The Theory ofScottish Fisheries Research Solid Spheres ... · transverse and longitudinal sound...
Transcript of The Theory ofScottish Fisheries Research Solid Spheres ... · transverse and longitudinal sound...
Department of Agricultureand Fisheries for Scotland
The Theory ofSolid Spheresas SonarCalibratlcmTargets
D N MacLennan
Scottish Fisheries ResearchReport Number 22 1981
ISSN 0308-8022
The Theory of Solid Spheres as Sonar Calibration Targets
D N MacLennan Marine Laboratory Aberdeen
introduction The measurement of acoustic echoes from fish is an important method for estimating fish stocks. The precision of such estimates depends inter alia
upon the accuracy to which the acoustic equipment has been calibrated. Two
techniques are cornmanly used to perform acoustic calibrations. The recipro-
city technique (Urick, 1967) requires certain assumptions about the response
of the transducer when transmitting and receiving, and it is difficult to
achieve a calibration accuracy better than 1dB by this method. The second
technique is based on the measurement of echoes from a standard target,
normally a sphere, and the response of the sonar is deduced from a know-
ledge of the acoustic properties of the target. Clearly the accuracy in this
case depends upon how well these properties are known. However, the usual
experimental approach is to measure the target properties using another
transducer which has itself been calibrated by reciprocity. Therefore, to
improve the calibration accuracy, some other means is required for estimat-
ing the acoustic properties of standard targets.
Welsby and Hudson (1972) suggested that ping-pong balls could be used for
calibrating sonars. The stability of such targets is in doubt, however, and it is
now considered that solid spheres offer a better prospect as standard targets
with stable properties. This paper shows how scattering theory may be
applied to estimate the target strength of solid spheres. The theory of
acoustic scattering by a rigid sphere has been described by Rudgers (19691.
The scattering properties of a real sphere which has finite density and elasti-
city are quite unlike the rigid ideal, especially above a critical frequency
when the mechanical resonances of the sphere become important. Both
transverse and longitudinal sound waves propagate within the sphere.
Two waves propagate in the surrounding medium, namely the incident wave
and the scattered wave caused by the presence of the sphere, The interaction
between the various waves results in a complicated scattered sound field that
is highly frequency dependent. However, the theory of scattering by an
elastic sphere (Hickling, 1962) is now well established. Neubauer, Vogt and
Dragonette (1974) have compared the theory with measurements on
aluminium and tungsten carbide spheres, and the agreement is reasonable.
Theory
Steady State Theory
Consider a plane sine wave of constant amplitude travelling in the zdirection.
Using spherical polar coordinates (Fig. 1) and the usual complex notation, the sound pressure is:
p, (r, fJ,t) = p,, exp (L {at - kr cos O} 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...** (11
k is the wave number, ti is the angular frequency and t is the time.
The constant pO is the amplitude. If the wave is incident upon an elastic
sphere, the interaction results in a scattered pressure field:
ps b, 8, tl = Pso b-, 01 exp Ii 4 . . . . ..**.................................................. (21
1
where JQ+ w (XI and YQ+ H (x) are Bessel functions of the first and second
kinds respectively. Hickling’s equations for ‘r?* contain typographical errors,
so the equations are given in full below in a form convenient for program-
ming.
Three sound velocities have to be considered. The velocity in water is c, and
the velocities of longitudinal and transverse waves within the sphere are c,
and c2 respectively. If c is the density of water and p, is the density of the
sphere, then:
q = ka ; q 7 qclc, ; q = qclc 1 2 2 ......................................................... @aI
A2 = N2 + 2-2) je (q,2 1 + q: j”g h2 1 ..................................................... WI
4 = X 02+1) Lq, j’Q Iq, 1 - jQ h, 11 ........................................................ 6~1
a= 2 tp, /pl k2d ............................................................................... VW
B = b, /PI (Cl /cl2 - a ............................................................................ 69
B2 = .A2 q: l/3jQ (q,) -cvj”* (q,Il -A, CY LjQ (q21 -c~~j’~ fq2)l ....... .WI
Bl = q LA2 q, j’Q (q, 1 - A, jQ h211 ...................................................... 0%)
tan vQ = -IB2 YQ kd - B,& h)l /fB2 Y’~ (cd - B, yQ hll ......................... (6h)
The prime symbol denotes differentiation of the function with respect to the
argument.
An important feature of the equations is that the physical properties of the
sphere appear in ratio with corresponding properties of water, ie
p, /p, c_, /c and c2/c. Thus the scattering is determined by these dimension-
less ratios and by q which is the ratio of the sphere circumference to the
sound wavelength in water.
In the far field, kr > >1, the form function is:
f=(q) = - (2/qI QzO(-lIQ (2Q+lI sinTQ exp li7Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)
The acoustic cross section u is defined to be 47r times the far field back-
scattered intensity, normalised to r - 1 m, divided by the incident wave
intensity. For the elastic sphere:
e=ra2 I f_(q)1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*................ (81
and the target strength in dB is:
TS = 10 Log,,, (u/4~) (91 . . . . ..*.....................................................................*....
The above theory applies when the incident wave source is far enough distant
for the incident field in the vicinity of the target to be reasonably described
as a plane wave. The case of a source near the target was discussed by
Hickling (19621, but it will not be considered here. For calibration purposes,
the distance between the target and the transducer should be large compared
with both the target diameter and the wavelength, when the plane incident
wave and the far field scattering approximations would be adequate.
3
T r a n s i e n t T h e o r y P r a c t i c a l s o n a r s t r a n s m i t p u l s e s , n o t s t e a d y s t a t e w a v e s , b u t a p u l s e m a y b e
d e s c r i b e d a s a s u m o f s t e a d y s t a t e f r e q u e n c y c o m p o n e n t s . T h u s t h e s c a t t e r e d
p u l s e w a v e f o r m m a y b e d e d u c e d f r o m t h e s u m o f t h e s c a t t e r e d c o m p o n e n t s
o f t h e i n c i d e n t p u l s e s p e c t r u m . C o n s i d e r t h e f o l l o w i n g i n c i d e n t w a v e t h a t i s a
m o d u l a t e d p u l s e h a v i n g a r e c t a n g u l a r e n v e l o p e a n d a c a r r i e r f r e q u e n c y
0 ~ = q O c l a :
pi (r 1 = p0 exp (i q0 71 for IT I < ~~ /2
I pi (71 = 0 for I T I > = 7. /2
_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 0 1
w h e r e + r i s t h e t r a n s f o r m e d t i m e p a r a m e t e r :
7 = ( c t - r c o s 0 ) / a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 1 1
T h e p u l s e d u r a t i o n i s T V a / c . N o t i n g t h a t q i s p r o p o r t i o n a l t o f r e q u e n c y , t h e
p u l s e m a y b e e x p r e s s e d a s a F o u r i e r i n t e g r a l : c a
P i ( 4 = P O s
g ( q ) e x P I i q 4 d q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 2 1
w h e r e
g k l l = s i n ( { q - q O } r 0 / 2 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 r ( q - q 0 1
( 1 3 1
T h e s c a t t e r e d f i e l d i s o b t a i n e d b y i n t e g r a t i n g o v e r t h e f r e q u e n c y c o m p o n e n t s
i n t h e p u l s e s p e c t r u m ( H i c k l i n g , 1 9 6 2 ; D r a g o n e t t e , V o g t , F l a x a n d N e u b a u e r ,
1 9 7 4 ) . T h e f a r f i e l d b a c k s c a t t e r e d p r e s s u r e i s : m
f _ ( q ) g ( q ) e x p ( j q ~ ~ 1 dq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)
W h e r e t h e f o r m f u n c t i o n f o r t h e e l a s t i c s p h e r e t a r g e t i s g i v e n b y ( 7 ) a n d r s i s
a s e c o n d t r a n s f o r m e d t i m e p a r a m e t e r :
TV = (ct - r) /a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 5 )
T h e P e r c e i v e d T a r g e t S t r e n g t h S u p p o s e t h a t a s i g n a l v i i s a p p l i e d t o t h e t r a n s m i t t e r i n p u t o f a s o n a r s y s t e m
( F i g . 2 1 , a n d t h a t r e f l e c t i o n f r o m a t a r g e t r e s u l t s i n a s i g n a l v 0 a t t h e r e c e i v e r
o u t p u t . T h e e l e c t r i c a l s i g n a l s v i a n d v 0 a r e m e a s u r e d , a n d t h e i r r e l a t i o n s h i p
d e p e n d s u p o n t h e a c o u s t i c p r o p e r t i e s o f t h e t a r g e t . a 0 i s t h e c e n t r e ( t u n e d )
f r e q u e n c y o f t h e s y s t e m , a n d q 0 = G . I ~ a / c .
I f v i = V e x p ( i m O t ) , a s t e a d y s t a t e s i g n a l , t h e n a c a l i b r a t i o n f a c t o r K 0 i s
d e f i n e d b y t h e e q u a t i o n :
C J = K 0 I v 0 I 2 / h i I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 1 6 1
K 0 d e p e n d s u p o n s y s t e m c h a r a c t e r i s t i c s s u c h a s t h e t r a n s d u c e r e f f i c i e n c y a n d
t h e a m p l i f i e r g a i n s . T h e c r o s s - s e c t i o n o i s g i v e n b y ( 8 ) i n t h e c a s e o f a
s p h e r i c a l t a r g e t .
4
vi *transmitter -
7r
7 pbf
e 4
transducer 1r
receiver -
Figure 2. Elements of a sonar system.
Pi
0 target
The definition of u given earlier is meaningful only for steady state signals.
When vi is transient, the intensities vary with time and another definition is
required. A convenient parameter for the purpose is the “perceived cross- section”, which is defined in terms of the ratio between output and input
signal energies:
t2 t4
uP = K0 ( ,/ iv0 12dt)/( f jvi 12dtI . . . . . . ..*...................*..*......... (171
t1 t3
The integration limits are chosen to include the periods when the signals are
significantly above noise level. up depends upon the electrical characteristics
of the sonar system as well as the acoustic properties of the target.
The electrical parts of the system are specified by two functions. F(q) is the
combined frequency response function of the transmitter and the transducer
in transmit mode, normalised so that F(qO) = 1. Similarly, G(q) is the
combined response function of the transducer in receive mode and the
receiver circuits, and G(qO ) = 1.
If vi is a pulse of carrier frequency a0 and duration r0 a/c, it can be shown
that the perceived cross-section of the elastic sphere is:
% = 7ra2 t j--l f_, ~q~FWG~q~g~q~ 12dqI/( ylg(qj 12dqj . . . . . ..a..* (181
-ca
where f_ (q) and g(q) are given by (7) and (13). As the pulse length tends to
infinity:
% -+ Ta2 If IqOI I2 ,.........................................................*.................. (18aI
Thus up in the limiting case of steady state signals is consistent with u as
previously defined.
By analogy with (91, the perceived target strength is defined as:
TSp = 10 Log,,, (up /47rI . . . . . . . . . . . . . . . . . ..~................................................. (191
5
Resui ts Acoustic Cross-Sections
The density and sound velocity ratios for brass, stainless steel and tungsten
carbide are shown in Table I. These ratios define the scattering properties of
homogeneous spheres, in particular the acoustic cross-section for steady state
scattering which is given by (6-8). The equations may be evaluated by
standard numerical methods, and the resulting form functions are shown in
Figure 3 against the parameter q = ka, which is proportional to the
frequency. The acoustic cross-section is proportional to the square of the
form function modulus,
For each material the cross-section increases as the fourth power of q at low
frequencies, in accordance with the Rayleigh scattering law.
Above q =1 the form function oscillates slowly at first. This is a geometric
resonance effect which is exhibited also by the rigid sphere (Rudgers, 1969).
Then, above some critical frequency, much sharper oscillation of the function
is evident, caused by mechanical resonances within the sphere. The first
mechanical resonance occurs at q = 3.6, 5.7 and 6.9 for brass, stainless steel
and tungsten carbide respectively. The frequencies of the mechanical
resonances are sensitive to small changes particularly in cz (Neubauer eta/,, 1974). Such changes might occur for example through work hardening or
temperature variation. To obtain a stable target strength, therefore, the first
mechanical resonance frequency should be well above the carrier frequency
a0 of the sonar. This implies that for a given a0 spheres larger than a certain
size will be unsuitable as standard targets. Among the materials considered
here, the size limit is greatest for tungsten carbide and smallest for brass.
Conversely, for a particular sphere, there is a maximum a0 above which the
sphere is unsuitable as a standard target.
L c e
“i
1 W !I ==
Ll Cl
-Iso
R
0
"l=Pi
Figure 4. Equivalent circuits for a sonar system with a transformer coupled transmitter - (aI transmitter and transducer in transmit mode, (b) transducer in receive mode and tuned receiver.
7
System Characteristics Consider a sonar system with a transformer coupled transducer for which
simplified equivalent circuits are shown in Figure 4. The transducer is
represented by the inductance L and the capacitance C. R is the radiation
resistance and the quality factor of the transducer is Q = Lao/R. If recipro-
city is assumed, L, C and R are the same in transmit and receive modes. The
system is matched so that the input and output resistances are the same
(R, = Rz = R), and the reactive components are tuned to the operating
frequency of the sonar; thus L, C, = LzCz = LC.
The transfer functions of this sytem are:
F(q) = (1 - %QQ, Xz + xi Q + Q, 1 I
X)-’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (201
G(q)=11 -%QQ2X2+%i b 1
+ Q2 Xl-” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)
where X = (q/q0 - qO/qI, Q, = L, aO/R, and Q2 = L2a0/R2
The particular system considered here represents a 38kHz sonar that has been
used for fish target strength experiments (Forbes, Simmonds and Edwards,
1980). For that system Q = 8.1, Q, = 1.27 and Q2 = 10.77. These values
have been derived from electrical measurements and the response of the
system to short acoustic pules in water.
-30
TSP tdB
UNGSTEN CARBIDE
i- 4!
----0.5 ms PULSE LENGTH ----- STEADY STATE SIGNAL
1 I ,
0 50 60 70 80 DIAMETER (mm)
Figure 5. Perceived targat strength of tungsten carbide spheres.
Perceived Target Strengths The perceived target strength (TSp) is obtained from the form function and
equations (18-21). Figures 5-7 show how TSp varies with the sphere size,
assuming that the material properties are those given in Table I. The calcula-
tions have been performed for the sonar system described above and a pulse
length of 0.5ms. Results for an infinite pulse length (steady state signal) are
shown for comparison. In the latter case, TSp is equal to the target strength
as defined by (9) and it is independent of the system characteristics.
More detailed calculations have been performed for three spheres which have
been used or considered for use as standard targets. They are (I ) a tungsten
carbide ball bearing, (2) a stainless steel sphere supplied by Simrad as a cali-
bration target and (31 a brass sphere which has been used by the Marine
Laboratory as a “sub standard” target. The densities quoted in Table I are
measured values for these spheres; however, the sound velocities in that table
are taken from published data (Kaye and Laby, 1973). The sphere sizes are
given in Table I I together with the calculated TSp for various pulse lengths.
Table Il. The perceived target strengths of three spheres when insonified by 38kHz
pulses of duration T. The results have been evaluated for the material proper-
ties given in Table I and the particular sonar system described in the text
Diameter
(mm) 38.1 60.0 69.9
% 3.05 4.81 5.60
Tbns) TSp (dB) TSp (dB) Tsp(dBI
0.5 -42.66 -35.90 -33.15
1.0 -42.49 -35.64 -33.43
2.0 -42.42 -35.52 -33.59
3.0 -42.40 -35.48 -33.65
-42.34 -35.40 -33.76
The sensivity of TSp to small changes in various parameters has been investi-
gated. For the first two spheres, the effect of varying the carrier frequency
by 5OOHz and the material property ratios by 10% is shown in Figures 8-9.
The changes in TSp are fairly small, around O.ldB for the tungsten carbide
sphere and 0.2dB for the stainless steel sphere.
Equivalent results for the brass sphere are shown in Figure 10. Here the varia-
tion of TSp is considerable, especially the variation with c*/c, which is an
order of magnitude greater than that exhibited by the other spheres. The
reason is that 38kHz is close to one of the mechanical reasonances of the
sphere. As stated earlier, the resonance frequencies are sensitive to small
changes in c*/c. In view of this, the TSp estimates for the brass sphere will
be useful only if c*/c is accurately known. The direct measurement of cZ in
a sphere is difficult. However, when the acoustic carrier frequency is close to
a mechanical resonance, the scattered pulse shape has certain features which
depend upon cZ. Thus cZ may be estimated indirectly by comparing observed
and calculated pulse shapes.
11
F i g u r e 1 1 s h o w s t h e e n v e l o p e o f t h e r e c e i v e r o u t p u t v o l t a g e o b s e r v e d o n a n
o s c i l l o s c o p e w h e n t h e b r a s s s p h e r e i s i n s o n i f i e d b y a 2 m s t r a n s m i t t e d p u l s e .
T h e e n v e l o p e h a s t w o w e l l - d e f i n e d c h a r a c t e r i s t i c s , t h e i n f l e x t i o n o n t h e
l e a d i n g e d g e a n d t h e r i n g o n t h e t r a i l i n g e d g e . T h e c a l c u l a t e d w a v e f o r m i s
o b t a i n e d f r o m a n e q u a t i o n s i m i l a r t o ( 1 4 ) b u t i n c l u d i n g t h e F a c t o r s F ( q ) a n d
G ( q ) w i t h i n t h e , i n t e g r a l . R e c e i v e d p u l s e e n v e l o p e s h a v e b e e n s o c a l c u l a t e d
f o r a s e r i e s o f c z / c v a l u e s a n d t h e r e s u l t s a r e p r e s e n t e d i n F i g u r e 1 1 . T h e
o b s e r v e d a n d t h e c a l c u l a t e d p u l s e s h a p e s a r e s i g n i f i c a n t l y d i f f e r e n t e x c e p t f o r
c z / c = 1 . 4 1 , w h e n t h e y a r e i n c l o s e a g r e e m e n t . T h e r e s u l t s i n d i c a t e t h a t c * / c
i s w i t h i n t h e r a n g e 1 . 4 1 * 0 . 0 1 , c o n f i r m i n g t h e v a l u e t h a t w a s o b t a i n e d
o r i g i n a l l y f r o m p u b l i s h e d d a t a .
T h e r e c e i v e d p u l s e s h a p e d o e s n o t c h a n g e m u c h w i t h c , / c , a n d t h i s p a r a -
m e t e r c a n n o t b e e s t i m a t e d b y t h e s a m e t e c h n i q u e . H o w e v e r , i t s e e m s r e a s o n -
a b l e t o a l l o w t h e s a m e p r o p o r t i o n a l e r r o r i n b o t h t h e s o u n d v e l o c i t y p a r a -
m e t e r s , a n d i n a n y c a s e t h e e r r o r i n c z / c i s m u c h t h e m o r e i m p o r t a n t . I t i s
n o w p o s s i b l e w i t h t h e a i d o f F i g u r e 1 0 t o s e t l i m i t s o n t h e p e r c e i v e d t a r g e t
s t r e n g t h e s t i m a t e s f o r t h e b r a s s s p h e r e . F o r e x a m p l e , f o r 0 . 5 m s p u l s e s ,
T S p = - 3 3 . 2 * 0 . 7 d B . W i d e r l i m i t s a p p l y a t l o n g e r p u l s e l e n g t h s a n d f o r s t e a d y
s t a t e s i g n a l s , T S p = - 3 3 . 8 ? 1 . 4 d B .
T h e l i m i t s o n t h e T S p e s t i m a t e s f o r t h e t u n g s t e n c a r b i d e s p h e r e a r e l e s s t h a n
* O . l d B f o r s o u n d v e l o c i t y p a r a m e t e r s w i t h i n * 1 0 % o f t h e a s s u m e d v a l u e s .
V a r i a t i o n o f t h e s p h e r e d e n s i t y a n d t h e c a r r i e r f r e q u e n c y n e e d n o t b e c o n -
s i d e r e d s i n c e p r e c i s e m e a s u r e m e n t s o f t h e s e p a r a m e t e r s a r e e a s i l y o b t a i n e d
a n d t h e T S p e s t i m a t e s m a y b e c o r r e c t e d i f n e c e s s a r y .
A c c o r d i n g t o K a y e a n d L a b y ( 1 9 7 3 ) t h e s o u n d v e l o c i t i e s f o r m i l d , h a r d e n e d
a n d s t a i n l e s s s t e e l s l i e w i t h i n * 2 % o f t h e v a l u e s s h o w n i n T a b l e I . T h e r e i s n o
r e a s o n t o s u p p o s e t h a t g r e a t e r d i f f e r e n c e s s h o u l d o c c u r b e t w e e n m a t e r i a l s i n
t h e s t a i n l e s s s t e e l g r o u p , b u t i t i s s u g g e s t e d t h a t l i m i t s o f ? 5 % o n t h e
a s s u m e d s o u n d v e l o c i t y r a t i o s w o u l d c o v e r a n y v a r i a t i o n l i k e l y t o a r i s e i n
p r a c t i c e . T h e c o r r e s p o n d i n g l i m i t s o n t h e T S p e s t i m a t e s f o r t h e 6 0 m m s t a i n -
l e s s s t e e l s p h e r e a r e t h e n ? 0 . 2 d B d e r i v e d f r o m t h e r e s u l t s i n F i g u r e 9 .
T h e p o s s i b i l i t y o f e r r o r i n t h e s y s t e m t r a n s f e r f u n c t i o n s h a s n o t b e e n
c o n s i d e r e d s o f a r . T h e e q u i v a l e n t c i r c u i t s u s e d t o d e r i v e t h e s e l f u n c t i o n s
( F i g . 4 1 a p p r o x i m a t e t h e p e r f o r m a n c e o f r e a l t r a n s d u c e r s . F o r e x a m p l e ,
s e c o n d a r y r e s o n a n c e s o f t h e t r a n s d u c e r a r e n o t t a k e n i n t o a c c o u n t . T h e
s i g n i f i c a n c e o f t h e s y s t e m f r e q u e n c y r e s p o n s e i s i n d i c a t e d b y t h e d i f f e r e n c e
A T S b e t w e e n t h e s t e a d y s t a t e a n d t h e 0 . 5 m s p u l s e l e n g t h c u r v e s i n
F i g u r e s 8 1 0 . S i n c e 0 . 5 m s i s t h e s h o r t e s t p u l s e l e n g t h t h a t w o u l d b e u s e d
w i t h s u c h a s o n a r , c o r r e s p o n d i n g t o t h e b r o a d e s t s p e c t r u m , e r r o r i n . t h e
t r a n s f e r f u n c t i o n s w o u l d c o n t r i b u t e s o m e p r o p o r t i o n o f ATS t o t h e
u n c e r t a i n t y i n T s p . H o w e v e r , g i v e n m e a s u r e m e n t s o f t h e s y s t e m c h a r a c t e r i s -
t i c s u s i n g s t a n d a r d t e c h n i q u e s , i t i s c o n s i d e r e d t h a t t h e t r a n s f e r f u n c t i o n s c a n
b e d e f i n e d w e l l e n o u g h t o r e d u c e s u c h e r r o r s t o i n s i g n i f i c a n c e .
T h e t a r g e t s t r e n g t h o f s o l i d s p h e r e s d e p e n d s u p o n t h e c a r r i e r f r e q u e n c y , t h e
d i a m e t e r a n d t h e d e n s i t y o f t h e s p h e r e . T h e s e p a r a m e t e r s m a y b e o b t a i n e d
f r o m s i m p l e m e a s u r e m e n t s , b u t t h e t a r g e t s t r e n g t h a l s o d e p e n d s u p o n t h e
s o u n d v e l o c i t i e s i n t h e s o l i d , w h i c h a r e l e s s e a s y t o m e a s u r e . H o w e v e r , i f t h e
c a r r i e r f r e q u e n c y i s w e l l b e l o w t h e f i r s t m e c h a n i c a l r e s o n a n c e , t h e t a r g e t
s t r e n g t h i s i n s e n s i t i v e t o t h e s o u n d v e l o c i t i e s . C a l c u l a t i o n s m a y t h e n b e b a s e d
o n p u b l i s h e d d a t a f o r t h e s o u n d v e l o c i t i e s a n d t h e v a r i a t i o n c o n s e q u e n t u p o n
t h e n o r m a l r a n g e o f a l l o y c o m p o s i t i o n i s n o t s i g n i f i c a n t .
1 4
38kHz is the sonar frequency normally used on fish surveys. For calibration
purposes, reference targets are required to have a TSp at this frequency
around -42d8 or higher. This implies a minimum sphere diameter around
4Omm. The larger the sphere, the lower are the mechanical resonance fre-
quencies. There is therefore a maximum diameter which should not be
exceeded for reference targets to have satisfactory acoustic properties. For
the 60mm diameter stainless steel sphere which has been studied in some
detail, the first mechanical resonance occurs at 45.1 kHz. Satisfactory results
were obtained for this sphere, but the use of larger stainless steel spheres is
not recommended. The corresponding diameter limit for tungsten carbide
spheres is 72mm. 8rass is, not considered to be a suitable material for
reference targets, since the mechanical resonance frequencies are relatively
low.
The acoustic properties of tungsten carbide have been shown to be superior to
those of stainless steel. The advantage is marginal however, and stainless steel
is much the cheaper material.
The calculation method outlined in this paper has been used to estimate the
target strength of a 60mm stainless steel sphere to within * 0.2dB, and that
of a 38.lmm tungsten carbide sphere to within ? O.ldB, at 38kHz. This is a
higher precision than would be practical from a direct measurement using
calibrated hydrophones.
For the quantitative assessment of fish stocks, it is the signal appearing at the
sonar output which has to be measured and calibrated. The target strength as
normally defined describes only the properties of the target, but the observed
echo from a pulsed sonar also depends upon the system frequency response.
To overcome this difficulty, the system performance is described by the
“perceived” target strength, which is proportional to the observed energy in
the sonar output signal. Thus the perceived target strength is a property of
the complete system, the sonar as well as the target. The definition of TSp in
terms of the observed energy is particularly suited to the calibration of echo
integrator equipment.
Summary One method of calibrating sonars is to measure the echo from a standard
target. The calibration accuracy depends upon knowledge of the acoustic pro-
perties of the target. The theory of acoustic scattering by solid spheres is
used to calculate the target strength of brass, stainless steel and tungsten
carbide spheres. The scattering is highly frequency dependent and the
“perceived target strength” (TSp) is defined, which takes account of the
frequency response of the target, transducer and electronics. The sonar is
modelled by equivalent circuits and the variation of TSp with pulse length
and the physical properties of the target material is examined. In order for
the target to have well-defined reflecting properties, the lowest mechanical
resonance should occur well above the acoustic carrier frequency. In the case
of 38kHz sonars, it is found that brass is not a suitabie material for standard
targets, that tungsten carbide is excellent but stainless steel is also satisfac-
factory. Calculations and simple measurements will allow the perceived
target strength to be determined more precisely than is practical by direct
acoustic measurement.
15
References Dragonette, L.R., Vogt, R.H., Flax, L. and Neubauer, W.G. 1974
Acoustic reflection from elastic spheres and rigid spheres and spheroids.
I I Transient analysis. Journal of the Acoustical Society of America,
55(6), 113Ckll37.
Forbes, S.T., Simmonds, E.J. and Edwards, J.I. 1980.
Progess in target strength measurements on live gadoids. Marine
Laboratory Working Paper No. 80/15,40 pp. (mimeo).
Hickling, R. 1962.
Analysis of echoes from a solid elastic sphere in water. Journal of the
Acoustical Society of America, 34 (101, 1582-1592.
Kaye, G.W.C. and Laby, T.H. 1973.
Tables of Physical and Chemical Constants (14th ed.1. London,
Longman, 386 pp.
Neubauer, W.G., Vogt, R.H. and Dragonette, L.R. 1974.
Acoustic reflection from elastic spheres. Journal of the Acoustica/
Society of America, 55 (6), 1123- 1129.
Rudgers, A.J. 1969.
Acoustic pulses scattered by a rigid sphere immersed in a fluid.
Journal of the Acoustical Society of America, 45 (4),900-910.
Urick, R.J. 1967.
Principles of Underwater Sound for Engineers. New York, McGraw-Hill,
342 pp.
Welsby, V.G. and Hudson, J.E. 1972.
Standard small targets for calibrating underwater sonars.
Journal of Sound and Vibration, 20, 399-406.
16
Appendix 1 a
L i s t o f P r i n c i p a l S y m b o l s c
s p h e r e r a d i u s
s o u n d v e l o c i t y i n w a t e r
l o n g i t u d i n a l s o u n d w a v e v e l o c i t y
t r a n s v e r s e s o u n d w a v e v e l o c i t y
b a c k s c a t t e r i n g f o r m f u n c t i o n
t r a n s m i t m o d e t r a n s f e r f u n c t i o n
p u l s e s p e c t r u m
r e c e i v e m o d e t r a n s f e r f u n c t i o n
s p h e r i c a l H a n k e l f u n c t i o n o f t h e s e c o n d k i n d
w a v e n u m b e r i n w a t e r ( 2 7 r / w a v e l e n g t h )
p a r t i a l w a v e m o d e n u m b e r
i n c i d e n t p r e s s u r e f i e l d
i n c i d e n t p r e s s u r e a m p l i t u d e
s c a t t e r e d p r e s s u r e f i e l d
s c a t t e r e d p r e s s u r e a m p l i t u d e
f a r f i e l d b a c k s c a t t e r e d s o u n d p r e s s u r e
L e g e n d r e p o l y n o m i a l
d i m e n s i o n l e s s f r e q u e n c y p a r a m e t e r ( k a )
v a l u e o f q a t t h e c a r r i e r f r e q u e n c y
e q u i v a l e n t c i r c u i t q u a l i t y f a c t o r s
r a d i a l p o s i t i o n c o o r d i n a t e
t i m e
t a r g e t s t r e n g t h
p e r c e i v e d t a r g e t s t r e n g t h
t r a n s m i t t e r i n p u t s i g n a l
r e c e i v e r o u t p u t s i g n a l
a n g u l a r p o s i t i o n c o o r d i n a t e
p a r t i a l w a v e p h a s e a n g l e
w a t e r d e n s i t y
s p h e r e d e n s i t y
b a c k s c a t t e r i n g c r o s s - s e c t i o n
p e r c e i v e d b a c k s c a t t e r i n g c r o s s - s e c t i o n
i n c i d e n t w a v e t r a n s f o r m e d t i m e
s c a t t e r e d w a v e t r a n s f o r m e d t i m e
f r e q u e n c y
c a r r i e r f r e q u e n c y
RE 83694 750 9f81 TCL 1 7