuations of a linear signal at the same range is given by xL, and therefore x~=~==~. A correlation coefficient may be defined as N3ij = X3iX3jI(XZiXZj)1’2,
IN3ijl~ 1.
(9)
Hence, the ratio between the fluctuations of the non-linear and linear signals is given by
ZIIXZ=
HiHjN3ij dvi dujl
HiHj dvi dC(.
(10)
Except for certain rare situations, the value of Nij is positive and varies monotonically as a function of the separation vector between dvi and dvj. Also HI and Hj are positive quantities for the most part with the exception of small regions within the near field of the transducer. In the idea1 case of a perfect end-fire array, Hi and Hi would always be positive. Therefore, it can be deduced that
z/z=
HiHjN3ij dvi dvjl
HiHj dvi dvj 4 1.
(11)
This is an interesting result because it shows that in the far field of a narrow beam parametric array the fluctuations of the difference frequency would be no worse than those of a directly transmitted linear signal. 3. IMPLEMENTATION OF THE THEORY Theoretical estimates of the parametric signal fluctuations have been calculated by using equation (6). Unfortunately, all of the terms in the equation are very difficult to evaluate.
400
N. P. CHOTIROS
AND
B. V. SMITH
Exact numerical answers were not possible. However, since they are integrated over a large volume, some approximations are possible. These are described below. By limiting the problem to on-axis signal fluctuations only, azimuthal asymmetry can be ignored without causing gross errors. Therefore, although the transmitting transducer in the experiment to be described was square, it was assumed that the primary signal profile was Gaussian and symmetrical about the axis of the primary beam. This gave an analytic expression for Hi. The primary beam was also assumed to be perfectly collimated in the near field, and beyond the Rayleigh distance the beam was assumed to be spherically spreading. The justification for adopting this simple, single lobe approximation of the directivity function is that the difference frequency signal is a function of an integral of the primary signals. Therefore, the overall average shape of the primary beam profile rather than its fine structure is what matters. Estimation of the various spatial correlation terms, xixi, etc., presented another problem. For infinite plane or spherical waves at long ranges, they depend only on the separation vector. For infinite plane or spherical waves at short ranges, they depend on both separation and range. In the case of the experimental model, the primary beam is narrow; therefore they depend on both position and separation vectors, and this is the case for which a theoretical estimate is required, However, since these terms are integrated over relatively large volumes, some approximations are possible, and theoretical results for infinite plane and spherical waves may be used with modifications to obtain approximate results. The correlation terms were considered as two separable components, an amplitude factor 7 and a correlation coefficient Nii, defined as follows:
The amplitude factors were approximated by the existing theoretical results for infinite waves. These results in turn were approximated by the simpler expressions applicable to the asymptotic range conditions [4] m >>LO and LO >>m, where L is the range, A the acoustic wavelength and Lo the upper limit of the homogeneous turbulence characteristic length [3,4,13]. Again, since the terms are integrated over a wide range of values of L, using these asymptotic expressions instead of the more cumbersome exact expressions introduced only marginal errors. As an example, asymptotic values of 2 as a function of range L3i are shown in Figure 2.
Figure 2. Linearized
amplitude
fluctuation
coefficient
as a function of range.
The correlation coefficients, Nij, were more difficult to evaluate since there are very few theoretical solutions available even for ideal infinite waves, and a review is given in reference [14]. Longitudinal and transverse components of Nij were assumed to be separable and independent, and the transverse component was assumed to be a range
PARAMETRIC
SIGNAL
IN
A
RANDOM
MEDIUM
401
dependent Gaussian function. Theoretical results based on the works of Chernov [3 j and Karavainikov [13] show that this is a reasonable approximation for most values of range and separation vector, except where the longitudinal separation is small compared to the dominant component of the refractive index spectrum. Again, since the term is integrated over a large volume, this discrepancy may be ignored. Lastly, finite beam width effects had to be taken into account, since the theoretical expressions used were derived primarly for infinite plane or spherical waves. As a consequence of the law of conservation of energy, the transverse component of correlation in a narrow beam is correspondingly narrower than that of an infinitely broad beam. Moreover, in the narrow beam there is a higher degree of negative correlation, and when integrated over the whole beam the positive and negative components tend to cancel. This problem has been discussed by Tatarski for the case of an infinite plane wave in reference [4]. By using equation (6), the above approximations, and a linearized refractive index spectrum of the form proposed by Medwin [15], a computer program was obtained which gave theoretical estimates of the on-axis amplitude fluctuations FN of the difference frequency signal. Further details are given in the Appendix. 4. LABORATORY TANK EXPERIMENTS
A random refractive index field was generated in a water tank, dimensions 1.8 x O-9 x description of the experimental arrangement has been presented elsewhere [l, 7, 141. A temperature microstructure was produced by convective currents, rising through a horizontal perforated metal sheet, generated by a heater array along the tank bottom. Investigations [l, 141 have shown that the temperature microstructure behaved as a passive additive of turbulence, and furthermore that it was the primary cause of the refractive index variations. Amplitude fluctuations of linear acoustic signals in this medium have been observed [l, 141 and were found to agree with theoretical estimates, to within the limits of experimental error. The parametric transmitter used had primary frequencies centred on 9 MHz and a 1 MHz difference frequency signal. The transducer itself was a 1 cm square piezoceramic plate. For comparison, a 1 MHz linear signal was also available from a 2 cm square piezoceramic plate. Small transducers with effective apertures of about 1 mm in diameter [ 1,141 were used for reception. By using measurement techniques outlined previously [l, 141, the normalized average sample variance, V$, of the axial amplitude fluctuations of the difference frequency signal were measured as a function of range. The result=e shomin Figure 3(a). The coefficient of variation, VN, was defined by V’, = {m(/~~/)~}/(Ip~j)~, and for small changes this approximates to V& -3. The subscript N indicates that the signal was non-linearly generated in the medium. A corresponding statistic VL may be defined for the linear signal. Theoretical estimates were computed from the results of section 2, with use of the known refractive index spectral function [l, 141. For comparison, V,, was also measured by using the linear 1 MHz signal, and it is shown in Figure 3(b). The spread in the experimental results, unfortunately, is large enough to cover up any small differences which may exist between VN and VI_,since the two sets of data have very similar values. The cause of the spread may be attributed mostly to the lack of stationarity of the temperature microstructure. Therefore, an alternative technique was required whereby the two types of signal could be observed under identical conditions: i.e., in the same part of the medium and at the same time. This was achieved by making use of the reciprocal behaviour of the refractive index field, as follows. O-9 m3. A detailed
402
N. P. CHOTIROS
AND B. V. SMITH
meowed
values
L(m)
Figure 3. Coefficient of amplitude variation of (a) a 1 MHz parametric signal as a function of range and (b) a 1 MHz linear signal as a function of range.
Both the parametric signal and the linear signal were transmitted simultaneously in opposite directions from the two respective transducers facing each other. Each transducer also functioned as a receiver when a signal arrived from the opposite transmitter. A block diagram of the experimental system is shown in Figure 4. The amplitudes of the two signals were compared, and the result is shown in Figure 5, in terms of the ratio of their coefficients of variation. The spread in the experimental results was small enough to show clearly differences between the two types of signal. Reciprocity was checked by transmitting identical linear signals in both directions.
-IF-
TRANSDUCER
TRANSDUCER
LINEAR TRANSMITTER
PARAMETRIC TRANSMITTER
PRF UNIT
ENVELOPE
ENVELOPE
DETECTOR
DETECTOR
t
t DATA
LOGGER
Figure 4. Block diagram of experiment to compare parametric and linear signals using reciprocity.
PARAMETRIC
1 Standard
SIGNAL
IN A RANDOM
403
MEDIUM
deviation 2
Average
of measured
values (reciprocity
r
check)
i
I
1
00
0.9
I
I
1 0
1
5
L(m)
Figure 5. Comparison of the amplitude fluctuations V,, V, and V,,, of a parametricsignal, a linear signal, and a linear signal transmitted in the reverse direction, respectively, as functions of range. At the longer ranges, the signal was transmitted along a diagonal of the tank in order to avoid multiple path propagation; consequently, the effective value of k, was significantly greater than the calculated value at the centre of the tank.
4.1.
DISCUSSION
OF LABORATORY
RESULTS
Under the given experimental conditions and at the ranges indicated, the experimental results and theoretical estimates agree in indicating that the parametric signal amplitude is subject to smaller fluctuations than the linear signal. From theoretical considerations, it
I’
1
f
-I
12-
1 l-
(#
:
1 o-
Asymptote
0
9-
0
8-
I
I IO
I
1
1
I00
L(m)
Figure
6. Theoretical
comparison
of parametric
and linear signal fluctuations
for long ranges,
K, = 179 rn--‘.
404
N. P. CHOTIROS
AND
B.
V. SMITH
can be shown that this is a consequence of the narrow primary beam. Theoretical estimates of the VN : VL ratio for other ranges show that at long ranges the ratio is less than but approximately equal to unity. At very short ranges, the validity of the theoretical estimates is doubtful, and the individual values of V, and VL would be extremely small and insignificant for most practical purposes. These estimates are shown in Figure 6. 5. EXPERIMENTAL 5.1.
DESCRIPTION
OF
THE
MEASUREMENTS
IN AN ESTUARY
SITE
The tidal estuary experiment took place near Weston-super-Mare, in the Bristol Channel. The main attraction of the site was the very high tides. This enabled the apparatus to be installed and optically aligned on land during low tide, and the underwater experiment to be carried out during the following high tide. The difference between high and low tide levels varied between 7 m during neap tides to 10 m during spring tides. The beach area contained a seaweed belt centred on the mid-water line, growing on the underlying rock and loose stones. Beyond the seaweed the bed rock was overlaid by sand and mud. The transducers were mounted on structures erected on the bed rock. A cross-section of the site is shown in Figure 7, with indications of the tidal conditions during the experiment. The maximum acoustic path length between transmitters and receivers was limited by signal-to-noise considerations, as well as multiple ray path and other logistical considerations. Hence, the transmitter-receiver separation used was limited to 43 m. 5.2. MEDIUM MEASUREMENTS In order to be able to make a theoretical estimate of the signal fluctuations, it was necessary to measure certain medium characteristics. Unlike in the laboratory tank where
Vertical Scale :
-5
?? nr
Tl.T3
I T2
metres
-3
Figure
7. Test site geometry
and water level at high tide.
PARAMETRIC
SIGNAL
IN A RANDOM
MEDIUM
40.5
the medium could be controlled, in this case the medium was very variable. An instrument package consisting of two thermistor probes and an acoustic velocimeter was mounted on a cantilever on the same structure as hydrophone Hl. The thermistor probes measured temperature fluctuations while the velocimeter measured a component of the flow velocity over a path length of approximately O-2 m. Measurements were taken simultaneously with the acoustic experiments, as far as possible. The temperature measurements show that over an interval of approximately 100 s the average sample variance of temperature was in the region of (O.O3”C)*; it was depth dependent and non-stationary, falling to a minimum at high tide. With just two thermistors, it was not possible to measure the structure function directly but from the knowledge of the flow velocity it was possible to make an estimate by deduction. It was also deduced from the flow velocity measurements that turbulent eddies would make a significant contribution to the acoustic signal fluctuations, unlike in the laboratory experiment. The effects of other scatterers were not measured (e.g., air bubbles), although, by using the velocimeter in a different mode, it was possible to measure the total refractive index fluctuations averaged over the 0.2 m path. The equipment and expense required to carry out a comprehensive measurement of the medium was beyond the scope of this experiment. However, by using the available measurements, and using the Medwin spectral model [14,15], it was possible to make linearized estimates of the effective refractive index spectral function caused by temperature microstructure and turbulent eddies. These are shown in Figure 8. These results represent upper and lower bound estimates of the average isotropic refractive index variations, and they can be regarded only as rough estimates since the refractive index fluctuations were in fact neither stationary nor isotropic. 10-‘0,
,
I
Upper bound
Lower bound
1
10-Z kn
1 hb
102
4,
f
f
kob’
kob
IO4
Wove number (rn-‘1
Figure 8. Estimated temperature variations
linearized one-dimensional and turbulence.
refractive
index
spectral
density
function,
caused
by
5.3. ACOUSTIC MEASUREMENTS The non-linear and linear signal fluctuations were measured and compared in a similar way as for the laboratory experiments previously described. The sequence of events was as
406
N. I’. CHOTIROS AND B. V. SMITH
follows, with reference to Figure 7. As high tide approached, preliminary medium measurements were taken; this occurred during the interval labelled A. Then, the acoustic measurements were taken. During intervals B and D, a non-linear pulsed signal was transmitted from Tl to H4, simultaneously with a pulsed linear signal from T2 to Hl. During intervals C and E, a pulsed linear signal was transmitted from T3 to H4 and an identical one from T2 to Hl, simultaneously, to test the reciprocity of the medium. As the tide receded, final medium measurements were taken in interval F. The coefficients of variation, V, and VL, are shown in Table 1. The results show that the medium was not reciprocal. The fluctuations for signals going from Tl or T3 to H4 were very much higher than for those from T2 to Hl. The cross-correlation coefficient was approximately zero. TABLE 1 Acoustic results from the estuary experiment
Period
Hydrophone
Signal type
B B B-C C C
H4 Hl
Non-linear Linear
H4 Hl
Linear Linear
D D D-E E E
H4 Hl
Non-linear Linear
H4 Hl
Linear Linear
0*0588 0.0210
2.80 1.45
0.0472 0.0245
1.93
0.0584 0.0191
3.06 1.59
0.0662 0.0344
1.92
With allowance for the lack of reciprocity, which was assumed to increase the signal fluctuations in one direction by a constant factor relative to those in the opposite direction, an estimate of the ratio V, : V, was made (see Table 1). This represents the ratio between the non-linear and linear signal fluctuations when the signals are propagating under identical conditions. 5.4. DISCUSSIONOF ESTUARY RESULTS
Due to a number of practical factors, the amount of data obtained in the estuary experiment was not as extensive as initially planned, but a number of interesting results were obtained. The apparent lack of reciprocity was very striking. This may be caused in two ways. Firstly, the medium was horizontally stratified to some extent [16]. From Figure 7, it can be seen that the ray path Tl or T3 to H4 is closer to the horizontal than T2 to Hl, although the difference is less than l-5”. Depending on the severity of the horizontal stratification, this may account for the difference in the fluctuation levels. Secondly, turbulence can also account for lack of reciprocity [17], but it is difficult to see how it can cause consistent differences on both sides of the tide peak, since the flow patterns before and after the tide peak must be quite different. By using the spectral estimates in Figure 8, a theoretical estimate of the VL : VN ratio was also obtained and is plotted as a function of range in Figure 9, along with the measured data. Since the medium cannot be described as isotropic by any stretch of the imagination, the theoretical results can be regarded only as order of magnitude estimates. The obvious conclusion, however, is that the difference between the fluctuations of parametric and
PARAMETRIC
I
SIGNAL
IN A RANDOM
I
IO
407
MEDIUM
I
1000
J 10000
Range Cm)
Figure 9. Comparison of the amplitude fluctuations of parametric and linear signals as a function of range, in k, = 40.0 m-‘, - - -. Empirical points: 0. the estuary experiment. Theoretical curves: k, = 21.7 m-‘, -;
linear signals is small beyond the parametric array length, for both upper and lower bound refractive index spectra. 6. CONCLUSIONS A thoretical expression for describing the fluctuations of a parametric signal has been derived based on the small perturbation method, and it has been implemented in the form of a computer program for predicting axial signal fluctuations. The theory was supported by laboratory experimental results. Experimental results outside the laboratory were also consistent with the theory, although these results were very limited. It is quite clear, however, that the fluctuations of a parametric signal would be of the same order of magnitude as that of a directly transmitted linear signal. It can also be deduced from the theory that, although there may be significant differences in the near field, their fluctuations in the far field would be approximately equal. Therefore, there should be no significant disadvantage in using a parametric signal instead of a linear signal, as far as signal fluctuations are concerned. ACKNOWLEDGMENT This work has been carried out with the support of the Procurement of Defence.
Executive, Ministry
REFERENCES N. P. CHOTIROS and B. V. SMITH 1976 in Proceedings of Institute of Acoustics Underwater Group Conference on Recent Developments in Underwater Acoustics, Portland Paper 4.1. The parametric end-fire array in a random medium. P. J. WESTERVELT 1963 Journal of the AcousticalSociety ofAmerica 35535-537. Parametric acoustic array. L. A. CHERNOV 1967 Wave Propagation in a Random Medium, (translated by R. A. Silverman). New York: Dover Publications.
N. P. CHOTIROS
408
AND
B. V. SMITH
4. V. I. TATARSKI 1961 Wave Propagation in a Turbulent Medium, (translated by R. A. Silverman). New York: McGraw-Hill. 5. B. J. USCINSKI 1977 The Elements of Wave Propagation in Random Media. New York:
McGraw-Hill. 6. B. V. SMITH 1971 Journal of Sound and Vibration 17,129-138. A theoretical study of the effect of an inhomogeneous medium upon a transmitter which exploits the non-linear properties of acoustic propagation. 7. B. V. SMITH, W. WESTON-BARTHOLOMEW and N. NAKLI 1973 in Proceedings of the Conferences, Finite Amplitude Wave Effects in Fluids, Copenhagen, 151-155. London: IPC. An experimental study of the parametric end-fire array in a random medium. Difference 8. R. L. ROLLEIGH 1975 Journal of the Acoustical Society of America 58,964-971. frequency pressure within the interaction region of a parametric array. 9. H. 0. BERKTAY 1965 Journal of Sound and Vibration 2, 435-461. Possible exploitation of non-linear acoustics in underwater transmitting applications. 10. H. 0. BERKTAY and D. J. LEAHY 1974 Journal of the Acoustical Society of America 55, 539-546. Far field performance of parametric transmitters. 11. R. T. AIKEN 1969 The Bell System Technical Journal 1129-1165. Propagation from a point source in a randomly refracting medium. 12. V. A. ELISEEVNIN 1974 Soviet Physics-Acoustics 19, 533-538. Longitudinal frequency correlation of fluctuations of the parameters of plane waves propagating in a turbulent medium. 13. V. N. KARAVAINIKOV 1956 Iaroslavl K.D. Ushkinskii State Pedagogical Institute. Fluctuations of amplitude and phase in a spherical wave. 14. N. P. CHOTIROS and B. V. SMITH 1979 Journal of Sound and Vibration 64,349-369. Sound and amplitude fluctuations due to a temperature microstructure. 15. H. MEDWIN 1974 Journal of the Acoustical Society of America 56, 1105-l 110. Sound phase and amplitude fluctuations due to temperature microstructure in the upper ocean. 16. V. M. KOMISSAROV 1964 Soviet Physics-Acoustics 10,143-152. Amplitude
and phase fluctuations and their correlation in the propagation of waves in a medium with random, statistically anisotropic inhomogeneities. 17. Yu. I. BELOUSOV and A. V. RIMSKII-KORSAKOV 1975 Soviet Physics-Acoustics 21, 103-109. The reciprocity principle in acoustics and its application to the calculation of sound fields of bodies. 18. N. P. CHOTIROS 1980 M.ScQ. Thesis, University of Birmingham. The parametric array in an inhomogeneous medium. APPENDIX Computation of the difference frequency mean square fluctuations was carried out by numerically evaluating the two-volume integral given by equation (6). Chernov’s results for a random medium with a Gaussian spatial correlation function were used because of their relative simplicity. These were reconciled with the Medwin refractive index spatial wave number spectrum by employing an equivalent Gaussian correlation coefficient N(r) = exp (-r2/U2,i), where the scale range (Li). This the fluctuations numbers [4]. By using the phase fluctuation
n = 1,2 or 3,
parameter a,i was dependent on both acoustic wave number (k,) and is a valid approximation since Tatarski has shown that at any given range of the sound signal are dominated by a narrow band of spatial wave same method as Chernov, the following expressions correlation functions were obtained [18]:
for amplitude
and
641, A2) Here z is the equivalent determined as follows.
mean
square
refractive
index
fluctuation,
and I1 and I2 are
PARAMETRIC
SIGNAL
IN A RANDOM
MEDIUM
409
When the longitudinal separation AL is greater than ani and the transverse separation component 1 is in the region I < Q,i {(2AL/ k,a ‘,i) + (k,a Ei/2AL)}p and these assumptions are satisfied for the greater part of the two-volume integral, then 11 is approximately given by 11 ={J?rk~a,iLi/(l+4AL*/a~ik~)}
exp (-/‘/Lz:~),
iA3)
where AL = Li - Li (it being assumed that Li < Lj), Li =x; if n = 1, 2 (primary signals), Li = R -xi if it = 3 (parametric signal), and 1’= (yi - yj)‘+ (zi - Zj)*. When the range L; is large compared with the Rayleigh distance, Li >>k,aEi, then II ~12, and 12 may be neglected. When the range is small, Li cc k,aEip then II= 12, and in the case of amplitude correlation, which is given by the difference between Ii and 12, it was necessary to consider second order terms; the resulting expression was approximately [lS] XniXnj-2Jiz
nrani -3 Li exp
(-Z*/U”,~)
{2L;AL +iLf}.
(A4j
These asymptotic solutions were used to form composite linearized approximations of the correlation functions required. Thus, the two-volume integrals were written as a summation of terms, each of the form
X
645, ‘46)
dxi dxj dyi dyj dti drj,
where Xi = Re (H;) or Im (Hi) =FH(Xi) exp {-2(y? + ~:)/bf}, and
-~YniYnj
=
XniXnj
or
SniSnj
=
Fn
(Xi9
Xj)
exp
I-[(Yi
-
Yj)*
+
(zi
-
zj)‘lla ti>,
FH(xi) and Fn(xi) being functions of the longitudinal position vector Xi+It can be seen that the assumption of Gaussian shaded primary beam profiles, which lead to corresponding Gaussian shaded source densities, allowed the integrals in yip yj, Zi and zj to be solved analytically. The remaining integrals in xi and xj were computed numerically, as integrals of the form R
II0
R
0
F(xo,
Xi, ani,
bi, 2)
dxi
dxp
647)
However, the expressions used to obtain the functions F(. . .) were strictly applicable only to infinite plane waves or spherical waves. The following corrective measures were used to take into account the effects of the narrow primary beams. Whenever the beam width bi was smaller than the transverse correlation width U,~,the transverse correlation components in equations (A3) and (A4) were changed to exp (-1*/b?), since correlation width can never exceed beam width. In the case of amplitude correlations, a consequence of the law of conservation of energy requires that the total amplitude fluctuation over a beam cross-section be zero. When the beam width is broad the effect would not be noticeable, but for narrow beams the correlation function F(. . .>in equation (A7) would be reduced by a factor Ki equal to [18] 77X interaction volume cross-section =1n-b? k3 Rayleigh aperture 2 2(R -xi)’
410
N. P. CHOTIROS
AND
B. V. SMITH
It is very difficult to assess the overall error introduced by the various approximations and assumptions made in the above evaluation of equation (6). However, in all cases they have been justified upon physical grounds and the final predictions do agree with experimental observations.