RESEARCH REPORT
REPORT 1082
28 NOVEMBER 1961

DIRECTIVITY FUNCTION OF A GENERAL
RECEIVING ARRAY FOR SPHERICAL
AND PLANE SOUND WAVES

K. K. Warner

U.S. NAVY ELECTRONICS LABORATORY, SAN DIEGO, CALIFORNIA
A BUREAU OF SHIPS LABORATORY

7801 34OdeYy/ TAN

THE PROBLEM

Evolve sonar array designs to optimize beam formation with
various types of conventional or novel arrays, with plane
and curved wavefronts; and derive directivity functions for
general arrays which may be very large in terms of wave-
lengths, and may have an arbitrary distribution of elements.
The elements may be arranged discretely or continuously;
the arrays may be one-, two-, ar three-dimensional.

RESULTS

Expressions are presented which allow the determination of
directivity patterns for any kind of array, conventional or
novel.

RECOMMENDATIONS

1. Employ the general directivity function to find array
directivity patterns not now available.

2. Program the general directivity function for an electronic
computer.

ADMINISTRATIVE INFORMATION

Work was performed by the author, a member of the Special
Research Division, under AS 02101, S-F001 03 04, Task
8051, NEL L3-2, as a portion of the sonar techniques pro-
gram. The report covers work performed mostly between
June and September 1960, and was approved for publication
28 November 1961.

The author wishes to thank Dr. C. J. Krieger for his val-

uable suggestions, many of which have been incorporated
into the report.

0 (HU MN

0301 0040553

NOMENCLATURE

D
Gear Yo> a)
(2c, Ys z,)

Glo (Ba 9

(cos a, COS 8B, cos 7)

D|

ay) ay). a)

BE, or E Cony)

B

7m

ar

I

distance from the sound source to the
origin of the array

coordinates of the sound source with
respect to the origin of the array

coordinates of the pu element with
respect to the origin of the array

the direction angles of the line joining
the origin with the source measured
frLOmIthelyc, psa ss eeSpecimvelly,

a unit vector in the direction of the
vector outward from (0, 0, 0) to

(xo, Yo» Zo) Where cos a, cos 8, and
cos y are its direction cosines

phase difference of the aoe element
with respect to the origin

angle between the vectors
(-x%0, -Yo» -Z0) and
(200 = 2, Yo — Y.> Zo 7 Z,)

distance from the source to the sus

element
.th
response of the 7° element
directivity function
the normalized directivity function

weighting factor assigned to the an

element or element (x, y, 2)
region in 2-space

region in 3-space
generalized region

dimensional differential

[e)

oN

ayaa

B N

W

)

9 or X

rora

Gan Be)
w

a> de> ds

Soy Ue

ni

sound pressure at a distance p from
the source

wave length of sound with velocity ec

wave number
se) = Aner
density of the medium

angle measured in the x-y plane from

the x-axis in a counterclockwise direction

radius of a circle

polar coordinates of the see element
separation between consecutive
elements) inthe 2-2 Yar dire cLion
respectively

substitution variables, defined
where used in the text

rectangular coordinate system; the
coordinate system whose origin is
the sound source and such that the
positive € -axis contains the line
segment from the origin of the array
to the sound source

CONTENTS

INTRODUCTION... page 5
PAR® I (GENERAL THRORY {2 5

IL,
2.
3.

Discrete Distribution of Elements... 5
Continuous Distribution of Elements... 9
Shading... 12

PART 1s VAP PEICAION SIO wr ORM ieee!

ls
2.

Discrete Distribution of Elements... J5
Continuous Distribution of Elements. . £4
APPENDIX A: NOTE ON A DOUBLE LIMIT PROCESS...

APPENDIX B: DIRECTIVITY FUNCTION IN VECTOR
NOTATION... .41

APPENDIX C: FORMULAS FOR EVALUATION OF
CERTAIN INTEGRALS... 43

APPENDIX D: PERTURBED CYLINDRICAL WAVE
FRONT...46

REFERENCES...50

ILLUSTRATIONS

Rectangular coordinate system showing a source and a
spherical wave front passing through a typical element of
a general array... 6

Source
Source
Source
Source
Source
Source
Source
Source
Source
Source

and a typical element of a plane array... 9

and elements of a linear array... 15

and elements of a circular array.. 17

and elements of a rectangular array...19

and elements of a rectangular parallelepiped.. .21
and a line array... 25

and a rectangular array... 26

and a solid rectangular parallelepiped... 28

and a circular ring... 29

and a circular disk...34

Element on spherical surface... 36

Source,

typical element, and point of compensation in

vector notation. ..42

Source

and perturbed cylindrical wavefront passing

through a typical element...46
Example of a perturbed cylindrical wave front... 49

39

INTRODUCTION

The work described here was done as part of the general
program to improve long-range sonar receiving arrays. It
had a twofold purpose: first, to show that previous work
on array directivity functions can be subsumed under a
more general method, and, second, to bring together
results that are scattered throughout the literature. The
advantages of the general results of this report are that
the directivity pattern can be determined for any distance
and direction of the sound source from the array, and for
any geometrical configuration of the array.

PART I - GENERAL THEORY

1. Discrete Distribution of Elements

If the source of the sound wave is at a distance D from the
origin of the array, then the sound wave will be spherical
in shape when arriving at the array. Furthermore, the
sound wave will expand radially from its source. Let

(x0, Yo. Zo) be the coordinates of the source with respect
to the origin of the array. Let (x;,y;,2;) be the coordi-
nates of the 7'% element of the array. A unit vector in the
direction of the line joining (0, 0, 0) and (xo, Yo» 20) will be
(cosa, cosb, cosy), the direction cosines of the line. Then,
from figure 1, the phase difference with respect to the
origin from the 7°" element is

ae + + + =
a, # [ x, cosa y, Coss Zz, Cosy p, (cosy, 1)] (1)

where & = Zing and } is the wavelength of the incoming
sound wave. Equation (1) is obtained as follows:

D=(x.,y.,2.) ° (cosa, cos8, cosy)+p, cosy,
U U U U U

SOURCE
(x0, Yo, Zo)

ELEMENT

69 W550
or, 4 U

x, cosaty, cos8tz, cosy WAVE FRONT

Figure 1. Rectangular coordinate system showing a source
and a spherical wavefront passing through a typical element
of a general array.

or, expanding the dot product:

D=y.co0OSaty,cosB8ts.cosyto.cosy.
a8, eg ae y ae Ve

ee
= = D-9. = x.cosaty,cosBts.cosytp.(cosy,-1)
aD D p, = %, cosa ty, cosB Za y p.,( v,

Equation (1) follows on multiplication by x.

The response of the see element is by definition 7,exp(jwt) -
exp( jo ,). * (See list of references at end of report. )

Letting the time variation factor of the response equal 1,
the response becomes

PR. = Hexp jo,

If a direction defined by the unit vector (cosq,, COS85, COSy7o )
should be compensated for, the response of the 7t4 element
becomes

R, = H,exp je, expr 8, = £,exp j(6,-6, ,)

1K

The directivity function is by definition”
Vv Vv
R= ye = \ Eexp j( 8% ;)
oil

BA Al
that is, the sum of the responses of the elements.

Hence, compensating for a spherical wave, the directivity
function for a finite number of discrete elements is

*Page 15 of reference 2.
**This function is also known as the directional characteristic
or the beam pattern of the array.

R= ) £,exp Jil, (cosa- COSa5 +y, (cosB-cos8p )

+ s + =il\e =
z, (cosy cosy, ) p, (cosy, 1) Po ,(cosy, , 1)] (2)
By simple geometry, it can be shown that
P, = \/D°-2D(x, cosaty, cos8t+z,cosy)+tr,? (2a)
t t t t t
where 7..° = x,*+y,2+z.2, and that
t hein!

D-(x,cosaty, cos8tz cosy)
U U U (2b)

cost. =
Ve 0;

If compensation is made for a plane wave front arriving
from a direction (cosa, COs, , COSy,) and if the actual
source is a finite distance D in a direction defined by
(cosa, cos8, cosy), it can be shown that the directivity
function is

WV

Re Ve, exp Jhlx., (cosa- cosa, ry, (cos8- cosa )

-_ U

Gal

+2, (cosy-cosyo +p, (cosy ,-1)] (3)

Now if D> =, in equation (3), the directivity function be-
comes in the limit that for a plane wave front:

MV

R= » B exp Jhl x, (cosa-cosay yy, (cos8-cos8o )
t=1

tz, (cosy- cosy, )]

because p .(cos,-1)50 as D>». This can be shown by use of
L'H6pital'S rule’

2. Continuous Distribution of Elements

By a continuous distribution of elements, it is meant that
the elements are in a region of space where they are every-
where contiguous and dense.

(a) Array in the Plane (fig. 2)

Z SOURCE
P(xo ’ Yo AO) )

W(x, y)

a B ELEMENT (x, y)

Figure 2. Source and a typical element of a plane array.

Let the array consist of the points of the boundary and the
interior of a region 8B. Let B be a connected and a compact
set. Let (AA porte AA.) be a partition of B such that the

responses on AA; are approximately equal. Then the

response on AA, is approximately

~ ; att + es
R, E(x,.y,) {exp jk [x, cosa y,COs8 p(x ,, y,)(cosy(x,, y,) 1)]}a4,

where (x., y,)is some point in AA,. Then the directivity
function fs approximately u

pew ial
U
i=1

or
N
ee ° + + —_
R ) Ble, y, texp jax, cosa y, cose p(c,. y, Kcosy(a,, y,) 1)]}a4,

i=l

Letting 7-0 as max |a4,|>0

R= { Fx, y) {exp jk xcosatycosstp (x, y)(cosy(x, y)-1) ]}dA (5)
B

This equation is for the uncompensated array. If the array
is compensated for a planar wave front whose direction is
defined by (cosag, COS85, COSy,), then the directivity function
is

R= i Ex, y) {exp jz[x«(cosa- cosa, )+y(cos8- CosB, )
B
+o (x, y(cos¥ (x, y)-1)]}d4A (6)

If a spherical wave front is compensated for, the directivity
function is

Be i E(x, y) {exp jx[ x(cosa- cosa, )+y(cos8- cosBo)
B

+o(x, ycosy(x, y)-1)-05 (x, y (cosy (x, y)-1) ] }aA

(b) Three-Dimensional Array

By reasoning analogous to that used in deriving equation (5),
it can be shown that, if 7 is a region in 3-space, then the
directivity function is

(®
R= | 1a ($6. vafemp [xcosatycosstzcosy
dE
+o(x, y, z(cosv(x, y, z)-1)] bday
Or, if compensated for a spherical wave front

r

R= / E(x, y, 2) {exp jk [x(cosa- cosa, )ty(cos8- cosB, )
df

+ z(cosy-cosy,)+ (x, y,z)(cos (x, y, z)-1)

-0o (x, y, z)(cosl> (x, y, z)-1) 5 | QV)

In Appendix A, it is shown that

Li P
D ae | IB Ges Ws &)) jo [xcosatycos8tzcosy
Te

to(x, y, z)(cosh(x, y, z)-1)] | dy

Li
i | Cae E(x, unferos [xcosatycos8tzcosy
oF

t+o(x, y, z)(cosv(x, y, z)-1)] jer
which is

R= { E(x, y, 2) | exp jk [xcosatycos8tzcos 7] | ay
Op

11

Hence, as D>», the directivity function becomes that of a
planar wave front. The corresponding result for a planar
wave front and direction of compensation can similarly be
derived and is

R= i Elx, y, 2) \exp.jit [x(cosa- cosa )ty(cosB- cosB, )
See

+z(cosy-cosy,)]}dV

(c) Generalization of Results (a) and (b)

Let F be a region containing elements (x, y,z). Then the
generalized directivity function is

Rs Jim. Ys Z) | exp jk[2(cosa- cosa, )ty(cos8- cosB,)

+z(cosy-cosy7o )to(x, y, z(cosW(x, y, z)-1)

-00 (x, ys z)(cosvo (x, y, z)-1)] } dF

where

(Ce Ube) = \f (oe- 200 P+(y- yo)? (2-2 )?
NxXo* tyo*t+Z07 -(xcosatycos8tzcos 7)

o (x, Yy> 2)

Nico +Yo* +Z0* =)

cosy(x, y, Z)

where

3. Shading

In the foregoing expressions, F(x, y,z) and #, are called the
shading coefficients. These functions are wéighting factors

such that if properly chosen will produce a high main lobe
and low side lobes in a given directivity function.™ If

E(x, y, Zz) = 1 or 1g C= 1, then the equations reduce to the case
of an unshaded array (i.e., all elements equally weighted).

So far the directivity function has been concerned principally
with the effect of phase differences. However, the sound
pressure will also affect the pattern in that it decreases
with distance from the source. This effect can be taken
care of, in the continuous distribution case, by letting

sles
Pd

hi D
E (x,y) =4 (x, y) G5)

Then, the uncompensated directivity function is given by

p= | awl, Woy fexps1D-olx, we
y) XH» |

In the above equation, w(x, y) is a shading factor, CS

is a weighting factor that takes account of the decrease in
sound pressure amplitude with distance from the source,
and D-o(x, y), the exponent of the exponential function, is
the phase difference of the element at (x, 1) with respect
to the origin. There is a corresponding, similar result
for the discrete distribution case:

N
= D ; —
R )) Dy exp jz[D p, |
ae @
Y
D
In most applications a 1, so that F(x, y)~wls, y).

Thus this correction factor for the decrease of sound
pressure amplitude is absorbed into the factor 7x, y).

*Reference 2, page 16, and reference 3, page 15.

**This treatment is for the 2-dimensional array, but it
can be extended to 3 dimensions. In either case, a spheri-
cal wave is incident on the array.

14

In order to ascertain when this correction term can be
approximated by unity, consider the sound pressure ata
distance o(x, y) from a point source emitting spherical
waves; then the sound pressure is given by

-

= -00 kCA 0 Hi 40 3K
Py = Ge, peers Lue: kote, yt 9

where the symbols have the meaning ascribed to them in
the nomenclature.

The sound pressure amplitude is

me |= Oo eA

fe) Oo)” Saw
and at a distance D-
Se OGiCA
ey Weg =

The sound pressure amplitude drop between D and P(x, y)
by the wave is given in db by

Fy H D
2010g, 5 em = 2010g, 5 o (x, y)
D ’

D : :
ao : etal
If 0 oS Ga) <1.1, then, the amplitude error is +ldb or
less, which is considered small enough in most cases so

D NRK eok
that 1 4
a Ga) can be replaced by unity

*Reference 4, page 30, e.g. (1.41)
**Reference 5, page 457

PART II - APPLICATIONS OF THEORY

1. Discrete Distribution of Elements

(a) Linear Array (Compensated)

Let V = 2m elements be placed a distance d apart on the
y-axis. Let the position of the ith element be given by
(0, (t+1/2)d, 0). Then by equation (3) (fig. 3),

+77 /2
P= ) Byexp,fal (¢#1/2)d(sine- sing, +p ,(cosy,-1)]

~=-/2

where
948 DP -2D(441/2)a singt+(1+1/2)? a?
and
_ D-(441/2)d sing
Pi

cosy.
Vs

LE ACOWUR CE
Plxo » Yo )

i (0. (-2)3)

Figure 3. Source and elements of a linear array.

15

16

If D-~, then

+N /2
RS y H, exp jk(7#1/2)d(sin6- sinh, )
t=-W/2

And if the E, are all equal, and I = 2m
m
ip = 25 )) exp ji(i-1/2)d(sine-sing, )
wel

Applying the geometric symmetry of the array

m
R= 2B, ) cos(ie(i-1/2)a(sing- sing, ))

aml
Then, using the identity*
ie sin oy
2 Y cos(i-1/2) x = ————, where = 2m

sin =
wail 2

the directivity function becomes,
Re Ey sine) o
where

= ns (sin§- sin, )

*Reference 2, page 16.

This is the equation for a linear array of / = 2m elements*
or, if the directivity function is normalized

) aK Kk

es sin(/e
Ry = By isin)

(b) Circular Array (fig. 4) (Compensated)

Let the circle have a radius 7. Then for a discrete set of
elements on the circumference, the position of the it
element is

x. = rcosg. . = rsing. = 0
i Di» U; D522,

» SOURCE
JAG 35 5 Uy 5 Bo)

ELEMENT
(x. y,)

Figure 4. Source and elements of a circular array.

*Reference 2, equation (A9), page 16.
**Reference 1, equation (51), page 56.

17

18

Substituting these polar coordinates in equation (3) and

FA 0
t
WV
R= > B,expJk jr I(cosp , (cosa- costo sing, (cos8-cos8o ) |
Gai

+ -
p, (7, ,\(cos¥,(r, 9 ,) 1)
It is easy to verify the relationships

cosa = cos@sin y

cos8 = sin§ sin y

obtained by introducing spherical coordinates. Substituting
into R and simplifying:
WV
R= ) B,expJjh | r[(sinycos(6-9 ,)- sinyo COS(Qo - ,)1
t=1

+o,(8,8, (cosy, (6, 0,)-1)

Let D-~; then, the directivity function becomes that of a
planar wave front for a circular array:

MV

R= ) B,expjkr| sin y(cos(6-9 ,)- sinyo cos(8o -9,)]

Call
Let WV = 2m elements be placed symmetrically with respect
to the x-axis with equal angular displacement between any
two consecutive elements. Let this angular displacement

be 7.5°; then 2; = (t+1/2) 7.5° for the ith element.
Further, let 6 = 0° and y = At WOOm sethen

W/2
R= ) Bvexpsjer[cos(-(i41/2)7. 5°)-cos(t+1/2)7.5°]*
i=-W/2

*Reference 3, pages 13-15

(c) Rectangular Grid (Uncompensated)

Let WV, = 2m be the number of elements in the x-direction, and
Nz = 2n be the number in the y-direction. Let the separation
of any two consecutive elements be d, and dz in the x-direc-
tion and y-direction, respectively (fig. 5).

Ah

P(x, Yo: Zo)

(x, Y,)

Figure 5. Source and elements of a rectangular array.

Then the coordinates of the jth element will be (7 + 1/2)d,,
(2 + 1/2)dz, +1/2 depending on whether 7 is negative or
positive, respectively . Then the uncompensated oar
corresponding to equation (3) becomes

+N, a
a7
R= \ y E, 7 OxP Je LGA / Addy cosat(2+1/2)dscoss
ye ae

ol (cosy, hall

3

20

Let D-o; then a plane wave front will be incident:

+11 +e
ae ee a
if = ) \ 2, pexp jk [(t41/2)d, cosat(t+1/2)dz cose J
pata 27M
2 2

Applying Euler's Formula and noting that because of
symmetry the odd function cancels in pairs (letting #, 28
be equal) ue

mie
R= 4B, my )cos(e(i-1/2)a, cosa) * cos(k(2-1/2)ds cosp)

OA, 2a.

Then by using the identity of Section (a)

stn
yx

2) cos(i- 2) 4 ——————
Lig Ot. OG
SEL oy

the directivity function becomes

PzE sin(W,é) , sin(Wsn)
i, 2 sin(é) sin(n)

where

bs kiddy ze kidz
Ee = 79, ©Osa Metis —z_cosB

Letting fi. ee 1 and normalizing

_ sin(W,&) sin(Izq) * (9)

Ry eee eames
N,sin(é) Nz sin(y)

*Reference 1, page 37, equation (29).

Similarly, one can derive a corresponding result when a
direction (cosa), COS85), COSy)) has been compensated for.
The result is similar to equation (9) except that

e = 7 (cosa - COSA )

and

Ade2
i 2

(cos - cos& )

(d) Rectangular Parallelepiped

Let a finite number of elements be arranged in the form of

a rectangular parallelepiped such that J, = 2m, Ns = 2n,

V3 = 2p are the number of elements in the x-direction, the
y-direction, and the z-direction, respectively. The separa-
tions in any axis direction between consecutive elements are
equaltaeleectathesseparcationtbeng nid en G sully thei ay sur=,
directions, respectively. Then the coordinates of the ith
hydrophone will be, if the origin of the coordinate system

is the center of the rectangular parallelepiped, (7+1/2)d,,
(Ce A10/F2)) api (310): 2)) Gain Ctl oo)

P(x0+ Yo: Zo)

Wit, 2, A)

Gans Vas Ba)
eae (go 9

Figure 6. Source and elements of
a rectangular parallelepiped.

ail

If no direction is compensated for, the directivity function is

SUB hy GM 9 Ns
2 2 2
ee ) y DBs i ,exp.se [(471/2)d, cosa
Sik Sis il
Op eh =z 2 = 3
Ua Coot ay

+(7+ +(At a a
(241/2)ds cosst(h+1/2)ds3 cosy Psie, (COs, | an 1)]

By taking account of the symmetry, and letting D>», and
assuming f, Lh are constant

we, \, jo

i 2 oye!) ) Y Lexp j(t-1/2)dy cosa] -
Cmte mlanml

[exp jz(2-1/2)d2cos8] - [expjx(h-1/2)d3cosy ]

Since each summation variable is independent of the other:

pes iB n ne) exp jk(i-1/2)d, cosa ] [2) exp jA(2-1/2)dz cos8]

[ 2) exp ji(h-1/2)dz cosy |

Using the identity in the previous section, the directivity
function becomes

R=E. sin(h §) : sin(W2 1) t sin(Wa( )
i, 4,hsin(e) sin(») sin(c )

where

any
i)
i}
(e)
2)
n
2

normalizing,

sin(W,¢) . sin(W2 n) sin(V3C)

sty a L,hM,sin(e) Wz sin(y) ' Wz sin(C) 0)

Note that in equation (10) each factor represents the nor-
malized directivity function for a linear array with the
respective number of elements in the respective axis
direction.

If a direction (cosa, COS85, COSy.) had been compensated
for, the substitution parameters in equation (10) would
have been

a5 Fa (cosa- COSA )

7 = “22 (cos8- cos8o)
C= EZ2 (cos y- C0S7o)

To show the validity of (11), below, let

a = B°
3 S97"
ya
and
Qo =) (Baie = O90)”
Bona om a0
and

24

Then the directivity function becomes

_sin( 1 cose’ ') sin(W.722(cosy’ - 1))

Tae TOD e ere —..0000 O———
A MN, warn cos8’ ) Nz sin(722(cosy’-1))

a Ts cosa eae)

R

W381

: ok
or rearranging,

2. Continuous Distribution of Elements
(a) Line Array

From the general directivity function for a continuous
distribution of elements, the directivity function for a
line array (fig. 7) (see Part I, section 2(c)) becomes:

LjP
R= ; rca exp ji[ x2(cosa- cosa +p (x) (cosy (x)- 1) ]
x=-L/2

When this integral is evaluated, the answer does not
appear ina closed form. However, if D-~, the integral

>
“Reference 6, page 15

(11)

P(xo , Yo )

Figure 7. Source and a line array.

is not difficult to evaluate. In this case (letting
E(x) = constant)

Lf[2 sin(T 1 (cosa- Cosa, ))
R= exp jizx(cosa-Ccosag) } dx = Lee ah Me AD
ee ct (cosa- Cosa, )

If this result is normalized, and if q = 90°-9, then

sin(Zz (sing- sing, ))
= F- (12)

wi (sing- sing, )

Ry

This result can also be obtained if equation (8) is norma-
lized and Wd>L; then, equation (8) approaches (12). This
can be shown by using L' Hopital's rule.

25

26

(b) Rectangular Area

Applying equation (6) to the rectangle (fig. 8)

aA fo)?

R= [ | me y))\ exp jz[x(cosa- cosas )ty(cos8- cosB, )
JOO ald
a Nahe ae)

+o(x, y)(cosy(x, y)-1)] | dxdy

Plxo; Yo, Zo )

yx, y)

o(x, y)

x

Figure 8. Source and a rectangular array.

Again, evaluation of the above double integral does not
give a closed form solution. However if D--, and if
E(x, y) = constant

GA 102
exp ji[x(cosa- cosa, )+y(cosB-cosB, )] | dxdy

This becomes

e/(OL IL é in(20 = mn
oe COSA )) sin( (cosR- COSBo ))

+ (cosa- COSA, ) + (coss- COs, )

or normalized,

: \ ; ~
sin( 2 (cosa- COSao )) sin(—"(coss- cose, ))

Ry = 2 5 (13)
7 (cosa- COSdo ) <"(cos8- cosa, )

If, however, the direction (cosa,, cos, cosy, ) had not been
compensated for, the directivity function would have been

new AGI ~ _ (bor \
e ee : sin( "oss )

My S eosa cose

R

which is equation(30)of reference 1, page 37. It is also
possible to obtain this equation from equation (9) by letting
VV, d,7a and Nzd2,-b as VW, 7, Nz>0. This is the manner in
which Stenzel derives the above equation.

(c) Solid Rectangular Parallelepiped (fig. 9)

The directivity function for a solid rectangular parallele-
piped is, from equation (7),

a b Cc
2 2 2

R = ; [ Pale Us cifexpjet ose COSA, )
an Oh -d Ie

+y (cos8 - cOS8, )+z (cosy- cosy)

+o(x, y, z) (cosy (x, y, z)-1) || dxdydz

27

Plxo » Yo 20)

Woe, Ys z)

p(x, y 2)

Figure 9. Source and a solid rectangular parallelepiped.

Because of the extreme difficulties in evaluating the above
triple integral when the source is a finite distance from
the array, the integral is evaluated for a source at infinity:

+y(cos8- cos8>o )+z(cosy- cosyo ) J | dxdyadz
Integrating (letting Z(x, y, z) = constant)

i at ; : bt ~
ia sin (Zi (cosa- COSA) )) sin (Gicoss- cos8o ))
~ (cosa- COS ) + (cos8-cos®o )

_ (en
sin( "(cosy -cosyo ))

Tr
7 (cosy-cosyo)

28

and normalizing

B (an iL ~ 4 ‘bu = N
sin( > (cosa cosa, )) sin + (cos8 COS8, ))

a foil
<n (cosa- cosa) =, (cos8- cosBo )

sin( “(cos y- COSY 5 ))
._ >= (14)
<1(cos8-cos®, )

If c>0 in equation (14), the equation for a rectangular area
is obtained, i.e., equation (13).

(d) Circular Array (fig. 10)

The directivity function for a circular array, obtained from
the generalized directivity function,is

Se

R= | 2G. y){ expj4[x(cosa-cosa, )+ (cos8-cos8, )

S25),
+o (x,y (cosy(x,y)-1)] tds

where s denotes arc length.

IG 5 Ue )

V(x, y)

OG

Figure 10. Source and a circular ring.

29

30

WER aGl =) 7a COSI OMY iig Sune ntlveta

Se2
R= jee. y) exp.falr(cosd (cosa- CcOSd )+sing(cosB- cosBo )

Sasi

*otai(cosy(s)-1)1}2¢
Let s = 7, so that

D>
R= | 20. y)
Q=2,

expjALr (cosp(cose- cosdo )+sing(cos8- cosBo ))

*o(@Nleosy(@)-171 ra

Using the relationships

cos8- cos8o

tan =
Po COSA- COSXo

_ COSA- COSAHo
COS & =

(cosa- cosa © +(cos8- cosBo )®

cos8-coso

sin @ = =.
Ni (cosa- cosdo )* +(cos8-cosBo )*

the integral becomes

R= | 20, y)
=D,

+o(g)(cosy(g)- 1) ] brag

expjklr N(cosa- cosa )* +(cos8- cos8o )* cos(Z- Zo )

so that

Let D-~2,
Ge
R= [aw exp jur a] (cosa- cosdo )* + (cos8- cos8o )° cos(g-Go na
D =r

and if

5)

In the following, let #(g) =
Z1 = 0, Ge = 2n

exp jkr VN (cosa-cosds )* + (cos8- cos80}” cos(g-Go Mee (15a)

aqTl

O
If
Ba 2 OL ga.
TI
R= ij expjarv (cosa-cosdo )* + (cos8- cosBo )* costars) (150)
Oo
If
fin ==> oe =m

TT
2
Rs \ [espns (cosa- costo )* + (cos8-cosBo )* cos(Z-Go | racase
TT
a

These integrals can be evaluated by the use of the formulas

(see Appendix C):

ei
| cos (@-p0 dap = Orien2 Oa be (27)

(a)

oO

cos” * (9-Bo dg =

oO

32

T
F cos (yrpo bap = Ga VGn 3). 8-3-1 (a) (b)
TT
B
J cos age Ia = Bam ay eae ()
os
D
The integrals then yield
R = 2rdJo (&) (16a)
(where Jo is the Bessel function of order zero)
TT
R= argo (@)4jr | sin[e(costp-go 11d? (16b)
oO
R= nro (2}+jr | sin €(costp-po 12 (16c)
TT

2
where

& = kr .f (cosa- cosao)* +(cos8- cosBo )*

These equations are solutions of 15a, 15b, and 15c, respec-
tively. If equation (16a) is normalized

R = Jo (&)

which is the same as the result obtained in reference 1,
page 58.

If equation 15c is evaluated for a, = 0°, Bo = 90°, 7, = 90°,
a = § then

il

2

R= r| exp 2jkrsin(g )sin(g-3 ) ag

TT

2

which can be written

exp,firr(cos(9-g)- cosy fa

y
iT
4
vw) a—, va

This integral yields*

T

2
R = Tro ( 2krsing tyr | sin| 2irsingsin( 9-5 )|ag

in
2

In general it; =-6@, Ga — 6 areythe limits of integration,
the directivity function will have the form

R = 28rJo (2)+g(2)+ inl)
For example, this is the case when § = 7/3.

(e) Circular Disk

Applying equation (6) to the case of a circular disk (fig. 11),
and letting 3 = 7m Cos?, Y = rsing

Rie ij | 20. y)
A

expjzLr{cos g(cosa- cosa )+sing(cos8-cosBo )}

+o(r, g(cos\(r, @)-1) ]}rdrdg

“Reference 3) page ts

33

34

P(x0, Yo, Zo)

v(7,@)

Figure 11. Source and a circular disk.

Let the disk have radius a; then

a et
Re J Vat, y) jexpjkLr {cosg(cosa- cosa )
r=0 g=0

+sing(cos8-cos8o )}+p(7r, g (cosy (7, )-1)] | rarda @

Let D>», and A(x, y) = 1, then

oH Bh |
Pe | I Jexnsrf const cosa- coms )+sing(cosB- cosBo ) ] rare
r=0 g=0

The integral becomes

a
R= an | ro (rd (cosa- cosco )? +(cos8- cosBo aa
O

Applying the identity

| set (x)dc = xJy (x)*

*Reference 1, page 119

The directivity function becomes

where

E = kas (cosa-cosdo )* +(cos8- cosBo )°

If this directivity function is normalized

_ 2d, (§)

Ry z

Let do = 85 = 90°; then & = Aasiny and

_ 2/1 (kasiny) ,,

Ry ~Fasin y

(f{) Spherical Surface Array

First, an uncompensated spherical surface array will be
considered, followed by a compensated hemisphere.

In the case of the uncompensated spherical surface array,
the directivity function is given by

R= fa, v2) exnstdveosntyoost zcosy]}ds
S

Let the sphere have radius qa. Then from figure 12,

x = asingcos6é
y = asingsiné
Z = acosg

oO

*Reference 1, page 20, equation (17).

(Up)

35

36

Figure 12. Element on spherical surface.

Also, the direction cosines expressed in spherical coordi-
nates are

cosa = sinycosy

cos8 = sinysiny

cos y = cosy

Where y is the angle measured from the x-axis to projec-
tion of the unit vector (cosa, cos8, cosy) on the x-y plane.
Let H(x, y,z) = 1, then substituting the above relationships
into the expression for #, and simplifying

sinddga 6

TT oT
R =a" \ j Jesse [singsin,{ cos(9-x)teospcos )|
B30 =O

Then, integrating with respect to 6,

TT
R = 20a” | f{expjkacosycos@} Jo (kasinysing)singdg
Bao

which becomes

R = 2ad\sinka*

Thus f# is a constant. This is to be expected since an
uncompensated spherical array would not favor any parti-
cular direction, because of its symmetry. This result
corresponds to the fact that a circular array, if uncom-
pensated, has a constant response for a plane wave arriv-
ing in the plane of the array.

For a compensated hemispherical array the directivity
function is

R= [ at, Ys B)

S!

exp Jk ((cosa- COSM )+y(cosB- cosBo )

+2(cOs y- COSYo )) | as

where the subscript zero indicates direction of compensa-
tion. If the radius of the hemisphere is a, and expressing
x“,Y,#, in spherical coordinates as before, R becomes

R= \ x0, §),expjzal singcos§ (cosa- cosdo )
+singsiné (cos8- cos8> )+cos@(cosy- cosyo) |} dS
_ cos8- cos8o :
Let tanvo i eanGCGaGs. Then the integral becomes
[at 6){ exp jxasing,/(cosa- cosas )* +(cos8- cosBo )*
S
cos(e- V5) esp ocossteos y- COSyo )} dS

*Reference 7, page 378-9

38

If #(g, 9) = 1, then integrating over a hemisphere, the
directivity function becomes

T
2
poe |

@-08=0

2 TT

3

expjzasing /(cosa- cosa )* +(cos8- cosBo )* cos(8- vy |

. ese cos @(cosy- cosyo | singdgd 8

Integrating with respect to 4

=

et
R = 2na* be (xa sing,/(cosa- cosdo )* +(cos8- cosBo )
(e)

. fexeseacosstoos COSyo | sing ag

Using the transformation & = cos@,

1 1
R = 2na" lhe (q /1- 2? )cospedét+j es (q./1-2° )sinp§dé
9 oO

where

g = ka J(cosa-cosdo )® +(cos8- cosBo )*

p = ka(cosy-cosyo )

A preliminary search of the literature has failed to produce
an evaluation of the integral. A simple method of evalua-
ting it is to expand the Bessel function into its series and
then integrate term by term. (For further details see
remarks in Appendix C.) Another method of approxima-
tion is to place a sufficient number of discrete elements

on the spherical surface and use the formula for the direc-
tivity function of a discrete element array for reception

of planar sound waves (the limit of equation (3) as D-=).

APPENDIX A:
NOTE ON A DOUBLE LIMIT PROCESS

Theorem. Let f(x, y) be a function defined on asx<b, c<y<~,
with the properties that
(i) ae y) = y(x) uniformly for each x on asx<b

(ii) f(x, y) and y(x) are continuous functions defined on
asx<b.

Then
b

b
vee \ re, y)ax = [Pe y)ax

a a

Proof: Since f(x, y) converges uniformly to g(x) for each x
on a<x<b, if e>0 be given, then there exists a Y (ce) (depend-
ing on c, independent of y) such that

| 7x, y)-9 a) | <==

whenever y2/(c) for each x on a<x<b(b>a)

This inequality can be written
s— - g(x) <f lx, y)< g(x)
EO 0 Gy b-a

whenever y>/ (ec).

Since g(x) and f(x, y) are integrable functions of x

[oy

b fo)
|G -at)ax< [rey dx < \@or eax
a

Qa a

whenever y=V(c). This reduces to the inequality
b

D
[rte y)dx- | oberon <e

a a

39

40

whenever y>/J(e), which is, by definition,

D Db
iim Jr: y) ax = [pee y)ax
Y CO Uae

a a

This result can be generalized to higher dimensions by
repeated application of the above theorem. Since Hexp(j6)
= Feos6+ j#sin§ and each function satisfies the condition of
the theorem, and since integration is a linear process,
the claim of Part I, Section 2(b) is seen to be true.

APPENDIX B: DIRECTIVITY FUNCTION
IN VECTOR NOTATION

The directivity function for a discrete number of elements
compensated for a direction (cosa), cos80, cosy.) anda
finite distance Do, with coordinates (x0, Yo,Z0) is

N
R= ) £,expsklx(cosa- COSA )+y(cos8- cosBo )+z(cos y- cosy )

gol

a i ha =
p, (cosy, 1) p (cosy, 7 1)]
This result can be written in vector notation as follows:
Let
v = (cosa, cos8, cosy)

ee
Vo = (cosdo, cos8o, cosyo)

ie
SO > (xo, Yo, Zo)

Then the directivity function becomes

i = v~ = o> |r |-7,-0
R= ) F,expjh Ta (v-v0)+ |r-r, | ————-1
,=] c ate 7 |
U
= = | 70 |-r., ~&
2 |raoirs | u -1
70-7, |

U
This can be simplified by multiplication inside the exponential
function to

M
R= ) FE, expel |7|-|7 |- [Py | o>

wall

>

|)

42

® POINT OF COMPENSATION

@ ACTUAL SOURCE

Figure Bl. Source, typical element, and point of
compensation in vector notation.

APPENDIX C: FORMULAS FOR
EVALUATION OF CERTAIN INTEGRALS

The recursion relationships used in evaluating the integrals
(15a), (15b), (15c) are proved by using mathematical induc-
tion. As an example the relationships

27
en x ts (2n-1)(2n-3)...5- 3-1
J cos Voll Ona Geman aN. (eu)
(@)
and
2
[cos 1(g-Go dg = 0, n=1,2... (C2)

Oo
will be proved. Equation (C1) is first proved. Forn = 1

2
ie (GGo dg = 1/2(9-Po )+1/4sin2(-Go) ait

fe)

Assume that equation (C1) is true for n = &

Then
eT] PAN
| costs V(9-Go )dg = | cos (Z-Bo dw
= 1 2hk+ . aM Spee ok *
= 3775 COS Xp fo )sin(g-po)| oper cos’ (2Bo )dg

(e)

“Reference 8, page 38, integral 267.

43

44

= —s (cos?#**(2n- o )sin(21-go )+ cos*** 196 singo )

2
2h+1
oS (oe (Q-Bo dw
{e)
2m

a J cos” (g- go Nap

(0)

But
2 i
2k — i Ae 8 ) 8
| cos (W-Go )aw = ses Bay (27)
O
by induction hypothesis; hence
aq
2+ 2) x= x Saas Ril
Se) OV open OSS AS: Cy)

{e)

Hence equation (C1) is established.

Similarly, equation (C2) is shown to be valid:
ILS = i

etl

eT
| cost gh to dg - sin(g-go)|_ =a

fe)

Assume that equation (C2) is true for n = &.

Then, forn =AtT1

an
| cos”***(g- go Jag =

(0)

i cos?” (g-go )sin(Q-o i

eT
TE | cos (p-go lag

= aaa [cos**(21-go0 )sin(2T7-Zo )+cos? "go sino ]

hii

| cos?*-*g-p )dg = 0

(©)

2k
2hF1

+

since

T
2,

' cos?*"1(g-Go dg = 0

by induction hypothesis. In a like manner the other integrals
are evaluated.

Since the integrals involved often contain Bessel functions
in the integrand, or some other function that must be
expanded in an infinite series in order to evaluate the
integrals, two conditions must be satisfied before the
integral can be evaluated:

1. The infinite series must be uniformly convergent, and the
individual terms must be continuous.

2. The limits of integration must be within the interval of
convergence of the series.

All the integrands treated in this report satisfy these con-
ditions when term-by-term integration is employed.
Furthermore, if the series is an alternating series that

is monotone nonincreasing, that is uniformly convergent

on its interval of convergence, then the error after trun-
cating the series after n terms is majorized by the magnitude
of the (n+1)th term. This is a convenient method of judging
the error.

45

46

APPENDIX D:
PERTURBED CYLINDRICAL WAVE FRONT

The function

A(r,@) = r+g(g)

represents the wave front of a radially expanding cylindrical
wave emitted from the source P(x, yo) after some time ¢
(fig. D1). If g(g) = 0, then A(r, g) = r which is a cylindrical
wave front inthe plane. The quantities r and g are defined
in the &y coordinate system.

Y

A

SOURCE
P(x0 , Yo )

ELEMENT
(xc, , y,)

Figure Dl. Source and perturbed cylindrical wavefront
passing through a typical element.

The distance D from the origin of the array to the source
is given by

= x.cosoty. singt Oo
D xy sé y,s 8 ee su.

Since
=r.+
nr,, 0) ins g(0)
and
zg) = D-h(r,, 0)
Is 0 oe
i :
pan = + ap - -
ay x, cosh y, sind p, cosy, r. g(0)
But
=r+t
Pa aa ey
hence

at f el A
6, kl, coséty, sindtp , (cosy, 1) g(t ,) g(0) ]

Now

. ro = =| °
au 6, Lo, cose y, sind |

since y,70 as D-~. Hence for large D, 6, approaches the
phase difference for a plane wave. if

The directivity function for an uncompensated discrete
array is given by
WV
= ; ar in§+ -1)+ =
R ) £, expselo, cose y, sine p, (cost, 1) gy.) g(0)]

sa

47

48

Note that if g(g) = 0 for allg, then the above directivity
function becomes that for a cylindrical wave front.

If a cylindrical wave front is compensated for (compensa-
tion indicated by subscript zero), the directivity function
for a discrete number of elements is

N
fe : : va Beate e
R ) B exp flor, (cos COsfo ) y, (sino sino ) p, (cosy, 1)

L=

- Poy (costoz -1 +9(vz)-9(0) |

The various quantities in this equation are easily obtained
from geometry:

p, = (2. = 20 Ea Ue) p or
ys /(x,-Deose +(y, -Dsiné zaman

D-(x;cos6+y sind)

Py

cost, =

An example of the function g(g) is sin 4g. Its effect ona
cylindrical wave front is shown in figure D2, when r =2-5
that is

WS

1 psu :
n (25.8) = 25+ sin4 9g

ke £3
nl25,0) = Be + sin4 g

Figure D2. Example of a perturbed
cylindrical wavefront.

49

REFERENCES

1. Stenzel, H., Guide for the Calculation of Sound Processes,
Navy Department, 1947 (NAVSHIPS 250-940)
2. Navy Electronics Laboratory Report 757, Simultaneous

Multibeam Phase Compensation, VI: Phase Compensator for

the Lorad Station- Keeping Receiver System, by C. J. Krieger
and R. P. Kempff, CONFIDENTIAL, 7 May 1957

3. Navy Electronics Laboratory Report 806, Simultaneous
Multibeam Phase Compensation, VII: Circular Array Phase
Compensator for the AN/SQS-11 Sonar, by C. J. Krieger
and R. P. Kempff, CONFIDENTIAL, 26 September 1957
4, Stewart, G. W. and Lindsay, R. B., Acoustics,

Van Nostrand, 1930

5. Beranek, L. L., Acoustics, McGraw-Hill, 1954

6. Navy Electronics Laboratory Technical Memorandum
397, Proposals for a Receiving Array for Project Artemis,
by R. P. Kempff, CONFIDENTIAL, 6 May 1960*

7. Watson, G. N., A Treatise on the Theory of Bessel
Functions, 2d ed., Macmillan, 1944

8.) Peirce, B. O., A Short Table of Integrals, 3d-ed.,
Ginn, 1929

*NEL Technical Memoranda are informal documents
intended primarily for use within the Laboratory.

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Bureau of Ships
Code 320

Code 360 Code 335
Code 670 Code 673 (3)
Code 688 (6)
Bureau of Naval Weapons
DLI=3 DMS EL (20,
RUDC-11 RUDC-2 (2)
Chief of Naval Personnel
Tech. Library (No SECRET)
Chief of Naval Operations
OP-O7T OP-O3EG
OP-73 (2)
Chief of Naval Research
Code 411 Code 455
Code 461 Code 466
Commander in Chief, PacFleet
Commander in Chief, LantFleet
Commander Operational Test and Eval. Force,
LantFleet
PacProj
Commander, Cruiser-Destroyer For, PacFleet
Commander, Destroyer For, LantFleet
Commander Submarine For, PacFleet
Commander Submarine Fors LantFleet
Commander, Anti-Submarine Defense For,
PacFleet
Commander Training Command, PacFleet
Commander Submarine Development Group TWO
Commander, Service For, PacFleet
Library
Commander, Service For, LantFleet
Commander Key West Test & Eval. Det.
Commander, Beach Jumper Unit ONE
Commander, Beach Jumper Unit TWO
Naval Air Development Center, Library
Naval Missile Center
Code 5320 Tech. Library
Naval Ordnance Laboratory
Library (2)
Naval Ordnance Test Station, Pasadena
Naval Ordnance Test Station, China Lake
Tech. Director Code 753
Charleston Naval Shipyard
Portsmouth Naval Shipyard
Puget Sound Naval Shipyard
David Taylor Model Basin, Library
Naval Engineering Experiment Station
Library
Navy Mine Defense Laboratory
Code 712
Naval Training Device Center
Navy Underwater Sound Laboratory
Code 1450 (3)
LantFleet, ASW Tactical School
Navy Underwater Sound Reference Lab.
Library
Naval Research Laboratory
Code 5120 Code 2027 (2)
Air Development Squadron ONE (VX-1)
Fleet Sonar School, Key West
Fleet ASW School, San Diego
Naval Underwater Ordnance Station

Office of Naval Research, Pasadena
Naval Submarine Base, New London
Naval Medical Research Laboratory
Naval Personnel Research Field Activityy
Washington, D. C.
Navy Hydrographic Office, Library
Div. of Oceanography

Naval Postgraduate School, Library (2)
Navy Representative, Project LINCOLN,
MIL.
Marine Physical Lab., San Diego
Scripps Institution of Oceanography,
TA TIOP yy
Asst. SECNAV, Res. and Dev.
Director of Defense Research & Engineering
Chief of Staff, G-2, US ARMY
Tech. Library Branch (3)
Army Electronic Proving Ground
Tech. Library
Redstone Arsenal, Tech. Library
Beach Erosion Board, US ARMY,
Corps of Engineers
Air Defense Command, Operations Analysis
Air University, AUL3T-5028
Cambridge Research Center, CRQSZI-1
Rome Air Development Center, RCRES-4C
Holloman Air Force Base, SRITL

National Research Council , Committee on
Undersea Warfare, Executive Sec. (2)
Brown University, Research Analysis Group
Pennsylvania State University, Ordnance
Research Laboratory
University of Texas
Defense Research Laboratory
Military Physics Laboratory, R.B. Watson
University of Washington, Applied Physics
Laboratory, Dr. JH. Henderson
“Woods Hole Oceanographic Institution
bell Telephone Laboratories, Murray Hill, N.
Bendix Aviation Corp., No. Hollywood, Calif.
Edo Corp., Long Island, N. Y.
General Electric Co., Syracuse, N. Y.
Raytheon Mfg. Co.,y Wayland, Mass.
Sangamo Electric Coe, Springfield, Ill.

Lab. of Marine Physics, New Haven, Conn.
Columbia University, Hudson Lab.

Harvard University, Dr. F.V. Hunt, Directory
Acoustics Res. Lab.

University of Michigan Research Institute

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