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Underwater acoustic target strength of nets and thin plastic sheets
Journal of Sound and Vibration (1973) 28(1), 139-149
U N D E R W A T E R ACOUSTIC TARGET STRENGTH OF NETS AND THIN PLASTIC SHEETS V. G. WELSBY AND (.J'. C. GODDARD
Department of ElectroMc and Electrical Engineering, Unirersity of Birmingham, Birmhzgham BI 5 2TT, England (Receired 13 January 1973) The purpose was to find the most suitable material to form a cage, itself having a very small acoustic echo strength, for use in experiments on the acoustic target strength of fish. A series of measurements have been carried out in the Acoustic Tank Laboratory at the University of Birmingham on samples of various materials and a theoretical analysis has been made which gives a useful insight into the way the echo strength depends on the dimensions and mechanical properties of nets and sheets. Sufficient agreement was obtained between theory and practice to show that the results will be valuable for future design work. Plastic sheets were shown to have no advantage over nets. The study has shown that significant effects are caused by surprisingly small amounts of gas adhering to the solid materials. It is this which is likely to set a practical lower limit to the target strengths which can be attained.
1. INTRODUCTION The present study arose in connection with research into the target strength of fish which required a cage strong enough to restrain a fish but having an echo level small compared with the signal from the wanted target. A series of measurements have been carried out in the Tank Laboratory at the University of Birmingham on samples of materials which might be used for the construction of cages for this purpose. A theoretical analysis has also been made which gives a useful insight into the way the target strength depends on the dimensions and mechanical properties of nets and sheets.
2. DESCRIPTION OF TESTS These fall under four headings: (a) (b) (c) (d)
straight filaments, filaments, net, /crumpled up. plastic sheets,)
Test (a) was intended mainly to establish the measuring technique and to confirm that there was some reasonable agreement between results and theory for this relatively simple case. Tests (b), (c) and (d) were then made with the idea of determining the mean target strength of a given quantity of material, when crumpled up so that it could be assumed to be randomly orientated with respect to the incident acoustic waves. The results could then be used to estimate the target strength of any proposed cage design using the material concerned. In each case two different sonars were available, working at 507 kHz and 986 kHz 139
140
v.G. W E L S B Y A N D G. C. G O D D A R D
and with 3-dB vertical beamwidths of 30 degrees and 5 degrees, respectively. The targets were placed at a range of 1.17 m for the 507 kHz tests and 2.54 m for the 986 kHz tests. Brass or steel solid spheres, suspended by fine nylon monofilament, were used as reference targets. The target strengths were calculated theoretically [1 ]. Adequately long pulse durations were used and the shapes of the echo pulses were checked to make sure that the readings did in fact represent "steady-state" conditions. A storage oscilloscope was used, the image of each sample pulse being retained for several seconds to enable an accurate estimate o f its amplitude to be made. In practice the cage is suspended in such a way that there is bound to be a certain amount of movement relative to the transmittel. The test objects were therefore allowed to hang freely in the water and a slight motion was imparted in order to ensure that the echo components from various parts of each target were randomized in phase. Each recorded value is the mean of 200 individual readings obtained in this way. In Figure 1 a histogram has been drawn for a typical set of readings. The fact that this forms an acceptable approximation to a Rayleigh distribution confirms that phase randomization was occurring. I
I
I
I
I
I
3
4
5
6
025 o2
Ol
~
iMe--~176
!
005
i
0
2
I
7
X
Figure 1. Distribution of a typical set of 200 measured values of pressure amplitude of target echo; (in this case for a sample of polythene net). x is an arbitrary variable proportional to pressure amplitude. An analysis of the relevant scattering theory is given in full in the Appendix and only the conclusions reached there will be summarized in this section so that they can be compared with the experimental results. (a)
Straigllt filaments
The usual definition of target strength is based on the fact that the value of the ratio of the received pressure p,, in the back-scatter direction, at a distance r, to the incident pressure p for a "large" sphere is p,/p = 89 where a is the radius of the sphere. For a sphere of 2 m radiusp,/p = l/r. So the target strength of any other object is obtained from its value of p,/p by means of the formula
TS=2Olog,or(~),
(1)
with r measured in metres. For a discrete object the "far-field" scattered pressure ratio p,]p is proportional to l[r so that the target definition given above gives a figure that is independent of r. In the case of the "infinitely-long" cylinder, however, it is proportional to ( I / r ) ~/2 so
141
TARGET STRENGTH OF NETS AND SHEETS
the target strength, as defined above, is not a convenient parameter to use. It is better to discuss the results directly in terms o f the ratio p,[p. F o r a straight filament at range r, with ka < 1, this is given theoretically [1--4] by
p,
1 ~
[a\ '12
= - 1 F /r=o ( k a ) 2(kr) -112 lq,
(3)
2,N/2
where
p,>1j"
+
(4)
A and p represent bulk modulus and density, respectively, subscript 1 refers to the water and subscript 2 refers to the filament. When k a >> 1 the ratio tends to a value P.Z"= (ka)U2(kr)-llz K~, (4a) P where K~ is a different factor, the value o f which depends on ka over the transition range where ka is roughly unity, and eventually settles down to the impedance mismatch coefficient (pzc2--plcl)/(p2c2 +plcl) when the wavelength becomes very small c o m p a r e d with the radius o f the cylinder. i
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"~
-IC
-201
-3(
o
/
i
f"
X~ /
x
-40
OI--
i
x
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02
05
IO
2
5
io
ko
Figure 2. Ratio of received pressure p, to incident pressure p for straight nylon filaments, at range r. Circles indicate measured values at 986 kHz, crosses indicate measured values at 507 kHz. Curve (a) gives predicted values, for ka < I, assuming material of filament to have infinite density and bulk modulus. The squares indicate some theoretical values (for similar materials), due to Faran [5]. Curve (b) is the corresponding result for ka ~- 1. Curve (c) is the predicted result when the actual values of density and bulk modulus for nylon are used. Vertical scale is 201og(p,/p) for kr = l. In Figure 2 the theoretical curves are plotted in a normalized form, with the zero o f the decibel scale taken f o r k a = 1 and kr = 1. Three additional theoretical points, due to F a r a n [5], have also been included. The samples used for the measurements were mono-filament fishing-line o f various diameters. Each sample was hung vertically, with a weight at the lower end chosen to ensure that the filament was as straight as possible without being over-stressed. A 6.3 m m diameter
142
v . G . x,VELSBY AND G. C. GODDARD
brass ball was used as a reference target which could be placed at the same range as the specimen to enable the system to be calibrated. The brass sphere was acoustically "large" at the frequencies concerned so its value ofp,/p was given by p,/p = 89 where al is the radius of the bail. The echo from the sphere appears on the normalized scale ofpr/p at a level
(kr)llz dB.
20 loglo
(5)
The target echoes were recorded in dB relative to that from the sphere and this enabled them to be transferred to the normalized graph so that they could be compared with the predicted curves. The results are shown in Figure 2. The specific bulk modulus and density were taken as 3.1 and 1-I respectively for nylon, giving Kz = 0.78. The mean level of the measured values is some 10 dB or so below the predicted figure. An encouraging feature of the results is that, although measured at different ranges and frequencies, with different sonars, they do agree reasonably well with the general form of the predicted curve. This tends to confirm that the measurements are probably reliable and that the discrepancy is due to some factor that has not been taken into account in the theory. A possible explanation for at least part of the difference is that the theory has taken no account of the fall in the incident field intensity towards the edges of the nominal beam, or of the directional pattern of the receiver. The tests were regarded as satisfactory and work proceeded with the rest of the programme of measurements. (b) Crumpled filaments For these tests, 20-m lengths of various nylon filaments were tangled into roughly spherical shapes each about 120 mm diameter. It is shown in the Appendix that the expected target strength TS, for ka ~ 1, is
TS = 20 loglo 88
ltz K2,
(6)
where L is the length of the filament and Kz is defined by equation (4). Measurements of TS were made at 507 kHz, a 19-mm steel ball being used as the reference target. The results are plotted in Figure 3, with the curve computed from equation (6) included for comparison. K2 has been assumed to be 0-78 for nylon. I -3C
/ -4C
/
/
/% -50
/'
/
/ -6C OI
I o 02
I O4
I 06
I I0
ko Figure 3. Target strengths of 20-m lengths of crumpled nylon filament at 507 kHz. a is radius of filament. Circles indicate measured values, dashed line gives theoretical prediction.
TARGET STRENGTH OF NETS AND SHEETS
(c)
143
Crumpled net
Tests were made at 507 kHz with samples of knotted polythene net which had a filament radius a = 0.3 mm and a mesh size o l d = 25 mm. From the Appendix the formula for target strength of netting is obtained as
TS= 201og,o88
.
(7)
Substituting assumed values of ARIA2 = 0.62 and P2/Pl = 0.94 for polythene in equation (4) gives K2 = 0.35. Hence the predicted target strength can be calculated as - 4 3 dB for S = 1 m 2. Effect of knots. This is discussed in the Appendix and it is concluded that the ratio of the scattered pressure due to knots, compared with that due to the filament of the net, is given approximately by
where a~ is the radius of each knot (assumed spherical) and /(2 and K4 are factors which depend on the compressibility and density of the material of the net. The values of these I
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-40
o
/
~45
/
0
/
0
-50
./
0
/
o -55 001
I 002
I 005
I OIO
I 020
I 050
Figure 4. Target strength of crumpled samples of 25-mm mesh knotted polythene net, as a function of area S, at 507 kHz. Circles indicate measured values. The dashed line gives predicted curve (neglecting the effect of knots).
factors would be very difficult to predict with any certainty but their ratio would not be expected to depart very drastically from unity for the materials likely to be used. The ratio (alia) could be assumed to be, say, 2 or 3. All that can really be said is that the effect of the knots cannot be ignored, so that at least part of the discrepancy between the measured target strength of the net samples and the predicted value for the filament alone (see Figure 4) may be due to this cause.
Crtunpled plastic sheets The Appendix gives the TS
(d)
in this case as 1 [ 1 /'/2
Ts= 2Olog,o - } kt(s)- K3,
(9)
144
v . G . WELSBY AND G. C. GODDARD
where t is the thickness of the sheet and
(A2--A, + p2--p, I
1':,=1 Z
(10)
. p, /.
Two sets of tests were made. In the first, samples of crumpled Melinex sheet, 0.5 mm thick and of various areas, were measured at 507 kHz. N o figures were available for the bulk modulus and density of this material so they were assumed to be the same as for polythene. This gave K3 = 0.35. The results are plotted in Figure 5 and compared with the theoretical curve, which gave - 4 9 dB for S = 1 m 2 for a lightly crumpled sheet or --43 dB for S = 1 m 2 if the sheet is crinkled, with a very small radius of curvature. Here again, there is reasonable agreement with the square-root law of variation of T S with area of sheet but the measured values are generally considerably higher than the predicted ones. I
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-40
o s
-as
/ (o )
/
/
-5C
I
002
I
005
I OtO S (m 2)
.// /
/ 001
/
I 020
/
'1 b) 050
IO
Figure 5. Target strength of crumpled samples of Melinex sheet, 0.05 mm, as a function of area S, at 507 kHz. Circles indicate measured values. Curves (a) and (b) are theoretical. They represent limiting cases where sheet is crinkled into very small radius of curvature [curve (a)] or only lightly crumpled up [curve (b)]. In the second series of tests the TS of samples of crumpled polythene sheet o f various thicknesses but the same area S = 0.40 m 2 were measured at 507 kHz. The value of K3 was taken as 0.35 and the theoretical value of T S was calculated. Table 1 shows all the results for sheets, with S = 0.40 m 2. The general conclusion to be drawn from these results is that, while they agree fairly well with theory for the thickest samples, they certainly do not do so for the thinnest one. It looks in fact as though there is some other factor, which has not been taken into account by the theory, becoming predominant for the very thin sheets. The most likely cause is a thin layer of gas, either continuous or in the form of very small bubbles, adhering to the surface of the sheet. There is no direct evidence for the truth of this assumption but it would explain the tendency for the target strength to become independent of the sheet thickness when the latter is reduced below a certain limiting value. (Incidentally, efforts were made by carefully rinsing the samples in a weak solution of detergent to ensure that they were properly wetted and that the amount of free gas was reduced to a minimum.) It is interesting to estimate what quantity of gas would be required to produce this effect. This can be done quite easily by noting that it is the compressibility mismatch of the sheet which matters, so the effect of a gas layer would simply be to increase the compressibility and to cause a proportional rise
TARGET STRENGTH OF NETS AND StIEETS
145
in target strength. Since the bulk modulus o f a gas would be about 10 -4 times that o f water while the bulk modulus of the sheet material is o f the same order as that o f water it follows that, to produce the observed effect, the gas film would only need to be about 5 • 10 -s millimetre thick. Or the same a m o u n t of gas would represent the volume o f a layer o f 0.2-mm diameter bubbles, scattered over the surface o f t h e sheet with a mean spacing o f a b o u t 10 ram. TABLE 1
Target strengths of samples of Melinex and polythene sheet, each 0.40 m 2 area, measured at 507 kHz Sheet thickness (ram) 0'050 0"025 0.013
Measured target Theoretical value strength (dB) (assumed the same , ^ . for both materials)i" Melinex Polythene (a) (b) -42 ---
-48 -47 -45
-47 -53 -59
-53 -59 -65
1"For the theoretical value (a) it is assumed that the sheet is crinkled into a very small radius of curvature while for value (b) it is assumed that it is only lightly crumpled. 3. CONCLUSIONS The most striking practical conclusion is that, irrespective o f theoretical predictions which take only the material o f a net or sheet itself into consideration, the real limit is almost certainly set by the presence o f gas bubbles on the surface o f the material. F o r this reason it was decided that, as far as the application described in the Introduction was concerned, plastic sheets seemed to offer little real advantage over nets. A surprising feature o f the theoretical analysis has been the smallness o f t h e a m o u n t s o f gas needed to produce significant changes in target strength. The experimental results have indicated that the limit m a y be at about - 4 0 dB for an area o f 1 m 2 o f net or sheet at 507 kHz.
REFERENCES 1. V. G. WELSBYand J. E. HUDSON 1972 Journal of Sound attd Vibration 20, 399--406. Standard small targets for calibrating underwater sonars. 2. I. MALECKX1969 Physical Foundations of Technical Acoustics. London: Pergamon Press. 3. LORD RAYLEIGH1877 Theory of Sound (two volumes). New York: Dover Publications (second edition, 1945 re-issue). 4. T. F. HtJE'rER and R. H. BOLT 1955 Sonies. New York: John Wiley & Sons. 5. J. J. FARAN 1951 Journal of the Acoustical Society of America 23, 405--418. Sound scattering by solid cylinders and spheres.
APPENDIX SCATTERINGTHEORY
A.l. Filaments and nets The theory o f scattering from a single small sphere provides a means of introducing ideas which apply generally to other shapes as well. Welsby and H u d s o n [1] give the derivation o f an expression for the scattered pressure p, at a distance r from the centre o f a sphere o f
146
v . G . WELSBY AND G. e . GODDARD
radius a, in a direction making an angle 0 with that of an incident planar wave field o f pressure p, It is assumed that the sphere is "small": i.e., ka < I. The results agree with those given by Malecki [2] :
p = \ 31 \r/K"
(A1)
where Kx
= [ A 2 - A, [- ~
3 ( p 2 - p,) 0]. 2 p 2 + p l cos
(A2)
J
A is bulk modulus, p is density, the suffix 1 refers to the water and the suffix 2 refers to the material of the sphere. For small objects it is not the mismatch between the impedances Pl cx and P2 cz which determines the scattering effect. There are two separate contributions to the scattering process; one is a "monopole" term produced by the mismatch between the compressibilities I[AI and I/A2 while the other is a "dipole" term produced by the mismatch between the densities Pl and P2. A small sphere made of a material for which pl cx = P2 c2 but Ax ~ A2 would certainly scatter sound, in spite of the apparent match between the impedances. The impedance concept is only valid when the incident and scattered fields have the same geometrical form; or at least can be assumed to have the same form over limited regions. This condition cannot be satisfied by "small" objects in a field with planar or nearplanar phase-fronts. Analysis for the case of an infinitely long cylindrical obstacle, of "small" radius a, lying perpendicular to planar incident waves is mathematically a little more difficult than that for the sphere but yields a result of the same general form: i.e., with a monopole contribution depending on the compressibility mismatch and a dipole term depending on the density mismatch. A solution is given by Malecki [2] but there is reason to think that this contains a numerical error in the dipole term. Reference to Rayleigh [3] suggests that the correct form should be
p
L "42
2
t92 + Pl
J
This is consistent too with the version quoted by Hueter and Bolt [4] for the special case when A2 >> A~ and P2 ~ 1. For the back-scatter direction cos 0 = - 1 and equation (A3) becomes
[aV/2 P--L"= ~l g ~ (ka)3'21r 1(2,
(A4)
where A2--AI
X'=L Z
2(p2-Pl)]
fAS)
Suppose now that the planar waves are replaced by an incident beam with a finite vertical angular beamwidth ~ and that the length of the cylinder, l, with its axis vertical, is reduced from infinity until it is equal to the insonified length 1 = ~r. The amplitude of the scattered pressure at the surface of the cylinder is still given by equation (A4) with r = a but its phase distribution is only partly coherent, with a coherence distance equal to the aperture size of the source of the incident beam. The resulting scattered power, instead of being radiated cylindrically, spreads spherically into a zone subtending an angle ~. This means that the
TARGET STRENGTH OF NETS AND SHEETS
147
pressure ratio p,[p, due to any length 2[ct < 1 < ctr of the cylinder, must be multiplied by a factor
2nrl ! 2rcr2 o: o~r"
(A6)
The numerator ofthis ratio is the area of a cylindrical zone ofaxial length I, and the denominator is the area of a spherical zone subtending an angle cc When lis made equal to the insonified length ctr the ratio becomes unity; in other words equation (A4) can still be used for finite incident beams, the predicted value ofp,/p being independent of the width of the incident beam. Suppose now that the cylinder is tilted relative to the vertical by a small angle ft. The scattered power will remain practically unchanged but the position of the zone into which it is scattered will shift through an angle 2ft. I f the beam is treated as ideal, with no sidelobes, it follows that as soon as Ifll exceeds ~/4 the scattering zone will miss the back-scatter direction altogether. So if there are a large number of short cylinders, at random orientations, the scattered power has to be multiplied by a weighting factor giving the probability that a cylinder will happen to assume a tilt angle for which the back-scattered power is not zero. This is the probability that e i t h e r - ~ ] 4 < f l < c q 4 or ( n - c q 4 ) < f l < ( r c + ~ [ 4 ) : i.e., 2(c~/2)[2n = ct[2n. If the length of each elemental cylinder is at least as great as the aperture of the transmitter, the factor liar will still apply. So the conclusion is that the value ofp,/p for a random distribution of cylinders of length l is given by equation (A4) multiplied by a factor
~-~xJ \-~r]
= 1~2~r1 "
(A7)
This gives the total for all the cylinders as
p,
7
[aL\ 1t2 = k(ka)
(A8)
where L = nl is the total length of all the cylinders. Provided the minimum radius of curvature is always sufficiently large and proximity effects between various parts of the filament can be neglected, the same result will apply to any filament of length L, irrespective of its shape. The condition to be met is that the filament must still appear practically straight over lengths at least as great as the aperture of the transmitter, so that coherent scattering occurs. The result should remain roughly true even when the filament is formed into a net cage provided the assumption regarding random orientation of the pieces of filament can be justified. I f d is the length of one side of one mesh of a net, the total length of filament in an area S of the net is 2S L = --. d
(A9)
A very rough estimate of the effect of knots, in a net with this form of construction, can be obtained by treating each knot as if it were a sphere with the same volume. Let al denote the radius of such a sphere and assume that kal >> 1. The ratio p,[p for the sphere is p - 2
[K,],
(A10)
v.G. W E L S B Y A N D G. C. G O D D A R D
148
where K4 is a numerical function of the ratio p~ cl/P2 cz (see reference [3]). So, from equation (A8), the ratio of the scattered pressure due to a knot to that due to a length d of filament is a 1/2 at [K4]
A.2. Sheets Consider the scattering effect of a thin, planar sheet with a normally incident planar wave field. In this case the conditions for the use of the impedance method are satisfied and this is in fact the best method of analysis for the purpose. An electrical transmission-line analogue for the two kinds of mismatch is shown in Figure 6. Under matched conditions the voltage d
] to)
L
(b)
Figure 6. Electrical transmission-lineanalogue of acoustic field perturbation by a thin sheet. (a) Effect of compressibility mismatch; (b) effect of density mismatch.
is equal to el2. The small perturbations of v caused by the presence of the shunt capacitance (representing compressibility mismatch) or the series inductance (representing density mismatch) are, respectively, 6v
-joJC
v
Z/Zo+ j ~ C
(A12)
)
and ~Sv v
jogL ZZo + ja)L
(A 13)
When this is translated into acoustics the corresponding expressions are found to be - j k t ( l l A 2 - IIAI) 2lAx + j k t ( l / A 2 - l / A , )
(A14)
and jkt(p2 - p~)
(A15)
2pl + jkt(p2 - Pl)
where t is the thickness of the sheet. For small perturbations it is justifiable to calculate the two effects separately and then to add the results together. So, where p, denotes the backscattered pressure at the surface of the sheet,
[p_,[. f lpl = kt
(A~-- A,)/A,
6o,-- p_,)/p~
]
~2 --jk---~'~2----~l)lA2] + 2 +jkt[(p2 - p,)lp~]J"
(A16)
TARGET STRENGTH OF NETS AND SHEETS
149
For the values ofkt, A and p which are of practical interest here this reduces to
p,]
kt
p ] =-~-Ka,
(AI7)
-p, I xa= (A2--At + p 20, )"
(At8)
where
For a finite incident beam, which will be assumed to have a square cross-section of angular width ct, the energy from an area 6S of the sheet is spread over an area ~Zr2 at range r. So the received pressure is proportional to (6S)t/2[~r. For random orientation of a large number of such sheets the same kind of argument can be applied as for the cylindrical elements. If /3 is the angle of tilt o f f S, in any plane, relative to the direction of the normal to the incident beam axis, there will again be a weighting factor to represent the probability that fl lies within one of two possible ranges of values each extending over ct]2. This is the probability that, of all possible positions over a hemisphere, the normal to 6S will lie within a square beam of angle ~/2. It is equal to (~r/2) 2
cc2
2z~r 2
8r~
(AI9)
So the factor to be applied to equation (A17) 0~
(aS) 1/2
(8,,)','
((~S~ 1'2 1
7'
(A2o)
and the total result for an area S is p,
1 [ 1 ~,/2 (S)ttz. et r -K3.
pl=~'~'n)
(A21)
This will remain true for any area S, even when the sheet is crumpled up, provided the radius of curvature remains large enough for pieces of sheet, comparable in size with the aperture of the transmitter, to be flat enough to produce coherent scattered waves. The effect of crinkling the sheet into a much smaller radius of curvature than this will be to replace the probability weighting factor ct2/8r~ by a number approaching unity. But, at the same time, there will be a reduction in the received power from each element 3S because it will be scattered over a hemisphere instead of the "square" solid angle ct. So the factor S[(~r) 2 will be replaced by S[(2nr2). The nett result will be to double the value ofp,/p given by equation (A21): i.e., crinkling the sheet into a small radius of curvature would be expected to increase the target strength of a given area of sheet by up to 6 dB, compared with the figure for a lightly crumpled sheet. It is assumed, incidentally, that even for the crinkled sheet the radius of curvature R is still always large enough to satisfy the condition kR ~ I.
U N D E R W A T E R ACOUSTIC TARGET STRENGTH OF NETS AND THIN PLASTIC SHEETS V. G. WELSBY AND (.J'. C. GODDARD
Department of ElectroMc and Electrical Engineering, Unirersity of Birmingham, Birmhzgham BI 5 2TT, England (Receired 13 January 1973) The purpose was to find the most suitable material to form a cage, itself having a very small acoustic echo strength, for use in experiments on the acoustic target strength of fish. A series of measurements have been carried out in the Acoustic Tank Laboratory at the University of Birmingham on samples of various materials and a theoretical analysis has been made which gives a useful insight into the way the echo strength depends on the dimensions and mechanical properties of nets and sheets. Sufficient agreement was obtained between theory and practice to show that the results will be valuable for future design work. Plastic sheets were shown to have no advantage over nets. The study has shown that significant effects are caused by surprisingly small amounts of gas adhering to the solid materials. It is this which is likely to set a practical lower limit to the target strengths which can be attained.
1. INTRODUCTION The present study arose in connection with research into the target strength of fish which required a cage strong enough to restrain a fish but having an echo level small compared with the signal from the wanted target. A series of measurements have been carried out in the Tank Laboratory at the University of Birmingham on samples of materials which might be used for the construction of cages for this purpose. A theoretical analysis has also been made which gives a useful insight into the way the target strength depends on the dimensions and mechanical properties of nets and sheets.
2. DESCRIPTION OF TESTS These fall under four headings: (a) (b) (c) (d)
straight filaments, filaments, net, /crumpled up. plastic sheets,)
Test (a) was intended mainly to establish the measuring technique and to confirm that there was some reasonable agreement between results and theory for this relatively simple case. Tests (b), (c) and (d) were then made with the idea of determining the mean target strength of a given quantity of material, when crumpled up so that it could be assumed to be randomly orientated with respect to the incident acoustic waves. The results could then be used to estimate the target strength of any proposed cage design using the material concerned. In each case two different sonars were available, working at 507 kHz and 986 kHz 139
140
v.G. W E L S B Y A N D G. C. G O D D A R D
and with 3-dB vertical beamwidths of 30 degrees and 5 degrees, respectively. The targets were placed at a range of 1.17 m for the 507 kHz tests and 2.54 m for the 986 kHz tests. Brass or steel solid spheres, suspended by fine nylon monofilament, were used as reference targets. The target strengths were calculated theoretically [1 ]. Adequately long pulse durations were used and the shapes of the echo pulses were checked to make sure that the readings did in fact represent "steady-state" conditions. A storage oscilloscope was used, the image of each sample pulse being retained for several seconds to enable an accurate estimate o f its amplitude to be made. In practice the cage is suspended in such a way that there is bound to be a certain amount of movement relative to the transmittel. The test objects were therefore allowed to hang freely in the water and a slight motion was imparted in order to ensure that the echo components from various parts of each target were randomized in phase. Each recorded value is the mean of 200 individual readings obtained in this way. In Figure 1 a histogram has been drawn for a typical set of readings. The fact that this forms an acceptable approximation to a Rayleigh distribution confirms that phase randomization was occurring. I
I
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I
I
3
4
5
6
025 o2
Ol
~
iMe--~176
!
005
i
0
2
I
7
X
Figure 1. Distribution of a typical set of 200 measured values of pressure amplitude of target echo; (in this case for a sample of polythene net). x is an arbitrary variable proportional to pressure amplitude. An analysis of the relevant scattering theory is given in full in the Appendix and only the conclusions reached there will be summarized in this section so that they can be compared with the experimental results. (a)
Straigllt filaments
The usual definition of target strength is based on the fact that the value of the ratio of the received pressure p,, in the back-scatter direction, at a distance r, to the incident pressure p for a "large" sphere is p,/p = 89 where a is the radius of the sphere. For a sphere of 2 m radiusp,/p = l/r. So the target strength of any other object is obtained from its value of p,/p by means of the formula
TS=2Olog,or(~),
(1)
with r measured in metres. For a discrete object the "far-field" scattered pressure ratio p,]p is proportional to l[r so that the target definition given above gives a figure that is independent of r. In the case of the "infinitely-long" cylinder, however, it is proportional to ( I / r ) ~/2 so
141
TARGET STRENGTH OF NETS AND SHEETS
the target strength, as defined above, is not a convenient parameter to use. It is better to discuss the results directly in terms o f the ratio p,[p. F o r a straight filament at range r, with ka < 1, this is given theoretically [1--4] by
p,
1 ~
[a\ '12
= - 1 F /r=o ( k a ) 2(kr) -112 lq,
(3)
2,N/2
where
p,>1j"
+
(4)
A and p represent bulk modulus and density, respectively, subscript 1 refers to the water and subscript 2 refers to the filament. When k a >> 1 the ratio tends to a value P.Z"= (ka)U2(kr)-llz K~, (4a) P where K~ is a different factor, the value o f which depends on ka over the transition range where ka is roughly unity, and eventually settles down to the impedance mismatch coefficient (pzc2--plcl)/(p2c2 +plcl) when the wavelength becomes very small c o m p a r e d with the radius o f the cylinder. i
i
i
y/
"~
-IC
-201
-3(
o
/
i
f"
X~ /
x
-40
OI--
i
x
I
I
I
I
I
02
05
IO
2
5
io
ko
Figure 2. Ratio of received pressure p, to incident pressure p for straight nylon filaments, at range r. Circles indicate measured values at 986 kHz, crosses indicate measured values at 507 kHz. Curve (a) gives predicted values, for ka < I, assuming material of filament to have infinite density and bulk modulus. The squares indicate some theoretical values (for similar materials), due to Faran [5]. Curve (b) is the corresponding result for ka ~- 1. Curve (c) is the predicted result when the actual values of density and bulk modulus for nylon are used. Vertical scale is 201og(p,/p) for kr = l. In Figure 2 the theoretical curves are plotted in a normalized form, with the zero o f the decibel scale taken f o r k a = 1 and kr = 1. Three additional theoretical points, due to F a r a n [5], have also been included. The samples used for the measurements were mono-filament fishing-line o f various diameters. Each sample was hung vertically, with a weight at the lower end chosen to ensure that the filament was as straight as possible without being over-stressed. A 6.3 m m diameter
142
v . G . x,VELSBY AND G. C. GODDARD
brass ball was used as a reference target which could be placed at the same range as the specimen to enable the system to be calibrated. The brass sphere was acoustically "large" at the frequencies concerned so its value ofp,/p was given by p,/p = 89 where al is the radius of the bail. The echo from the sphere appears on the normalized scale ofpr/p at a level
(kr)llz dB.
20 loglo
(5)
The target echoes were recorded in dB relative to that from the sphere and this enabled them to be transferred to the normalized graph so that they could be compared with the predicted curves. The results are shown in Figure 2. The specific bulk modulus and density were taken as 3.1 and 1-I respectively for nylon, giving Kz = 0.78. The mean level of the measured values is some 10 dB or so below the predicted figure. An encouraging feature of the results is that, although measured at different ranges and frequencies, with different sonars, they do agree reasonably well with the general form of the predicted curve. This tends to confirm that the measurements are probably reliable and that the discrepancy is due to some factor that has not been taken into account in the theory. A possible explanation for at least part of the difference is that the theory has taken no account of the fall in the incident field intensity towards the edges of the nominal beam, or of the directional pattern of the receiver. The tests were regarded as satisfactory and work proceeded with the rest of the programme of measurements. (b) Crumpled filaments For these tests, 20-m lengths of various nylon filaments were tangled into roughly spherical shapes each about 120 mm diameter. It is shown in the Appendix that the expected target strength TS, for ka ~ 1, is
TS = 20 loglo 88
ltz K2,
(6)
where L is the length of the filament and Kz is defined by equation (4). Measurements of TS were made at 507 kHz, a 19-mm steel ball being used as the reference target. The results are plotted in Figure 3, with the curve computed from equation (6) included for comparison. K2 has been assumed to be 0-78 for nylon. I -3C
/ -4C
/
/
/% -50
/'
/
/ -6C OI
I o 02
I O4
I 06
I I0
ko Figure 3. Target strengths of 20-m lengths of crumpled nylon filament at 507 kHz. a is radius of filament. Circles indicate measured values, dashed line gives theoretical prediction.
TARGET STRENGTH OF NETS AND SHEETS
(c)
143
Crumpled net
Tests were made at 507 kHz with samples of knotted polythene net which had a filament radius a = 0.3 mm and a mesh size o l d = 25 mm. From the Appendix the formula for target strength of netting is obtained as
TS= 201og,o88
.
(7)
Substituting assumed values of ARIA2 = 0.62 and P2/Pl = 0.94 for polythene in equation (4) gives K2 = 0.35. Hence the predicted target strength can be calculated as - 4 3 dB for S = 1 m 2. Effect of knots. This is discussed in the Appendix and it is concluded that the ratio of the scattered pressure due to knots, compared with that due to the filament of the net, is given approximately by
where a~ is the radius of each knot (assumed spherical) and /(2 and K4 are factors which depend on the compressibility and density of the material of the net. The values of these I
I
I
I
-40
o
/
~45
/
0
/
0
-50
./
0
/
o -55 001
I 002
I 005
I OIO
I 020
I 050
Figure 4. Target strength of crumpled samples of 25-mm mesh knotted polythene net, as a function of area S, at 507 kHz. Circles indicate measured values. The dashed line gives predicted curve (neglecting the effect of knots).
factors would be very difficult to predict with any certainty but their ratio would not be expected to depart very drastically from unity for the materials likely to be used. The ratio (alia) could be assumed to be, say, 2 or 3. All that can really be said is that the effect of the knots cannot be ignored, so that at least part of the discrepancy between the measured target strength of the net samples and the predicted value for the filament alone (see Figure 4) may be due to this cause.
Crtunpled plastic sheets The Appendix gives the TS
(d)
in this case as 1 [ 1 /'/2
Ts= 2Olog,o - } kt(s)- K3,
(9)
144
v . G . WELSBY AND G. C. GODDARD
where t is the thickness of the sheet and
(A2--A, + p2--p, I
1':,=1 Z
(10)
. p, /.
Two sets of tests were made. In the first, samples of crumpled Melinex sheet, 0.5 mm thick and of various areas, were measured at 507 kHz. N o figures were available for the bulk modulus and density of this material so they were assumed to be the same as for polythene. This gave K3 = 0.35. The results are plotted in Figure 5 and compared with the theoretical curve, which gave - 4 9 dB for S = 1 m 2 for a lightly crumpled sheet or --43 dB for S = 1 m 2 if the sheet is crinkled, with a very small radius of curvature. Here again, there is reasonable agreement with the square-root law of variation of T S with area of sheet but the measured values are generally considerably higher than the predicted ones. I
I
I
i
I
-40
o s
-as
/ (o )
/
/
-5C
I
002
I
005
I OtO S (m 2)
.// /
/ 001
/
I 020
/
'1 b) 050
IO
Figure 5. Target strength of crumpled samples of Melinex sheet, 0.05 mm, as a function of area S, at 507 kHz. Circles indicate measured values. Curves (a) and (b) are theoretical. They represent limiting cases where sheet is crinkled into very small radius of curvature [curve (a)] or only lightly crumpled up [curve (b)]. In the second series of tests the TS of samples of crumpled polythene sheet o f various thicknesses but the same area S = 0.40 m 2 were measured at 507 kHz. The value of K3 was taken as 0.35 and the theoretical value of T S was calculated. Table 1 shows all the results for sheets, with S = 0.40 m 2. The general conclusion to be drawn from these results is that, while they agree fairly well with theory for the thickest samples, they certainly do not do so for the thinnest one. It looks in fact as though there is some other factor, which has not been taken into account by the theory, becoming predominant for the very thin sheets. The most likely cause is a thin layer of gas, either continuous or in the form of very small bubbles, adhering to the surface of the sheet. There is no direct evidence for the truth of this assumption but it would explain the tendency for the target strength to become independent of the sheet thickness when the latter is reduced below a certain limiting value. (Incidentally, efforts were made by carefully rinsing the samples in a weak solution of detergent to ensure that they were properly wetted and that the amount of free gas was reduced to a minimum.) It is interesting to estimate what quantity of gas would be required to produce this effect. This can be done quite easily by noting that it is the compressibility mismatch of the sheet which matters, so the effect of a gas layer would simply be to increase the compressibility and to cause a proportional rise
TARGET STRENGTH OF NETS AND StIEETS
145
in target strength. Since the bulk modulus o f a gas would be about 10 -4 times that o f water while the bulk modulus of the sheet material is o f the same order as that o f water it follows that, to produce the observed effect, the gas film would only need to be about 5 • 10 -s millimetre thick. Or the same a m o u n t of gas would represent the volume o f a layer o f 0.2-mm diameter bubbles, scattered over the surface o f t h e sheet with a mean spacing o f a b o u t 10 ram. TABLE 1
Target strengths of samples of Melinex and polythene sheet, each 0.40 m 2 area, measured at 507 kHz Sheet thickness (ram) 0'050 0"025 0.013
Measured target Theoretical value strength (dB) (assumed the same , ^ . for both materials)i" Melinex Polythene (a) (b) -42 ---
-48 -47 -45
-47 -53 -59
-53 -59 -65
1"For the theoretical value (a) it is assumed that the sheet is crinkled into a very small radius of curvature while for value (b) it is assumed that it is only lightly crumpled. 3. CONCLUSIONS The most striking practical conclusion is that, irrespective o f theoretical predictions which take only the material o f a net or sheet itself into consideration, the real limit is almost certainly set by the presence o f gas bubbles on the surface o f the material. F o r this reason it was decided that, as far as the application described in the Introduction was concerned, plastic sheets seemed to offer little real advantage over nets. A surprising feature o f the theoretical analysis has been the smallness o f t h e a m o u n t s o f gas needed to produce significant changes in target strength. The experimental results have indicated that the limit m a y be at about - 4 0 dB for an area o f 1 m 2 o f net or sheet at 507 kHz.
REFERENCES 1. V. G. WELSBYand J. E. HUDSON 1972 Journal of Sound attd Vibration 20, 399--406. Standard small targets for calibrating underwater sonars. 2. I. MALECKX1969 Physical Foundations of Technical Acoustics. London: Pergamon Press. 3. LORD RAYLEIGH1877 Theory of Sound (two volumes). New York: Dover Publications (second edition, 1945 re-issue). 4. T. F. HtJE'rER and R. H. BOLT 1955 Sonies. New York: John Wiley & Sons. 5. J. J. FARAN 1951 Journal of the Acoustical Society of America 23, 405--418. Sound scattering by solid cylinders and spheres.
APPENDIX SCATTERINGTHEORY
A.l. Filaments and nets The theory o f scattering from a single small sphere provides a means of introducing ideas which apply generally to other shapes as well. Welsby and H u d s o n [1] give the derivation o f an expression for the scattered pressure p, at a distance r from the centre o f a sphere o f
146
v . G . WELSBY AND G. e . GODDARD
radius a, in a direction making an angle 0 with that of an incident planar wave field o f pressure p, It is assumed that the sphere is "small": i.e., ka < I. The results agree with those given by Malecki [2] :
p = \ 31 \r/K"
(A1)
where Kx
= [ A 2 - A, [- ~
3 ( p 2 - p,) 0]. 2 p 2 + p l cos
(A2)
J
A is bulk modulus, p is density, the suffix 1 refers to the water and the suffix 2 refers to the material of the sphere. For small objects it is not the mismatch between the impedances Pl cx and P2 cz which determines the scattering effect. There are two separate contributions to the scattering process; one is a "monopole" term produced by the mismatch between the compressibilities I[AI and I/A2 while the other is a "dipole" term produced by the mismatch between the densities Pl and P2. A small sphere made of a material for which pl cx = P2 c2 but Ax ~ A2 would certainly scatter sound, in spite of the apparent match between the impedances. The impedance concept is only valid when the incident and scattered fields have the same geometrical form; or at least can be assumed to have the same form over limited regions. This condition cannot be satisfied by "small" objects in a field with planar or nearplanar phase-fronts. Analysis for the case of an infinitely long cylindrical obstacle, of "small" radius a, lying perpendicular to planar incident waves is mathematically a little more difficult than that for the sphere but yields a result of the same general form: i.e., with a monopole contribution depending on the compressibility mismatch and a dipole term depending on the density mismatch. A solution is given by Malecki [2] but there is reason to think that this contains a numerical error in the dipole term. Reference to Rayleigh [3] suggests that the correct form should be
p
L "42
2
t92 + Pl
J
This is consistent too with the version quoted by Hueter and Bolt [4] for the special case when A2 >> A~ and P2 ~ 1. For the back-scatter direction cos 0 = - 1 and equation (A3) becomes
[aV/2 P--L"= ~l g ~ (ka)3'21r 1(2,
(A4)
where A2--AI
X'=L Z
2(p2-Pl)]
fAS)
Suppose now that the planar waves are replaced by an incident beam with a finite vertical angular beamwidth ~ and that the length of the cylinder, l, with its axis vertical, is reduced from infinity until it is equal to the insonified length 1 = ~r. The amplitude of the scattered pressure at the surface of the cylinder is still given by equation (A4) with r = a but its phase distribution is only partly coherent, with a coherence distance equal to the aperture size of the source of the incident beam. The resulting scattered power, instead of being radiated cylindrically, spreads spherically into a zone subtending an angle ~. This means that the
TARGET STRENGTH OF NETS AND SHEETS
147
pressure ratio p,[p, due to any length 2[ct < 1 < ctr of the cylinder, must be multiplied by a factor
2nrl ! 2rcr2 o: o~r"
(A6)
The numerator ofthis ratio is the area of a cylindrical zone ofaxial length I, and the denominator is the area of a spherical zone subtending an angle cc When lis made equal to the insonified length ctr the ratio becomes unity; in other words equation (A4) can still be used for finite incident beams, the predicted value ofp,/p being independent of the width of the incident beam. Suppose now that the cylinder is tilted relative to the vertical by a small angle ft. The scattered power will remain practically unchanged but the position of the zone into which it is scattered will shift through an angle 2ft. I f the beam is treated as ideal, with no sidelobes, it follows that as soon as Ifll exceeds ~/4 the scattering zone will miss the back-scatter direction altogether. So if there are a large number of short cylinders, at random orientations, the scattered power has to be multiplied by a weighting factor giving the probability that a cylinder will happen to assume a tilt angle for which the back-scattered power is not zero. This is the probability that e i t h e r - ~ ] 4 < f l < c q 4 or ( n - c q 4 ) < f l < ( r c + ~ [ 4 ) : i.e., 2(c~/2)[2n = ct[2n. If the length of each elemental cylinder is at least as great as the aperture of the transmitter, the factor liar will still apply. So the conclusion is that the value ofp,/p for a random distribution of cylinders of length l is given by equation (A4) multiplied by a factor
~-~xJ \-~r]
= 1~2~r1 "
(A7)
This gives the total for all the cylinders as
p,
7
[aL\ 1t2 = k(ka)
(A8)
where L = nl is the total length of all the cylinders. Provided the minimum radius of curvature is always sufficiently large and proximity effects between various parts of the filament can be neglected, the same result will apply to any filament of length L, irrespective of its shape. The condition to be met is that the filament must still appear practically straight over lengths at least as great as the aperture of the transmitter, so that coherent scattering occurs. The result should remain roughly true even when the filament is formed into a net cage provided the assumption regarding random orientation of the pieces of filament can be justified. I f d is the length of one side of one mesh of a net, the total length of filament in an area S of the net is 2S L = --. d
(A9)
A very rough estimate of the effect of knots, in a net with this form of construction, can be obtained by treating each knot as if it were a sphere with the same volume. Let al denote the radius of such a sphere and assume that kal >> 1. The ratio p,[p for the sphere is p - 2
[K,],
(A10)
v.G. W E L S B Y A N D G. C. G O D D A R D
148
where K4 is a numerical function of the ratio p~ cl/P2 cz (see reference [3]). So, from equation (A8), the ratio of the scattered pressure due to a knot to that due to a length d of filament is a 1/2 at [K4]
A.2. Sheets Consider the scattering effect of a thin, planar sheet with a normally incident planar wave field. In this case the conditions for the use of the impedance method are satisfied and this is in fact the best method of analysis for the purpose. An electrical transmission-line analogue for the two kinds of mismatch is shown in Figure 6. Under matched conditions the voltage d
] to)
L
(b)
Figure 6. Electrical transmission-lineanalogue of acoustic field perturbation by a thin sheet. (a) Effect of compressibility mismatch; (b) effect of density mismatch.
is equal to el2. The small perturbations of v caused by the presence of the shunt capacitance (representing compressibility mismatch) or the series inductance (representing density mismatch) are, respectively, 6v
-joJC
v
Z/Zo+ j ~ C
(A12)
)
and ~Sv v
jogL ZZo + ja)L
(A 13)
When this is translated into acoustics the corresponding expressions are found to be - j k t ( l l A 2 - IIAI) 2lAx + j k t ( l / A 2 - l / A , )
(A14)
and jkt(p2 - p~)
(A15)
2pl + jkt(p2 - Pl)
where t is the thickness of the sheet. For small perturbations it is justifiable to calculate the two effects separately and then to add the results together. So, where p, denotes the backscattered pressure at the surface of the sheet,
[p_,[. f lpl = kt
(A~-- A,)/A,
6o,-- p_,)/p~
]
~2 --jk---~'~2----~l)lA2] + 2 +jkt[(p2 - p,)lp~]J"
(A16)
TARGET STRENGTH OF NETS AND SHEETS
149
For the values ofkt, A and p which are of practical interest here this reduces to
p,]
kt
p ] =-~-Ka,
(AI7)
-p, I xa= (A2--At + p 20, )"
(At8)
where
For a finite incident beam, which will be assumed to have a square cross-section of angular width ct, the energy from an area 6S of the sheet is spread over an area ~Zr2 at range r. So the received pressure is proportional to (6S)t/2[~r. For random orientation of a large number of such sheets the same kind of argument can be applied as for the cylindrical elements. If /3 is the angle of tilt o f f S, in any plane, relative to the direction of the normal to the incident beam axis, there will again be a weighting factor to represent the probability that fl lies within one of two possible ranges of values each extending over ct]2. This is the probability that, of all possible positions over a hemisphere, the normal to 6S will lie within a square beam of angle ~/2. It is equal to (~r/2) 2
cc2
2z~r 2
8r~
(AI9)
So the factor to be applied to equation (A17) 0~
(aS) 1/2
(8,,)','
((~S~ 1'2 1
7'
(A2o)
and the total result for an area S is p,
1 [ 1 ~,/2 (S)ttz. et r -K3.
pl=~'~'n)
(A21)
This will remain true for any area S, even when the sheet is crumpled up, provided the radius of curvature remains large enough for pieces of sheet, comparable in size with the aperture of the transmitter, to be flat enough to produce coherent scattered waves. The effect of crinkling the sheet into a much smaller radius of curvature than this will be to replace the probability weighting factor ct2/8r~ by a number approaching unity. But, at the same time, there will be a reduction in the received power from each element 3S because it will be scattered over a hemisphere instead of the "square" solid angle ct. So the factor S[(~r) 2 will be replaced by S[(2nr2). The nett result will be to double the value ofp,/p given by equation (A21): i.e., crinkling the sheet into a small radius of curvature would be expected to increase the target strength of a given area of sheet by up to 6 dB, compared with the figure for a lightly crumpled sheet. It is assumed, incidentally, that even for the crinkled sheet the radius of curvature R is still always large enough to satisfy the condition kR ~ I.