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