Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) + alpha(2), for the cases of a finite cylindrical region and a region of rectangular parallelepiped. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r  r' and r  4', where r and r' are distances from the origin to the point of observation and source, respectively. When the wave number approaches zero, the results reduce directly to the potentials of variable densities

Fu, L. S.

Zhong, W. F.

English

NASA Technical Reports Server

NASA Contractor Report 3750 
Volume Integrals Associated Wit 
the Inhomogeneous Helmholn-E$ 
II - Cylindricd Region; Rec~mzgdur Region 
W. F. Zhong and L. S. Fu 
GRANT NSG3-340 
DECEMBER 1983 
LOAN COPY: RETURN TO 
AFWL TECHNICA!. X3RARY 
KIRTLAND AFB, NM 87117 
25th Anniversary 
1958-1983 
https://ntrs.nasa.gov/search.jsp?R=19840006458 2020-03-21T01:19:54+00:00Z
TECH LlBRAtiY KAFB, NM 
1lnlll~luI~I~Illl~l~l 
OOb2351 
NASA Contractor Report 3750 
Volume Integrals Associated With 
the Inhomogeneous Helmholtz Equation 
II - Cylhzdrical Region; Rectungdur Region 
W.F. Zhongand L.S.Fu 
The Ohio State University 
Cozumbus, Ohio 
Prepared for 
Lewis Research Center 
under Grant NSG3-340 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Branch 
1983 
I. INTEGRATION OVER A FINITE CYLINDRICAL REGION, Fig. 1 
The integrals, Eqs. (12,13) [l], see also [2], are of either one of the 
following forms: 
4O = ~~l(x')p(y')q(z')sdv' (11 
I$' = jIjp(xf,yf,zl) 
(21 52 
k "", Ilk 1-1 dv' 
axfLay’ ax’ - - 
4= = yPw~Y’,z’l ax,gay,;;x,n-bk {y+ dv’ (3) 
Letting x1=x', y'=5cos8, z'= <sin0 and dv'=dx'dy'dz'=gd<d0dx', these 
integrals can be further evaluated as follows: [3] 
(a> 
0' = ~~~(x')P(y')q(z')sdv' 
n 
= (q-l)! !(s-l)! !(2r) (a2+q+s) (2 %l+p) if p q s all even - - 
(q+s)!! 2+q+s l+p s , (44 
=o if any one of p,q,s is odd.(4b) 
where according to the definition of factorial, 
(q-l)!! = (q-l)! 
2(q-1)(q-11! 
= l-3.5.. .(q-1), q even 
(q+s)!! = 2 (w(w! = 2.4.6...s(s+2)...(s+q), q,s even 
and a is the radius of the cylinder, 1: the length. 
(b) n=O, I$': 
0' = [/(x')p(yf)q(z')s F dv' 
(Sal 
(SW 
= ~j~(~~)~(y@>~(z~)~ y (-l)m(';~~::!'r1)2mdvf 
s-2 m=o 
1 
= y (-Urn g;;! Smtp 
m=o 
where 
S m,p = ~~~(~~lp~~~lq(~11s(r~~2mdv~ 
Using the multinomial formula, [4] 
(6) 
(71 
the integral 
s = 
m,p c 
I 
~]/(x')2ml+P(y')2m2+q(z~)2m3+sdv~ 
ml,m2,m3 ml!k!!m3! Q 
ml,m2,m3 ml!':!m3! 
1 
(p-l)! !(u-1)!!(2a) 2+Jl+v 2R1+X = (p+v) ! ! (g&(r) 
m=0,1,2,... , if A,p,v all even 
if any one of x,~,v is odd (9) 
(p+u)!! are defined as the same as (Sa) and (Sb), 
=o 
where (p-1) !!, (v-l)!!, 
and, 
Y= 2ml+p 
p = 2m2+q 
v = 2m +s 3 
m = ml+m2+m3 
(c) n=O, Qc: 
$ = ~/~(x')~(~')~(x')~ F dv' 
n 
(10) 
where 
X = 2ml+p 
p = 2m2+q 
v = 2m3+s 
ml+m2+m3 = m 
(13) 
The integral on the right hand side of Eq.(12) can be evaluated as 
follows: 
Ipx’l~(Y’)w % = 
= (j-l-l>!!(w-l)!! (2T).I 
(u+v) ! ! if u,v all even 
(144 
= 0 if any one of P,V is odd. (lab) 
In the above expression 
+k-1) (k-l)" ' 
+ (-1>2 k! ! SW !, (xlX+k-l (a2+xf2) 1/2-xlX+k)dx~ 
where 
k= l+y+v 
is odd, and 
(k-l)!! = 2 ! = 2*4...(k-1) 
(k)!! = k! = 1*3...(k) 
(15) 
(161 
C17d 
(17bl 
3 
The integrals 
A+2t-2 2 (a +,f2)1'2dx' 
k, 
I 
= +t7l(-1) 
tl+l (k +l)(kl-l)...(kl-2t1+3) 
(k,+2)k,...(k,-2t,+4) (a) 
2t1-2 
- (El 
kl-2tl+l 
. ( a2+Q2) 3/2 
= 
1 
kl 
+ c-11 
2 (kl-l)!! 
(kl+2)(kl)! ! 
2ak'(am + a2she1 3 ) if X,p,v are even 
= 0 if any one of X,p,v is odd. (18) 
where 
kl=X+2t- 2 (19) 
is even, and 
(kl-l)!! = 
(kl-l)! 
kl kl 
= 1.3.5...(kl-1) 
2(F-l).(z_-1) ! 
5 kl 
(kl)!! = Z2 '(+ = 2.4...(kl) kl is even 
(20al 
(2Obl 
The integral 
X+k-1 2 (a +x12)1'2dx' 
-2 
-a. k, 
L 
= + ttl (-l)t2+1 :~~~~~::i:1:;;::::;::2+3)(a)2t2-2.(1)k2-2t2+1. (a2+g2)3/2 
2 
k2 
= (-1) 2 
(k2-l)!! 
(k2+2).(k2) !! 
.2ak2(p,m + a2shm1 k) if X,p,v are even (21al 
=o if any one of X,p,v is odd. C21bl 
4 
where 
k2 =A*k-1 =k+p+v 
The integral 
= 2aX+k+1 
X+k+l 
= 0 
if X,p,v are even (2W 
if any one of X,u,v is odd. (i-1 
II. INTEGRATION OVER A RECTANGULAR PARALLELEPIPED, Fig, 2 
In the case of a rectangular parallelepiped, the integrals in (14), 
(lS), (16) can be evaluated as follows: 
(4 
(to = ///(x')P(y')Q(zr)Sdv' 
a 
= (p+l) (Q+81] (S+l) (a) 
Q+l(b)s+lcQ)P+l if p,Q,s all are even Wa) 
= 0 if any one of P,Q,S is odd. Wb) 
where R,a,b is length of the rectangular parallelepiped toward xl,yt,z'- 
direction, respectively. 
(b) n=O, 4' 
$' = ~//(x')~(~')~(z')~ 9 dv' 
n 
= Ill(X')P(Y'>Q(Z')s i (-l)m~~(r)2m 
52 Ill=0 
= : wm g;)! 
Ill=0 
sm p , 
and 
S ,,, p = !//(x')P(y')q(z1)S(r1)2mdv' , n 
c Bm! 
x+1 
= 
ml ,m2 ,m3 ml!m2!m3! 
m=O,1,2,... if X,u,v all are even 
= 0 if any one of X,n,v is odd. 
(26a) 
Wb) 
where 
m = ml+m2+m3 
A = 2ml+p 
p = 2m2+q 
v = 2m +s 3 
(cl n=O, $'. 
4' = ~;~(x')~(~')~(z')~ w dv' 
= ; wm & cm,p 
m=o 
(27) 
(281 
and 
c 
I 
c = 
m,p ml,m2,m3 ml!L!!m3! 
~pX’)~(Y’lw 5 
= 
ml,m2,m3 ml!mTfm3! -II 
c JL(xf)A[ja(y’l~(P - dz')dy']dx' (29) 
-a -b m 
where 
A = 2ml+p 
p = 2m2+q 
v = 2m +s 3 
(r2 2 2 = x' +y' 
ml+m2+m3 = m 
The integrals in (34) can be evaluated as follows: 
(30) 
= nv (,,)V-2t+1(x,2+y’2)&~ + (ml)‘)/* 2(w-1)~~(x~2+y~2)V~2sh-1JX,~~y,2 
(v) ! ! 
if v is even (314 
=o if v is odd. (31bl 
where J$ is defined as an operator as follows: 
TJ 
= & ,“:’ C-1) 
t+1 (v+l)(v-l)...(v-2t+3> 
= v(v-2)(v-4)...(v-2t+2) 
(v-l)!! = (v-l)! 
,(:-l)($l)* 
= 1.3 . ..(v-1) 
(“)I! = $3 * = 2*4...v , v is even 
(324 
(32bl 
(32~1 
(ii) 
la ~~(~y'(~q' dy'dz' 
-a -b Lpi7 
= nv n~,n2(-llv'2 &(b)v-2t+1(x1)2nl In 
+ . 1. (-l)"'& 2(;;;i;!(b)(x']2i1 I2Wl 
ll'l2 
- illm2(-1)v'2 +-g :';;;;;!(b)3(x']2i1 13(x’l 
,l 
+ 1 ! (-l)v’2’j & 2~~;:~*’ 22j~~~~~~2j+ll (b)2j+1(xl)2i1 
il,i2 j=2 
14(x’) (33) 
where 
IIWl = Ja (Y'l 2n2+p ,f,--- dy' 
-a 
~-2tl+1(b2+x,2)t1-1/(a2+b2+x,2)3 + (4)5/2 2(5-11!! 2+x,2]</2 
. (am + (b2+xf2)shW1 a ) 
m 
8 
= 0 
In the above expression 
if u is even C 34al 
ifu is odd. C34bl 
2 52 
% = - v+l f c-11 
tl+l (E+l)(E-1)...(c+2tl+3j 
tl=l (5+2X . ..(E-2t.+4) 
(E-l)!! = 1*3...(5-1) 
(5+2)!! = 2.4...(5+2) 
5 = 2n2+u 
5 is even 
5 is even 
(35) 
(364 
C36bl 
(371 
The integral 
12(X’) = la(Y'l 
2i2+u d , 
-a w 
= nE (al 
S1-2t3+1 
(xJ)t3-1Jav + C-11 
Cl/2 2(5,-l)!! (x,2)51'2sh-1 a 
1 x' 
if p is even (38al 
= 0 if u is odd. ( 38bl 
where r[ 
El 
is defined as the same as(32a), only if replace v with El and 
5, = 2i2+u (394 
(6,-l)!! = 1*3...(51-1) 51 is even (39b) 
(cl)!! = 2.4...$) Cl is even (39cl 
The integral 
51 
13(x’) = L: 4* W' 
= ";,&(a> 
cl-2t4+1 
Cx'2) t4-1/Q17 
ST2 
22$-l)!! 51 -- 1 
+ c-11 (51-2) !! (x’2)2 (sh 
-1 a 
x’ - &.&I 
9 
=o if p is odd. 
where 
5, 
if u is even 
2 +; 1(ml) II’ =- t4+1 (51+11(51-1)...(51-2t4+3) 
51 cl+1 t4=1 (El-21 (5-4). . . ($-2t4) 
( 4oa) 
( 4Obl 
(41) 
The integral 
14(x') = !a (Y+ dy, 
-a d(~~Z+y’~)~J+l 
= iTff (a) 
cl-2t2+1 
51 
2 t2-~~Ca2+x,212j-1 (x' I 
(j-1)(j-2)...(j-to) 
(2-2j) (2j-1)(2j-3)...(2j-2to+l) 
E. 
+ c-11 
s,/* 2(C1-l)!! 
-1 -- 
(Cl-2j) (El-2j-2)...(2-2j)(2j-l)(a)(x'2)2 
1 1 
' J(a2+x,212j-l 
if p is even ( 424 
=o if u is odd (42bl 
where 
(l/2 
II" = 2 1 (-1) 
t2+1 (~l+l>(~,-1>...(sl-2t2+3) 
5 51+l to=1 (Cl-2j)(Cl-2j-2)...(cl-2j-2t2+2) 
(43 
10 
- 
(iii) 
= "v"s,l~n2(-l)v'2 --&(a)'-2t1+1~(b)v-2t+1~Il 
+ (-l)E/2, Z(S-II!! (b)V-2t+l~I 
v (5+2)!! 2 
+If 5l i Ii (-1)v’2 & w(a)E1-2t2+1. (b) .I3 
1' 2 
2(5-I)!! 
ts,)!! (b) -I4 
- II’ Ii 
5l i 
(-1)V'2 
1' 2 
$+ :~~:~/![a)~1-2t4+l~b)'.Ig 
. . 
51+v-1 
- ill.2(-1J 
2 i! 
qq 3(v)!! 
2(5-l)!! 
91 (v-l] (b)3*16 (5-2)!! 
2(v-l)!! (2j) ! 
(v) ! ! 22j(j!J2(2j+l) 
(a> 
cl-2t2+1 
.(b)2j+1.1 
7 
+ c 
il,i2 j=2 to=1 I- 
* 2j 
(2j) ! 
2 (j!)2(2j+l) 
(2j)! 
il,i2 j=2 22j(j!)2(2j+l) 
2(51-l)!! 
(E,-2j)(5,-2j-2)...(2-2j)(2j-l)(a)(b)2j+1'19 
The integrals in (44) can be evaluated as in Appendix. 
11 
(44) 
FfG f. 
Y 
3 
FIG 2 
REFERENCES 
[l] L.S. Fu and T. Mura, Volume Integrals Associated with the Inhomogeneous 
Helmholtz Equation: I. Ellipsoidal Region, NASA Contractor Report XXXX, 
1983. 
[2] L.S. Fu and T. Mura, Volume Integrals of an Ellipsoid Associated with 
the Inhomogeneous Helmholtz Equation, Wave Motion, A(2), 141-149, 
(April, 1982). 
[3] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and 
Products, Academic Press, N.Y., 1965. 
[4] P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 
N.Y., 1277, 1953. 
13 
APPENDIX 
The I-Integrals and 'Ir-Functions 
(a) The I-integrals 
W 
I1 = ~e(x1)h+2s~b2+x~2)t1-1~(a2+b2+x~2)3 dxl 
-II 
1 2 (b) 
2r1 = 71 . (a2+b2) t3-1 * CL> 
n-2t3+1 
Or1,r2 Y2! 
J(a2+b2+ 2)5 
+ rllr2(-1)11’2 2 8(rl-1)!! (b)2r1. (a2+b2Jn’2[~~.~(a2+b2+Q2)3 + 
, rl!r2! (11+4)!! 
+ +(a2+b2) (Qia2+b2+i2 + (a2+b2)shS1 a_>] 
da2+b2 
if X is even 
= 0 if X is odd 
in which the a-function r 
rl is defined in (b) of the Appendix. 
(ii) I 2 = I' 2 + 1" 2 
L 
I; = / a(x') 
A+279 2 
(b +x t2)5'2~a2+b2+xf2 dxf 
-9. 
=JI c 
0! 
------(a> @I 
2e1 
'I1 e1,e2 Y2! 
. (a2+b2) t4-1 -(II 
ll1-2t4+1 
- d(a2+b2+Q2) 3 
+ %182(-11 
n1/2 01 
L 
2(3-l)!! (a> @I 2e1 
e1!e2! (r11+2) !I 
(a 2 +b 2 1 q1'2 
> 
. (g,da2+b2+ * + (a2+b2)sh-1 YZ& 
da2+b 2 
if X is even 
=o if X is odd. 
14 
1; = ((x1) h+2n1(b2+x’Z) 35/2sh-1 Jb2;x, 2 dx’ 
= 2 1 e! a(b)2el(E)q3+1 _ 5 1 t6+1 
el,e2 el!e2! n3+1 * el! e2! 
. (b2jt6-1(a) 
'13-2t6+1 
n3-2t3+1 + c-11 
713/2 
&I 
n3-1 
-t -' {)(a)3(b) 2el + 
g 
+ c ; (-$1 
(3 1> !
-$&(a)2J +' '1 
el,e2 j1=2 22jl(jl)!(2jl+l) 1' 2' 
2 t7-1 
(b I (21 
n3-2t7+1 
- (n 
n3 (b2+e21jl-1 
+ c-11 
n3/2 (03-l)!! 
(n3-2jl+l)(n3-2j-1)...(3-2jl) 
- @I n3 (n' (11) + (-1) 
j 1-l 
ll3 
2t 2el b g -’ ;IltW 
if A is even 
= 0 if X is odd. 
(iii) 
I3 = J%(xl) 
h+2il+2t3-2 - 
Ja2+x 1 2 dx' 
-II 
= I[ (a21tsm1 (%I 
112-2t5+1 
J(a2+a2)3 + t-11 
r12/2 2(n2-l)!! rl /2 
n2 (n2+2)!! 
(a21 2 
- (EJa2+Q2 + a2shm1 $) if X is even 
= 0 if X is odd. 
(iv> 
I4 = !:(xfJA+2i1+‘1sh-1 5 dx’ 
-1 
(El 
l19-2t15+1 - 
Ja2+Q2 + (-1) 
n9/2 2(n9-l)!! 
(l-l,)! ! (a1 
T19 
sh-1 ' al 
if X is even 
=o if A is odd. 
'15 
(VI 
15 = Ja(x,) 
X+2il+2t4-2 dx, 
-II Ja2+x' 2 
=II (a) %4-l 
'I8 
* (111 
l18-2t14-1 - 
- Ja2+Q2 + c-11 
'lg/2 2(T-lg-l)!! 
(ng) ! ! 
(a2)n8'2sh-1 II 
a 
if X is even 
=o if A is odd. 
16 = 1; - 1; 
I;, = ~~(x')h+2il+~l-2sh-l $ dx' 
= +[IInlo(a2)t16 
-1 -2t16+1S~a2+k2 
+ c-11 
7110/2 2(yo-l)!! 
. 
(yol ! ! 
= 0 
if X is even 
if h is odd. 
=r[ (a) t9 . 
n5 
= 0 
(~21 
n5-2$-l - 
.Ja2+R2 + (- 
if X is 
if A is 
!il*<l-2 
dx' 
I .I 
7l5/2 2(Tl5-l)!! 
(aJ 
n5/2 
(n5) ! ! 
sh-' 5 a 
even 
odd. 
(vii) I7 = Jk(x,) 
X+2il+2t2-2 dx' 
-11 J(a2+xf2) 2j-1 
= fl (a2) t10 -l (I) 6 
I-l -2t10+1 
+ c-11 
2(n6-1)!! 
'6 J(a2+Q2)Lj-3 (n6-2j-2) (n6-2j-4). . . (-2j) 
16 
=o 
an6Q 
(2j -3) a2J(a2+E2) 2J-3 
j-2 
Cl+ c 
kll=l 
8kll (j-2)(j-3)...(j-kll-1) (a2+ 2)k11 
(zj-5)(2j-7)...(2j-2kll-3) 
(4a2)k11 
I 
if A is even 
if X is odd. 
(viii) I8 = ~~(x,.J~+2il+51-2fo-2 ;2+x.2)todx1 
a2+x12) 2J-1 
=ll (a2Jt12 s$! 
-1++1 
n7 (,5, W2! 
(El 
n7-2t12+1 
+ I: 41 
J(a2+a2)2j-3 V2 
@,W,! 
* C-11 
n7/2 2(n7-l)!! 
(n7-2j-21 (n,-Zj-4). . . (-2j) 
(a2)n7'2+% & 
(2j -3) a2J(a2+a2) 2J-3 
= 0 
j-2 8k13 
Cl+ 1 
(j-2)(j-3)...(j-k13-lJ(a2+hL2Jk13 
k13=1 (2j-5)(2j-7)...(2j-2k13-3)(4a2)k13 
if X is even 
if A is odd. 
(ix) 
I9 = Ia 
h+2il+El-2 dx' 
-11 J( a2+xr 2) 2j -1 
(a2)t12 
-1 
(III 
n8-2t12+1 
=ll 
'8 J(a2+k2) 2j - 1 
+ c-11 
n8/2 2(Ilg-l)!! 
(n8-ij-2) (n8-2j-4). . . (-2j) 
an8Q 
j-2 8k12 
(2j-3)a2J(a2+a2)2J-3 
Cl+ 1 
(j-2)(j-3)...(j-k12-')(a2+g2Jk12 
3 
k 12=1 (2j-5)(2j-7)...(2j-2k12-3)(4a2Jk12 
if h is even 
=o if X is odd. 
17 
(b) The T-Functions 
The n-functions in the I-integrals listed in (a) of this Appendix are 
defined as follows: 
Q = X + 2n, + 2r, (is even) 
L L 
rl + r2 = r = tl-i 
(ii) II 
Y 
as the same as (35), if replace 5 w ,i.th r11 
Ol = X + 2nl +2e2 (is even) 
e1 + e2 = 8 = 5/2 
ii,-1 
II' = 4 
t8+1 t8-1 [3-2(jl+l)][3-2jl]...[3-2(jl-t8+2)] 
n3 
3-2(jl+l) -(a - (j 1-l) (ii 1-2> - -. (j 1-t8) 
n3 = X + 2nl + 2e2 (is even) 
el + e2 = e = +s 
(iv) II 
n2 
as the same as (35), if replace 5 with n2 
?l’2 =X+2i 1 + 2t3 - 2 (is even) 
(VI rI 
n9 
as the same as (32a), if'replace v with ~~ 
n9 = X + 2il + 51 (is even) 
18 
(vi) II 
'8 
as the same as (32a), if replace v with n8 
'8 =X+2i 1 + 2t4 -2 (is even) 
(vii) II 
rllo 
as the same as (32a), if replace v with r11~ 
nlo =A+2i 1 + 51 - 2 (is even) 
(viii) II 
n5 
as the same as (32a), if replace v with n5 
n5 = X +2il + 51 -2 (is even) 
(ix1 
II $lJ 
tlO+l (n6+1)(n6-1)...(n6-2tlo+3) 
'6 ($-2j-2>(06-2j-4)...(n6-2j-2tlo) 
n6 = A + 2i +2t 12 - 2 (is even) 
(xl n: 
n7 
as the same as II 
'6' 
if replace 06 with o7 
rl 7 = X +2il + 5, +2$* - *to - 2 (is even) 
$1 + $2 = I$ = to 
(xi) II as the same as II 
'8 '6' 
if replace n6 with n8 
'I8 = X + 2il + 51 - 2 (is even) 
19 
1. Report No. 
NASA CR-3750 
2. Government Accession No. 
4. Title and Subtitle 
Volume Integrals Associated With the Inhomogeneous 
Helmholtz Equation. II - Cylindrical Region; 
Rectangular Region 
7. Author(s) 
W. F. Zhong and L. S. Fu 
9. Performing Organization Name and Address 
The Ohio State University 
1314 Kinnear Road 
Columbus, Ohio 43212 
‘2. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, D.C. 20546 
3. Recipient’s CatalogNo.- - 
5. Report Date 
December 1983 
6. Performing Organization Code 
6. Performing Organization Report No. 
None 
10. Work Unit No. 
11. Contract or Grant No. 
NSG3- 340 
13. Type of Report and Period Covered 
Contractor Report 
14. Sponsoring Agency Code 
505-53-12 (~-1838) 
5. Supplementary Notes 
Final report. Project Manager, Alex Vary, Materials Division, NASA Lewis 
Research Center, Cleveland, Ohio 44135. 
6. Abstract 
Problems of wave phenomena in fields of acoustics, electromagnetics and 
elasticity are often reduced to an integration of the inhomogenous Helmholtz 
equation. Results are presented for volume integrals associated with the 
Helmholtz operator, v2 + cr2, for the cases of an ellipsoidal region, a finite 
cylindrical region, and a region of rectangular parallelepiped. By using 
appropriate Taylor series expansions and multinomial theorem, these volume 
integrals are obtained in series form for regions r > r' and r < r', where r 
and r' are distances from the origin to the point of observation and source, 
respectively. Derivatives of these integrals are easily evaluated, When 
the wave number approaches zero, the results reduce directly to the potentials 
of variable densities. 
7. Key Words (Suggested by Author(s)) 
Acoustic waves; Elastic waves; Ultrasonic 
waves ; Helmholtz equation; Material 
inhomogeneities; Cylindrical/rectangular 
inhomogeneities 
16. Distribution Statement 
Unclassified - unlimited 
STAR Category 38 
9. Security Classif. (of this report) 20. Security Classif. (of this bage) 21. No. of pages 
Unclassified Unclassified 21 
.-. 
‘For Sale by Ihe National Technical Information Service, Springfield, Virginia 22161 
22. Price’ 
A02 
NASA-Lan9ley. 1983 


Volume integrals associated with the inhomegeneous Helmholtz equation. Part 2: Cylindrical region; rectangular region

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