Problems of wave phenomena in fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) to alpha(2), for the case of an ellipsoidal region. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r  4' 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

Fu, L. S.

Mura, T.

English

NASA Technical Reports Server

NASA Contractor Report 3749 
Volume Integrals Associated With ‘: 
the I&omogeneous Helm-h&z 
L. S. Fu and T. Mura 
GRANT NSG3-269 
DECEMBER 1983 
LOGAN COPY: RETURN TO 
AFWL TECHNICAL LIBRARY 
KIRTLAND AFB, N.M. 87117 
25th Anniversary 
1958-1983 
https://ntrs.nasa.gov/search.jsp?R=19840006457 2020-03-21T01:19:51+00:00Z
TECH LIBRARY KAFB, NM 
0llb2337 
NASA Contractor Report 3749 
Volume Integrals Associated With 
the Inhomogeneous Helmholtz Equation 
I - EZZipsoidul Regiolz 
L. S. Fu and T. Mura 
The Ohio State University 
Cohw.&us, Ohio 
Prepared for 
Lewis Research Center 
under Grant NSG3-269 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Branch 
1983 
- 
1. INTRODUCTION 
Volume integrals associated with the integration of inhomogeneous 
Helmholtz equation are of practical interest in determining physical 
quantities in acoustic, electromagnetic and elastic fields. The inhomo- 
geneous scaler Helmholtz equation takes the following form: 
v24 + a24 = -4lrpQ (11 
where P(E) is the source distribution or density function, V2 and a are 
the Laplacian and wavenumber, respectively. It is well known [1,2] that 
a particular solution to Eq. (1) is: 
q = ~(~'1 R -1 exp iaR dV' 
in which (4nR) -1 exp iaR is the free space Green's function, and Cl is the 
region where the source is distributed. The source distribution function 
p($ can in general be either expanded or approximated in a polynomial 
form and hence p(zl) is normally written as 
P (:‘I = (x’p (y’p (2’) v 
where X, IJ, v are integers. The integration of the vector Helmholtz 
equation is analogous, [2]. 
The reduction of time harmonic fields of frequency w in the acoustic 
and electromagnetic fields to that of integrating the inhomogeneous 
Helmholtz equation over a given volume can be found in many standard texts 
[3,41. A formulation that leads to the required form of volume integration, 
(31 
Eqs. (2,3) such that the elastic fields can be determined has recently been 
given in [S-7]. Using the dynamic version of the Betti-Rayleigh reciprocal 
theorem, an integral representation of the displacement field ui in an 
elastic medium containing an inhomogeneity can be given in terms of the 
eigenstrains E?. (11 
=I 
and eigenforce IT; as: 
urn&‘) =- C jkrs 'jm,k' W'U (r r'l ~~~ *(+) dV 
n (4) 
where g. 
3m 
are the spatial part of the free space Green's tensor function. 
For a linear isotropic elastic medium, 
o2 6 exp iBR 
jm 
I-- 
R 
_ [""P ; R _ exp i" R ] ,jm ] (51 
in which p o is mass density, a and B are wavenumbers for longitudinal and shear 
waves, respectively. Expanding the eigenstrains and eigenforces in a poly- 
nomial of position vector2 yields: 
=A.+A 
J jk 'k 
+ Ajkl XkXR + . . . (ha1 
E;~(L) = Bij + Bijk xk + BijkL xkxa. + . . . 
and substituting it and (5) in (4), the displacement field is found to be 
urn(~) = fmj(;) A. + f 
3 mjk(~) Ajk + '.. 
+F mij (~1 Bij + Fmijk(~ Bijk + . . . 
WI 
(7) 
where 
41rPoW2 f mjk(r) = -B2 @k ~mj + ek,mj - $k,mj 
--- 
4rrpow2 Fmij (rl = - [ha2 JI,, ‘ij + 2~ B2g,ibrnj - 2~ JI,,ij + 2~ ~,,ij] 
4npow2 F mijk(r) 
-2' 'k,mij + 2p @k,mijl 
Here, X, P are LamG's constants, and 
$(r) = , R-l exp(iaR) dV', iif 
R 
Qk(r) = ~ 
I ff 
xiR-' exp(iaR) dV' , 
R 
'kl...s = I I I 
"i";r. ..x;R 
-1 exp(iaR) dV', 
n 
@J(r) = JJJ R-l exp(iBR) dV', 
R 
akb) = 111 xiR -' exp(i@R) dV', 
R . . . 
JJJ 
1 
'kl...s = xix;, . . .xSR -' exp(iBR) dV'. 
si 
This paper presents results for the volume integrals over a region 
(84 
CabI 
(8~) 
Pa) 
(-1 
@cl 
(9d) 
(gel 
(9f) 
that is either an ellipsoid, a finite cylinder or a rectangular parallelpipe 
3 
with semi-axes al, a2 and a3' Fig. 1. The integrals in (9) are subsequently 
referred to as the #-integrals and they are obtained in series form by expanding 
R-l exp (iaR) in appropriate Taylor series expansions for regions r > r' and 
r < r', and by using the multinomial theorem with also the assistance of the 
classical result of Dyson [8] in the case of an ellipsoid. Certain derivatives 
of the Q-integrals that are of interest are also presented. 
Professors C. P. Yang and C. Saltzer of the Department of Physics and 
Mathematics at The Ohio State University, respectively, participated in many 
helpful discussions. 
2. SERIES REPRESENTATION OF THE O-INTEGRALS 
Let R -1 exp iaR be expanded in a Taylor Series expansion for ?l as 
R-l exp iaR = 
n=O 
+ [xi &Jn[r-l exp iar] , 
for r > r' 
and in a Taylor Series expansion for i! as 
R-l exp iaR = i C-lln 
n=O 
-7 (xi $7)” [ (r’>-l exp iart] , 
(101 
(111 
for r c r' 
in which the summation convention is observed and i = 1,2,3. 
Employing the multinomial theorem as suggested in Ref. [l], 
the o-integrals can be explicitly written as triple sums: 
WI” n 4, (21 = #a 
n=O J?=O k-0 11! K! (n-k-k)! '* ax II k n&-k ay a2 
l ',II (X1>" (~'1~ (z')~-'-~ p(x',y',z') dV' , 
for r > rl 
4 
(12) 
and 
@< (21 = 11 yk rn-2-k . 
n=O 11=0 k=O 
exp iar' \ 
n-g-k r' f 
dV' , 
The Taylor series representations given in Eq. (10) and Eq. (11) 
converge for the region r > r' and r < r', respectively. The 
integral 0, (,r) in Eq. (12) is normally used to evaluate physical quan- 
tities measured at large distance from the region R. The apparent singu- 
-1 -2 larities present in Eq. (13) appear as I?n E, E , E ,... where E is a 
small positive number. These singularities disappear, however, if E is 
taken to be the radius of a sphere centered around the origin. In evalu- 
ating Q< (z) f or an ellipsoid, care must be taken in determining the con- 
tribution to the integral from the lower limit E. A further note on this 
is given at the end of Section 3. 
3. INTEGRATION OVER AN ELLIPSOIDAL REGION 
The integrals in (12, 13) are of either one of the following forms: 
0' = 1;' (x')' (Y,)~ (z')' dV' , 
QS = JJl P(X’,Y’,Z’) a” 
R axfLayfkaZf n-!Z.-k 
8 = JJJ P(X’,Y’,Z’) a” 
R axteay~kaz~n-'-k 
(13) 
(14) 
(15) 
(16) 
5 
These integrals can be further evaluated as follows: 
(al 0' = 111 (x')~ (Y*>~ (z')~ dV' 
Q 
I al 
f+l g+l h+l 
a2 a3 Nf/2)Ng/2)R(h/2) 
(2m+3) R(m) (171 = 
0 Tf any one of the superscript power f,g, or h is odd. 
where a 1' a2' a3 are the axes of the ellipsoid, and 
2m = f +g+h 
(2m)! R(m) = - m! 
This result was first obtained by Moschovidis [9]. 
(b) n = 0, 0': 
2 - 111 p(x',y',z') . 
sin ar' 
r' dV' R 
= ,;(, p(x',y',z') T(J) mD?& (rt)2m-2 dV' 
m=l 
= tn~l(-ll 
m-1 a2m-1 
(2m 'm,p 
where 
S m,p = I;I' (xl)' (y')' (z*)~ (~'~+yf~+z~~)~-~ dV' 
(181 
(191 
x+1 u+l v+l 
al a2 a3 4rr = 
(x+~+v+2m+l) (X+p+v+2m-1) 
6 
: 
2ml 2m2 
(m-l)! 
ml!m2!m3! ;: 
a2 a3 
2m3 
(2ml+A)! (2m2+u)!(2m3+v)! [2(mbl)+X+u+v]/2! 
ml,m29m3 ml+X)/2! (2m2+u)/2! (2m3+v)/2! [2(m-l)+X+p+v]! 
= 0, if L,u, or v 
in which the multinomial 
ml+m2+m3 = m-l , if )r,u,v all even, Wal 
odd. (19b) 
formula 
= c 
mlsm2,m3 ml ! 
$1 m , w12ml (y9)2m2 (z')2m3 
3' (20) 
is used. In (20), the sums are taken over all non-negative integers m 1' m2 
and m3 for which ml+m2+m3 = m. 
(c) n = 0 , 8 
8 = 111 p(x',y',z') co;,ar' dV' 
s-l 
(21) 
= mFo (-l)m &r,, (x’)’ (y’)’ (z’)’ l + dV' 
R 
Using the multinomial formula and letting the integral in (21) be denoted by 
C 
m,p = !isI (x’)x be (z’P w12m-1 dV’ 
The volume integral in Eq. (22) may be viewed as the potential of variable 
densities observed at the origin,_r=O. Applying the results on volume 
7 
integration over an ellipsoid given by Dyson [8],Eq. (22) can be 
written as 
c = 1 (ml ! m4 mlsm2,m3, ' "laZa3 l ml! m2! m3! al 
2ml+X 2m2+p 
a2 a3 
2m3+v 
OD 
I 
JI m+p 
ax 
m+p 
1 
1 . 
2(m+p)(m+p)!(m+p+l)1 
IS - 
0 2 a:+$ 
d$ 
l Q , if X,II,V, all even 
= 0 , if X,p,v, is odd (23.b) 
where 2p = (A+lr+v) 
d2L . 
dx'2L1 dyf2'2 &.f2+ ' 
2ml+? 
a2y 
, 
( ) 
2m2+u 
2 f a +JI 2 
(23.a) 
(24) 
8 
in which the sums are taken over non-negative values of Cl, E2, t3 for which 
II1 + E2 + c3 = 11. Using the definition of 6', (24), and noting that 
F = (x') 2ml+A (Y’> 2m2+lJ(z,12m3+v 
a 2' ( ) 
2m3+v 
3 .- 
a$+$ 
it can be easily shown that 
6 m+P 
= h+pl ! 
(ml+X/2)!(m2+~/2)!(m3+v/2) ! (2ml+X)!(2m2+p)!(2m3+v)! l 
. (.$“‘” (#2+‘i’( g)“““” 
Finally, for n = 0 
8 = ,Fo (-Urn .& cmtp = 
where 
(25) 
(26) 
m! (2ml+A)!(2m2+p)!(2m3+v)! 
c = 
m,p 1 ml,m2,m3, ml!m2!m3! (ml+X/21 !(m2+p/2)! (m3+v/2)! 
al 
2ml+,I+l 
a2 
2m2+p+l 
a3 
2m3+w+l 
. . 
22m+2p (m+p+l)! 
0 J, m+P d$ 
. (ai+$)ml+X/2 (a:+$)m2+u'2 (a~+$)m3+v12Q ' 
OS = ~;lp(x',Y',z') ax,fiaya:az,ne~-k Si;,ar' dV' 
where . 
n-ll-k= (rt)2m-2 dV' 
= c 
(m-l)! (2ml)! (2m2)! (2m3)! 
. 
ml ,m2,m3 II! k! (n-E-k)! 
. J;’ (xtlA+2mi-~ (y,lu+2m2-k (Z,lv+2mw+fi+k dV, 
(27) 
(281 
WI 
in tiich the multinomial formula is used and m 1' m2* m3 are summed over all 
integers greater than and equal to unity and m +m +m 123 are summed over all 
integers greater than and equal to unity and m +m +m 1 2 3 = (m-13. The integral 
in (29) can be obtained by using the formula given in (17), and is easily 
shown to be 
10 
S" 
(m-l)! (2ml)I (2m2)! (2m3)! 
m,p = c a! k! (n-t-k)! 4n ml ,m2,m3 
al 
A+2ml-a+1 
a2 
u+2m2-k+l 
a3 
v+2m3-n+l+k+l 
. 
(2p+2m+l-nj (2p+2m-n-1) . 
=o 
if (x-e), (u-k), (v-n+L+k) all even 
if (A-L), (p-k) or (w-n+ll+k) is odd 
where 
2p = A+u+v 
(e> n 0, ac # 
8 = ‘ii’ P(x’,Y’,z’) ax,,;;,kaz,nmQ cosr:r’ dV’ 
where 
(30.a) 
(30.b) 
(31) 
WI 
11 
When n # 0, it is not as easy to find a compact form for these integrals. 
For the determination of the elastodynamic fields of an ellipsoidal inhomogeneity 
as formulated in [6,7] it is sufficient to determine a<(r) for a finite number 
of n's in determining the B.., 8.. l]k' and A., A. 7 Ik' -'- 
in [6,7]. For example, 
11 * 
LI if it is necessary to determine the eigenstrains E.. up to a second order 11 
distribution, it is then sufficient to find ac for 1 < n < 6. - - 
The integral a,(r) in (13) can be replaced for n = k by 
where 
The substitutions of the derivatives of (r') 2m-1 in Eq. (34), 
lead to integrals that can be easily evaluated by using 
Eqs. (23,24,25), for the cases m > 1, k = 1, 3 and m > 2, k = 4, etc. - 
Special attention must be given to the cases m = 0, k = 2,3, and m = 0,l 
k = 4, 
Using the notations given in Ref.[lO] and noting that 
2 dV' = dx' = dr' dS = dr'*r' dw , 
(33) 
(34) 
(35) 
we obtain 
where 
r'(l) = 
1 l/2 
0 
i? 
12 
(36) 
8 = &f/a: , Iii = xi/r' . (371 
and f = 0, e = 1 due to the fact that here we consider the source point 
being situated at the origin, i.e. 2 = 0. Volume integrals associated with 
Eq. (32). _ _ 1 < n c 4, can be written into surface integrals by using Eq. (35). 
and finally reduced to simple integrals through the work of Routh [ll], e.g. 
lLJo*(r')‘3 dV' = //p l (r')'3 l dr' l i-l2 l du 
= Ijj P - (r')-l dr' dw 
If p = 1, r11, p. 9011, 
I/J (rfle3 dV' = /I{Ln r'(ai) - Ln ~1 dw 
1 
= Al 
J;(, (r')-' dV' = -iJl{[r'@..]-2 - em21 dw 
c 
= -$/ g du + A2 
2a 1 =a-. 
3 a.a. 
+A 2 
11 
(38) 
(391 
The surface integral of the type II II: !Z; IL: g-l dw can be reduced to simple 
c 
integrals as well by using the work of Routh [7] in the same manner as 
listed in Ref. [6] and therefore will not be repeated here. me constants 
13 
The constants Al and A2 are equal to 4a(llna - 'ln~) and +(27r/3)(eS2), 
respectively for a sphere of radius a, where E is a small positive number. 
-1 -2 The coefficient of these types of terms, En E, E , E , . . . . in the Q-integral 
can be shown to be identically zero in a straight forward manner if $I is a 
sphere. When .9 is an ellipsoid, the lower limit of integration should be taken 
from the surface of a small sphere with radius E, (38,39). The contribution 
to the &integral from the lower limit can therefore be identified as zero. 
14 
x.. 
3 matrix: A, p ; ,, 
inhomogcncity : 1; II*; p * 
/ 
I 
I a, 
I 3 
h incident lC3VC 
Fig. 1 An ellipsoidal region 
of integration. 
15 
111 
PI 
[31 
[41 
[Sl 
[61 
[‘I 
181 F.W. Dyson, Quart. Pure Appl. Math. XXV, 259-288 (1891). 
PI Z.A. Moshovidis, Ph.D. Dissertation, Northwestern University (1975). 
PO1 T. Mura, Micromechanics in Defects in Solids, Nijhoff, 1982. 
1111 R.J. Routh, Phil. Trans. Roy. Sot. 186-11, 897-950 (1895). 
REFERENCES 
P.M. Morse and H. Feshbach, Methods of Theoretical Physics (1953). 
Chen-To Tai, Dyadic Green's Functions in Electromagnetic Theory, 
Intext Educational Publishers, San Francisco (1971). 
J.J. Bowman, T.B.A. Senior and P.L.E. Uslenghi, Electromagnetic and 
Acoustic Scattering by Simple Shapes, North-Holland, Amsterdam (1969). 
D.S. Jones, The Theory of Magnetism, Pergamon Press, New York (1964). 
L.S. Fu, Fundamental aspects in the quantitative ultrasonic determination 
of fracture toughness" General equations, I1 NASA Contractor Report 3445 
(July, 1981). 
L.S. Fu, Method of Equivalent Invlusion in Dynamic Elasticity, in 
Review of Progress in Quantitative Nondestructive Evaluation, D.O. Thompson 
and D.E. Chimenti, Vol. 1, 145-155, Plenum Press, NY, 1982. 
L.S. Fu and T. Mura, The Determination of the Elastodynamic Fields of an 
Ellipsoidal Inhomogeneity, J. Appl. Mech., Trans. ASME, 50, 390-396, 
(June, 1983). 
16 
.-- . 
1. Report No. 2. Government Accession No. 
NASA CR-3749 
4. Title and Subtitle 
Volume Integrals Associated With the Inhomogeneous 
Helmholtz Equation. I - Ellipsoidal Region 
7. Author(s) 
L. S. Fu and T. Mura 
9. Performing Organization Name and Address 
The Ohio State University Il. Contract or Grant No. 
1314 Kinnear Road NSG3-269 
Columbus, Ohio 43212 13. Type of Report and Perfod Covered 
1 2. Sponsoring Agency Name and Address Contractor Report 
National Aeronautics and Space Administration 
Washington, D.C. 20546 14. Sponsoring Agency Code 
506-52-62 (~-1836) 
1 5. Supplementary Notes 
Final report. Project Manager, Alex Vary, Materials Division, NASA Lewis Research 
Center, Cleveland, Ohio 44135. 
3. Recipient’s Catalog No. 
5. Report Date 
December 1983 
6. Performing Organization Code 
6. Performing Organization Report No. 
None 
IO. Work Unit No. 
I 
1’ 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 + cc2, 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 re,gions 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)) 18. Distribution Statement 
Elastic waves; Acoustic waves; Helmholtz Unclassified - unlimited 
equation; Volume integrals; Material STAR Category 38 
inhomogeneities; Ellipsoidal inhomo- 
geneities 
9. Security Classif. (of this report) 20. Security Classif. (of this page) 
Unclassified Unclassified 
21. No. of pages 
18 
22. Price’ 
A02 
*For sale by the National Technical Information Service, Springfield, Virginia 22161 
NASA-Lang1 ey , 1983 


Volume integrals associated with the inhomogeneous Helmholtz equation. Part 1: Ellipsoidal region

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