N
E!
N
E!
N
NSQ UAR E
! AS H I G T O
N
Y OR !
U IVE R
L
L
EGE
CO
MATH E MAT IC S
S IT Y
N
OF AR T S A D S
R E S E AR C H
R E S E AR C H R E !OR T
NVER SIT Y
Y OR! I
N
o.
G R OU!
E M-5 4
N TR ANSMISS ION
E C TI ON TH E OR E M S
AN
D R E FL
C E R T AI
y
b
VIC
T! E R S ! Y
L
JU Y 1953
CT
N
o.
AF l9 ( 1 22 ) 42
-
-
CE RTAI
SMI SS I ONAN
NTRAN
D
L
N
RE F E C TI O
y
b
VI C T!E RS !Y
U
NNERSTT
‘”
E
!ro !e c t D i re ctor
re se a r c h repo rte d in thi s do cume nt h as b ee n ma de p o s sibl e
th rou gh suppo rt and spons or ship e x te nde d by the G e o phys i cs
Re s e a r ch D i re ct o ra t e o f the Air F o rce Cambridg e Re s e ar c h C e nt e r ,
o
I t is pub l i s he d for t e c hni c al
u nde r Co nt r a c t N
AF l 9 (l 22 ) h2
info rmati o n only ) and d o e s no t ne c e s s arily re p re s e nt re comme ny
d a t io n s or conclu si on s of t he spons o r ing ag e ncy
The
‘
.
.
-
-
.
.
J u ly , 1 953
Abstr a ct
The we ll known
-
s c att e re r
a
an a
and
re l ation b e twe e n
p aral lel cyl inde r s
An
.
b i trary cyl ind ri c al bo s s on
ar
riv e d ; he re
is
t he
a
t he
.
ap proxim a te
dis tr ibu tio n o f cyl indri c al
xte nde d to ob t ai n
uni form
.
pl an ar di st r ibu
of
b o s s is
re
re fl e ct io n o f
the
pl ane
re fle ctio n co e ffici e nt for
b o s se s on
a
a
2%
I ntroduction
1
2.
The Tran s nd s s ion C a s e
1
3.
The Re fl e ction C as e
6
.
de
l at e d to the s c a tte r
Tab le of C ont e nt s
1
an
pe r fe ctly re fl e cti n
g p l ane i s the n
p l itu de in the s pe cul ar di re ction
an
e
n al ogou s re fle ct ion the o rem for
-
of
-
a
to t al c ro s s s e ct i on of
xt e nde d t o ob t ain
ne r gy c ro s s s e ctio n
am
am
e
e
plitu de is
its forward s c att e re d
ppro xim at e tr ansmi ssion co e fficie nt for 3
t io
n of
ing
tot al
the
.
u
This
n i form
I n tro du c ti o n
1.
In
wi t h
th i s p ap e r
s c a tt e r ing
t he
In
e
wav e by
pl ane
a
phy s i c al con s t a nts
re fl e ct io n o f e le c tro m ag ne t i c
an d
.
fo rwa rd s c a tt e re d
pe r
uni t l e ng t h o f cyl ind e r ,
he a t
of
t h i s t h e o re m
fo r
an d
as
In
am
an d
a
S e c t ion 3
As
in
an
a
b0 3
89 3
2.
t he
T
t he
he
1
.
as
a
n al o go u s
a
-
p o we r
cyl ind e r
t he
the n d e riv e
we
.
t he n de r iv e
we
re fl e c t ion co e ffi c i e n t
fe r
di s s i p a t e s
co ro ll a ry
a
.
re fl e c t ion p r o bl e m f o r
re fl e c t ing pl ane
an
t he
a
a
c o ro ll a ry
u ni fo rm
a
re al p art
pply i t
an d
rb i tr ary
de rive
we
.
a
di re c t ion o f s pe cul a r re fl e c t io n o f
th e
rb i tr a ry
t o ta
l " e ne r gy c ro s s s e c t i o n "
the
the
a
re al p a rt o f
the
-
0 8 88 ,
a nd
the
pp roxim a t e tr a ns m i s si o n co e ffi c ie nt
s mo o th p e rfe c tly
l i tu d e in
T r an s m i s s i on
-
a
t he
pl an e
t o o b t a in
d i s t r ibut ion of i de nt i c al
Case
.
c
d i me n s io na l p r o bl em o f
-
e
re l a t io n th a t
t o t a l c r o s s s e c t ion i s p ro p o rt ion al t o
rb i tr ary c ro s s s e c t ion
yb
ma
an
co ns i de r
we
,
.
.
The two
a
a
t he
am p
a
.
of
k nown
-
w av e s
c oust i c
;
u ni t l e n g t h
pe r
to o b t ai n
co ns i d e r
tr an s mi s s ion
pp roxima t e
a
-
d i stribu t ion o f i de n t i c al cyl inde rs
we
s c att e r ing
a nd
rb i tr ary c ro s s s e c t i o n
w hi ch i s de fine d
pply i t
the o re m st a t in g th a t
t he
a
pl i tu de i s p ro p o rt ional to
cyl indri c al p ro t ub e r a nc e on
of
we ll
the
s c att e re d r ad i at ion
u ni fo rm pl an a r
a
cylin de r of
a
p re s e nt
!e
‘
the
as
o f in t e re s t in co nne c t io n
a re
whi c h s e rv e s p r ima ri ly to int ro du ce no t a t io n
S c t ion 2 ,
s c a tte ring o f
tre a t c e rt a in t h e o re m s w hi c h
we
fo rmul at e d
as
Thi s the o re m
was
c as e s ,
S.R.
e
.
g.
,
d e r iv at ion i s
,
we
an d
a
t he
s
c a t te rin g of
pl an e
a
w ave by
a
cyl inde r
rb i tr a ry d i e l e ct r i c p ro p e rt i e s ( a s in Fi g
.
1
)
fo ll ow s :
f i rst p rov e d by s e v e r al
!a pp a s ,
b e l ie v e ,
J
.
Appl .
du e
to
wri t e rs f o r
!hy s .
M
.
Lx
a
El
,
,
a
v a r i e ty o f
31 8
!hys .
Re v
The
.
g
,
306
s p e c i al
m o s t g e n e r al
.
z
-
seek
!e
solu t i on of the re du c e d wave
a
whi ch s ati s fi e s
Y
Y
is
1
a
a )
m
E
whe re Y
e
,
in Fi g
.
.
l:
1
1
i
e
c
at
a
s sume d to
be
exp
2
tion (V + k )Y
the
0 for k
cyl inde r ,
a nd
an
an
gle
)
’
(
-
an
iw t ) .
o u t go in g wav e
as
coo rdina te s
r
-
> cc ; the s uppre
we u s e
are
s se d
i llu s t r a t e d
.
S c at te r in g of a p l an e w ave by a cylin de r
po l ar coo rdina te s o f an ob s e rv at io n po in t
8 re sp e c t iv e ly
n is the no rmal ou t o f S
S
r,
0
an d
a
an d
s
.
and
o( s ) ,
hav e in ge ne r al
S e e !app as ,
2
re f .
1
,
f o r de t ail e d tre a tm e nt o f
a
! e)
are
the
poin t o n t he su r fa ce
i s the c oo rd in a te al ong
.
2.
i s o f the fo rm
a:
.
!e
211A
,
ik r co s (O- a
the s c a tte re d w av e , is
time f a cto r i s
Fig
I
pl ane w ave inci de nt
(2)
and
e qu a
2
p re s crib e d boundary conditions on
(1 )
whe re
a
sp e c ia l c a se
.
y
(3)
/
5
o
G
whe re
is t he two
(h)
Fo r
mm )
-
G (r , o)
kr
1,
5
c
ik r
th e
to
a
wave in ci de n t
r
bs o rp t i on
a
flow in t o
r
0
pl i tu de "
an
th e
e
ikpm
1
‘
-
-
) indi c ate s
ngle
a
a
nit l e ng t h of
qa
-
Re
l
Ei
-
boun d a ry co ndi tions
d
divide d
e
writ te n
2 11
Y
fir fi
by
th e
From
e
and
s e ek
so luti o n
a
plitude
.
f (c
am
,c
)
t h e cyl inde r .
as
t ime
t im e
the
-
-
v
a e
r ag e d powe r
v rage d flux
a e
pe r
nit
u
as
rdo
-
de .
Re
S
0
.
fie ld re s po ns e in
the f a r
-
cyl in e r
yb
ma
/
0)
.
-
a u
fi C C OS (¢
de
b so rption cro s s s e ct i o n , whi ch is de fin e d
a
is t ake n ove r S
a
c i rcle of ra d iu s
l eme nt ary c o n s id e r ations
r e n cl o sing S ;
we know
th a t
the
the
s e c o nd
i nte g ral s
are
qu al .
S imi l arly the s c at te rin g c ro s s s e ct io n ,
-
s c at te re d
e
fur )
'cros s s e ctions ' of
The fi r s t int e g r al i s t ake n o v e r
e
at
5°
-
re l at ionship o f the f orward s c atte re d
s c at te ri ng
an d
of inc ide nt wave ,
( 5)
f (0 , c
am
we de monst ate the
The
a ea
%
do no t int e nd t o sp e c ify
!e
R ath e r
to
2
's c a t te rin g
d ire c tion
Gre e n '
s func tion
(k l;
~ H r
( )
whe re the
to
o
co n s e que ntly (3 ) re duce s
Y
a
1)
(
H
ds ,
hav e
p we
g
and
gn
Y (p )
dime ns io nal fre e sp a c e
i
l]
G
é
ne rgy
(6 )
po we r
de n s i ty ,
q 3 Re
per u
m
1}
nit le ng th o f cyl in de r
be
writ te n
w hi ch is de fine d
d ivi de d by
as
the time
-
the
—
f
r dO
a
Re
7
5
-
v rag e d
a e
v r ag e d in cide nt
a e
as
‘
t ime
ds .
I n ge ne ral we ha ve
7Q
( )
n
'R9
i
!
-
a(!
it
)
c
i
+1
c
an
h ave
we
s qu a re
a
t he
te rm I
7
( )
Fr om
.
i
qa q
Com p ar ison
-
cyl inde r
—> oo in
now
u se
se
ctio n
y
of
a 9 a.
the
)
the
.
Q
f a c to r in
is
Our
a
y
a. ,
'i s
r
t he
no rm al i ze d
as
he at
be
re
r gy flux
e ne
s c atte re d
and
fo rwar d s c a t te re d
( 8) i s t o
of
am
plitu d
3
e
p l a c e d by q o f
ob t a i n
we
If (0 c ) I d0
3
coro l l ary
o rigi n ,
we
.
0
I f the cylind e r is l o c ate d
r
me e ly t he
di ffe rs by
a
at
x
-O
,
have
ph as e of
slowly v a ryi ng f un c t i o n o f
de finit i on o f
re l a t i on
the
r
p has e
1
2
a
a
aY
an
d i s s ip at e s
an d
q o f (6 )
1
.
5
co sO'
If
3
w av e
re al p a rt o f
fo rm o f
( 8) t o de r ive
at
.
cylind e r , i
the
( 9)
tu
7
)3
b s o rp t ion , the n
the fi rst
'ra t he r than
y
c
e
a
Re f (
!e
ik pc o s (g
ob t a ins from t he in ci de nt
th e re is no
le tt ing r
.
-
-
r a di atio n , is pro po rtio nal to
If
(Y
to t al cro s s s e c t io n "
yie l ds the we ll k nown
i
f
-
"
an
Re f (a . a
Q
thu s t he to t al c ro s s
t he
TI?
an d
of
( 8)
+
(who se in te g r ate d v al u e v an ishe s ) wi thin
o b t a in the
we
e
Re
1
5
-
+
Re
d8 ,
dd e d
b ra ck e ts
Q
!
R!
q
8Y
whe re
d8
a!
I
de
)
1 81
41
x
;
in ci de n t w av e
t he
y
'comp ar e d to
f a cto r if i lb
,
r
y
at
' '
s im
-
from tha t o f
yy
x= 0 ,
,
re f .
r
the n
1
.
'
.
5
-
N
(1 0 )
2—
rco sa
2
Y
m
whe re the in te gral
in the
ri g ht half
-
(11 )
pl an e
0, n
(
1
-
v al u a te d by
)f( n
t he
e
k
me tho d o f st a t io n a ry ph as a
have
we
p/kc o sa
flux ,
uni
pl ana r
a
s c atte re rs
tim e s the le ft side of (1 1 )
ve r ag e singl e
-
h
s c a t te re d w ave s )
tr an smi s sion co e ffi cie nt
(1 2 )
T
c
i t le ng th
u
a
p ar al le l cyli nde rs
of
yxis
-
and
a
pproxim a te ly
nity co rre sp o nd s to
2:
He nc e by
.
e
t he
Q is
.
s c at te re r ,
and
p
t he
s e c 0.
e
u
Q p
l
ne r gy
p/k
se c a ,
I
'rem ove d '
incide nt
h
Twe r slc
.
e ne rg
co s
<< 1
a
e
an d
th e
e
.
l eme n ta ry
the
e ne
c ircul ar cylinde rs ;
.
se e
!e
u
nit
rgy s c at te re d into
On the S c atte re d Re fl e ction
1 950 ) p . 1 5 ff
y
ph si cal
from the transmi tte d wave by
I
m uis s ing '
from
y
,
we ma
( the numbe r of s c a tte re rs illu mina te d by
(!e ignore he re
wave
ni ty
as
inc ide rt wave ) t ime s Q is t he
mi tt e d
.
if
qual to
ywrit
o f (1 1 )
me an s
oul d h av e b e e n wri t te n imm e di ate ly from
side rations
of
un
transmi s s io n of
the
l e ft s ide o f (1 1 ) to t he int e rfe re nce te rm s o f the inci de nt
t he
Thi s re sult
in
the n the t r ansmi s s i on co e ff i cie nt is
,
a
ca se
se c
form di stribu tion o f i de ntic al
a
1
an d
V.
Q
xp re s s i on follows from
v e ra ge numb e r o f
I f p is t he
Cons e que ntly
.
(1 1 ) is of p hys ic al int e re st in co nne ction wit h
wav e thro u gh
min us p
0,
for x
se c 0.
l as t
the
Eq .
.
c
2Re
whe re
a
was e
0
for x
of
r
a ea
unit
u
nit
one
ar e a of
o f the t rans
r
a ea
of the
S c al ar Wav e s , RIM 22 (Math . Re s
-
have shown (1 0 ) to be
e
xa c t for
the
Multip le S c at te ring o f Wav e s by !l ana r
Rand om D is t r ibutio ns o f Cyl ind e rs
a nd
con
!o s s e s ,
to
b e p ub li s he d .
.
-
tr an smit te d wave
3
s ince it i s in g e ne r al sma ll comp are d to
,
The Re fl e c t i o n
.
su
a
fe c tl
Fig
rf a c e com po s e d o f
yr
.
e fl e
pl ane
ctin g
n alo gou s re fle ctio n
a
an
a
pr obl em o f
Fi g
as
re fl e c ti on o f a p l ane
a b o s s o n a smooth
p e r fe ctly re fl e c t i ng p l ane
y
t he
(1 3 )
whe re
to t al f i e ld
1 (0 , a
I
i
,
t he
)
I
Fi g
.
3:
pl ane w ave in ci de nt
1
p
a
sm ooth pe r
im a g e p robl e m : s c at te r ing
o f two pl an e w av e s by a
cyli n de r
.
as
i
a
2.
.
2 : S c a tt e re d
wav e I i b
wr i te
.
rbitr ary cyl indr i c al p ro tub e r an ce on
.
we
Q )
Cas e
we now c o n s i de r t he
on
6
v
b
,
in c i de n t wave , i s g ive n by ( 2 )
an d
whe re
7
-
(1 h)
-
1
t
?
t
is the wave re fle cte d from the
si gn is to b e t ake n for I
at
/2 ( c as e
0
the
!) .
n
0
pl ane in
p l ane
b se nc e of
the
a
b o s s ; the
th e
at
infinity
prob lem
%
g
0
»
bo s s , s atis fie s
.
2 is
of
m inu s
n
The fhnction ! , the w ave s c att e re d by the
b
The
.
)
/2 ( c as e A) and the plu s si gn for
0
at
The p rObl e m s hown in Fig .
-
e
n+a
‘
radi ation c o ndition
l e ft hand
ikr co s (O
e
Fig
qu iv al e nt t o tha t shown in Fi g
3
.
yb
ma
e
e
formul at d
.
3 int he
e s s e nti ally a s
in
S e ction 23 we have
(1 5)
r
The
of
v alu e
g
a
e
p ],
(1 5) in
The fun ctio n 1
as
from
I
ea
0
c h othe r
am
pl itu de
The
.
we mu st
n
an
y
pl ane is i de nt i c all
e
s
,
a
t
e s s e nt i al
‘
I
yid ntic l wi th t
s
o f co s
ay
g
i s fo rmal l
/Q ,
0
at
the le ft- hand
h
s e nt tw o s c atte ring
4
v a
i ?
e
ur
,
di ffe re nce i s
o
of ( 3 )
and
bu t the pre
diffe r from f of
one
we ll
as
Thu s for c a s e
o f symme try .
have
(1 6 )
and
s imil arly for c as e ! ,
3
0 at o
re quire
we
8+ (osa )
v
Fo llowing the
pro ce dure of
tot al c ros s s e ction
-
re fl e c tion ' ( s e e
and
Fig .
The ab so rp tion
t he
2)
and
S e ction 2 we
s c atte ring
wav e s ,
and
a
c count
-
the
a
7
n alo gy to ( )
e
quations
f a ct t h at
fa c il it ate s exte n s ion
we now have in
.
plitude in the
am
s c a tte ring c ro s s s e c tio n s
t h at t he ri ght hand side s o f the
.
a re l at ion
"
be twe e n the
spe c ul ar
di re ct i on of
g (n
-
i z at ion t ake s into
now se e k
we
are
are
t o be
d e al
y(5)
d by
2;
giv e n h
‘
he re
divi de
wi t h
o f t he f in al re sults t o the
'
and
e
xce p t
this no rmal
two i nci de nt pl ane
probl e m o f Fi g
.
2.
8
-
(1 8)
Re
q
1
+
-
p
3-
*
)
b
1
(
1
55
Y
I
p
de
)
h
6 (I
an
2ik
-
whe re
a
dde d
the
Q
-
qa+ q
F or c a s e A
-
Re
1
te rm
b ra cke ts
v anishe s ) with in the s qu are
(1 9 )
For
case !
( 22 )
( 21 )
we
Q:
( 22 )
and
§ {E
we now
4 19 9 00 5 614
9
-
an d
thi s
i
n ge the si gns o f I
we
for
e
q8
e
ds .
s
;
-
re l a t i on
1
wi t h
ik pco s (¢
y try p
mm e
ro
pe rty ( 1 6 )
we
o b t ain
s e cond te rm of ( 20 )
Re g ( n
+
)
e
ob t a i n
and
x ce p t for the si gn
an d
of
t he m n ctio n
Y
fo r c a s e
p
He n c e , for
.
u
ni fo rm ity,
de f ine d in
A) .
on
e
ls e
suppre s sing the
n
.
in ci de n t t o th at for I
no co nne ct ion
i
Re g (
h ave no t s im ply
4
-
it he r c a s e
1
q
o bt aine d
ha s
a ,c
a
Q
I
)_e
to b e t he ne g ativ e
whi c h is the de sire d
with
the
fo rm al ly ide nt i c al
yw ri t
N
ot e t ha t
of
-
t we ma
( 23 )
(Y
Re g_ ( n
1
s m
+
g
%
de fine
sub s c r ip t
L
the
and
ch ang e t he s i gn
y i plych
we m a
inte g r ate d v alue
(wh
H e nce
.
an
E
are
)
—
-
H e nce from th e de finition
Q:
p
de ,
6(
”
J
(s )
)
p
)
have
we
( 20 )
( 21
Y
(ria p )
Re
q
h ave
we
(I i
D
d de d
incide nt
f (n
-
the
a
c , n- a
)]
t he
.
re sul t fo r
a
y
s
m me t
r i c al cyl ind e r
I n th a t c a s e we wou ld have u s e d
Q1
p re s e n t p ro bl e m
Q
p
.
S
Re
( 8)
2Q , howe v e r ,
1
9
-
I n t e rms of the
r e sult fo r
%
1
Q+ u
1
-
fi rs t fl inc t ion i s
the
whe re
f
[ (u
!e
re s p e c t t o the pl ane o f s
s c atte re d d i re c t io n
)
fl it
4
-
v al u a t e d
yt
m me
ry ,
G am
and
S e c t i on 2 is cle ar fo r t he c a s e o f
we ma
yr wri t
e
( 22 )
e
as
a] :
s p e cul
t he
at
t he
s e c ond i s
e
th e
diffe re n c e b twe e n
The
.
e
a ,a
p l ane wave
single
a
re fl e c tion w ith
n gl e of
ar
a
v alu at e d in
the
f o rwa rd
p re s e nt prob le m
and
th at of
e
In
non- abs o rpti on .
a al
n
o gy to
we
now
we
h ave
o b t ain
( 214)
ia
l
R e g (n
t
t he
am
[
Fo r c a s e
A
a nd
-
Y
-
c , u- a
in c i de nt
i
a
dde d
e
( 23 )
y
h ave onl
one
in Q o f
8
singl e pl ane w av e ,
a
i
i
a
«
f (0 , n
—
fl
simply c h ang e
we
ne r g i e s wh ic h
d e r iv e d fo r
wa s
.
in c i de nt
but
f a ct o r of
2
qu a dr a t i c
ar e
w i th o nly
le ft
t he
s i gn s o f
t he
) t e rms
t he f (0 , c
E qua t i on
.
If (0
t he
t he
p roble m o f Fig
.
e na
b le s
2
) l t e rm o n
,a
.
wav e ( th i s
e
limi nate s
inte g ra te only ov e r
we
Ho we v e r , i t
3.
.
by v i rtue o f ou r no rma l i z at ion ;
He n ce in us in g
i t he r int ro du ce the f a cto r
t he
we
or
e
n
e
.
t ake q
a
and
q
as
i n (5)
lse int e g rat e o nly ov e r
t he
.
2 we
in tro du ce d
we
3n /2 ( t hi s
<: O
l so ho l d s
in Fi g
,
f a c to r o f
t he
/2
i.
a
re s t o re s
t he
(6 )
a nd
a nd
l e f t hand
-
.
E q.
of
the
)]
s e co nd te rm o n
t he
for t he p ro ble m of Fi g
pl ane
to
2
.
E q.
e
plitu de s c o rre sp on d ing
f(n
have no t
we
us t o c o rre l ate
r i gh t
o
!
Re f ( n
C l e a rly
é;
« -
He nce in t e rm s of
fo r c as e
11
( 23 )
The
t o t al
e
is
a
re fl e c t i o n th e o re m
p re s e n t
a
n al o g ou s
to
t he
"
t r a n sm i s s io n
t he o re m "
t he o re m st at e s t ha t
ne rgy c r o s s s e c t ion of
-
an
a
rb it r ary b o s s on
a
pe r fe c tly
re fle c t in g
10
re fl e ction
in t he dire ctio n of spe cul ar
!e
( 23 ) to de riv e
now
u se
and
( 1 0 ) wi th
for ( 9 )
the Y
re sul t o f (1 0 ) fo r x
t he
( 25)
I
p
t he
or
em
o rol l a ry
o
0
a
pp li e s
in
of
a
by
.
a
Y
b
ane .
l
p
and
a.
a al
to the transmi s sion co e ffi ci e nt o f (1 2 )
og
Q
inte re st in conne ct i on
bo s s e s on
a
as
se e a
conve n tio n intro du ce d for
di stribu tion
n
1
-
unifo rm
of
p ro c e e d
re pl a ce d by g ; he re only
‘
Re
the
!e
nal o go us to
we find
c cordanc e with
' s i c al
l
pv
the
He nc e
h
N
ii
is
c
re p l a ce d
2 Re I
whe re Y
Thi s
a
of
p lane
wi t h
a "
re fl e ction fr om
stri ate d su rf a ce "
now hav e the
we
a
In
re fle ction
c o e ffi cie nt
( 26 )
R
whe re Q
at
p
se e c
the spe cul a r
The
:8
Q p
l
is t he
an gl e
e ne
p re se nt re sul t s
dim e n sional p r o blem s .
r gy
o f t he
c an
p/kco sa
s e c :1 ,
.
1
'
mi s s ing "
p la ne
be
,
are a
o f th e w ave re fl e c te d
.
re adi ly
ex te nd e
d to
the
a al
n
o gou s t hre e
LE
T O
T T E R S
now sho w tha t E q (3) is a conse q uence of E q (9 ) an d E q
! e m ay apply (I ) of E q (9 ) to a prac tical se t up provided
tha t p /le c 0 5 <<1 and provided tha t the wid th and heigh t of t he
bosses are smal l co m pared to p" ; the range of i n tegra tion is to be
ta ken as the i llu m inated region If the incid n t beam wid th say (1
is ve ry much la r ger than al l wavelengths i nvolv d then we may
i n general use i n fini te li m i ts i n the n com ponen t because of the
'
expon n t of E (y
) the m ain con tribu tion of the in t gral arises
from the im m edia te vicini ty of the sp cular or stationary poin t and
is conse q uen tly p rac tically i ndependen t of the wid th of the
i llu m ina ted r gion H ence for the n componen t we may use E q
n i tion
(1 1) and the de fi
gp sea to w rite the leading term of E q
(9 ) as ( l
) n Fo the r co m ponen t (the incoheren t scat te r
howev r the in tegrand is slowly varyi ng and the in tegra tion
shou ld be ta ken on ly ove r the illu m ina ted region If this re gion
is s m all compared to the distance r of the observation poi n t from
i ts cen ter
if
then we may w ri te this ter m as
pd
| is the normali ed
[ 7 T he func tion
sca t tered energy flux of a boss a t the cen ter of the i lluminated
region 7is the uni t vec tor from this boss to the observation
poin t and p d sea is the nu mb r of i llu m inated bosses (For smal l
circu lar bosses IE I m ay be ob tained di rec tly from E qs
and
H ence we may w ri te
( 1 3)
(I) (1
)n + p d se t» E 7
!
e
.
.
.
-
,
.
a
e
.
,
e
,
e
e
e
!
e
.
.
'
a =
—
o
.
’
r
.
e
):
!
o
,
,
.
2
,
0
2
z
.
0
e
:
,
.
2
.
—
z
2
o
0
.
I
E D
is the abso rption c ross sec tion H oweve r i t is simpler for this
case to ob tai n R di rec tly from th e 7
! co m ponen t of E q
T hus
'
l
if w e w ri te E (y
is
)
flop ) where
a slowly varying func tion of 3 co m pared to the exponen t then
C onse q uen tly we have
.
his expression is to be used as follows a t the specular angle we
ignore the t er m i n E 1
the energy sca t tered into the te
flec ted beam ) since i t is on ly of the order of d/r ti m es ; thus we
have
:
2
0
,
(I)
~
1—
nz
0 =
R
,
6=
—
a
(14)
.
have the refore !usti fied th e use of E q
provided tha t
On the o ther hand i f 0 is n o t near
we
<1 an d
p /k<
ignore the first ter m of E q ( 13) and use
!
e
.
—
,
a
,
.
,
.
-
M
!
,
"
2
,
,
(D
)R e U (
com
( 1 6)
where f(
) is the value of the sca t tering ampli tude a t the
specular angle (M os t generally we exp ress f(0 ) as an in tegral
over the su rface of the boss ) I t can be shown that R of E q ( 16) is
iden tically l ( l ) T he leading ter m of R of E q (16) i n
volvin g lea (or an e q uivalen t para m eter) is of th e lowest order of
m agni tude fou nd i n E e g
for the ci rcular cy
lin
d ical case etc [ C o m pare wi th E q
! e have assu med i n the abo v e tha t the inciden t bea m is
perfec t ly collim ated and also tha t the pic k u p is highly di rec tional
I n prac tice one wou ld ta ke i n to accou n t the radia tion characte r
istic s of both the sou rce and pic k u p N
ote also tha t since ou r
expressions are averages they shou ld be compared wi th the average
of m easu re m en ts over a large nu m ber of approp riate su rfaces
T hi s w o r k w a s p e rf o r me d a t ! a s hing t on S q u a r e C oll e g e o f A rts a n d
Y o r k U ni v e rs i t y a n d w a s supp o rt e d in p a rt b y C on tr a c t N
S ci e nc e N
AF
42 w i t h t h e U S A i F o r c e t h r o u gh sp on s o rs hi p o f t h e G
p hy s ic s R e s ea r ch D i r e c t o r a t e i t h e A i F o r c e C am br i d g e R e s ear ch C e n t e r
A i R e s ea r ch a n d D e v e lo p me n t C o mma n d
V T w e rs k y J A pp l ! hy s 22 82 5
s ee a l s o J A t S A
22 53 9
2 3 3 36 ( 19 5 1 ) f o r t h e a co ust ic c a s e
by
!l
a r R a do D i t ib
V T w e rs k y M l tip l S tt r i g f ! a v
tio
d !o
M a t h ema t ic s R e s ea r ch G r o up N
f ! a r ll l C yl i d r
Y o r k U ni v e rs i t y ( t o b e pub li s h e d )
f o u n d (s ee r e f e r e nc e 2 ) t h a t b o t —h R a n d R — l a s
!
/Z furt h e r
m o r e t h e t o t al in t e n s i t y a s 6
/2 r e g a rd l e s o f p ol a r i a t ion !
us e d t h e pr o b a b ili t y d i str i but on fu nc t ion s o f li q u i d st a t e t h e o ryt o d e fin e
t h e e n s em b l e o f surf a ce s t h e r a n d o m n e ss o f t h e d i t biti b e ing
sp e ci fie d b y t h e r a t io o f t h e a v e r a g e se p a r a t ion o f tw o s c a tt e r e rs t o t h e i r
m ini m u m s e p a a t ion ! f o u n d t h e b e h a v io r a t 9
/2 t o b e in d e
p e n d e n t f t hi s r a t io ( d al s o o f t h e d e l e c tr ic pr o p e rt i e s o f t h e b o ss e s a n d
o f t h e v a l u e o f l ) pr o v i d e d t h a t t h e m ini m u m s e p ar a t ion w a s l a r g e co m
p a red t o a n d
No t e t h a t f o r t h e ci r c u l a r b o ss t h e n co m p on e n t o f E q (9 ) yi e l ds t h e
r e su l ts o f E q
pr o v i d e d t h a t t e r m s t o t h e o rd e r o f (k ) a r e r e t a in e d in
ni
t e ! d i d no t g e t
E (y
) a n d pr o v i d e d t h a t t h e li m i ts on t h e in t e g r a l a r e in fi
t h e pr e se n t r e su l t in r e f e r e nc e 1 s inc e w k e pt o ly t h e (t ) t e r m s o f
t h e r e su l ts o f r e fe r e nc e 1 a r e in g e n e ral co r e c t only t o t h e o rd e r
i
o f (tI f)t hiosr con
(k )
d i t ion i s no t fu l fill e d t h e n we ma y us e t h e r a d i al e n e r gy flu e s
ot hf rsecfae sree nct hee 1 f o rmat hye sfihoni wt e ad mi strinii but
ion
s t o t h e o rd e r o f (t ) o r (k ) F
m u m a t t h e sp e c u l a r a ngl e i t h e w i dt h
fl
or ef prt heese nint etde rfbeyr e1ncine b eam i sannadrrcon
o wseer tuheannt lyt h et wh ei dtv ahl uoef taht e0 p l mabyeam
be
Eq
q
s ma ll e r t h a n t h e v a l u e s in t h e i mme d i a t e v ic ini t y o f
—
a
a
,
a
.
,
.
—
a
—
-
ao
r
,
.
.
.
,
.
.
,
.
.
-
.
,
-
.
.
ew
.
o
-
.
r
.
r
I
r
.
.
.
.
.
.
.
2
u
.
ns o
a
-
roz p d seca
lE
(15)
2
,
ou tsid e of the re flec ted beam we consider on ly the incoheren t
scat tering
! e note tha t the resu l ts can be ex tended to the case of a b
sorbing bosses by replacing by
wh re
seC
ie
.
.
,
.
a
a
,
e
oc
.
m
.
.
.
.
.
cou s
.
.
n
e
ca
e
n
e
e s an
sses
n
an
e:
o
m
s r
u
ew
.
.
.
3
e
;
= a
—
.
u
nr
a —n r
t
:
s
z
e
.
i
!
r
—
e
.
an
o
ea
a
= a
-o
1r
i
,
.
.
.
a ‘
.
’
e
ea ?
e
.
.
.
.
a 3
ea 2
n
e
.
on
s ru
.
.
(D
.
eo
o
.
T
T OR
qa
,
,
:
,
T H E
r
.
6
x
.
ea 2
l
ux
i
o 3
.
.
"
!
.
S
e
.
,
e c u ar
— a
.
— a
.
.
or