EPA-600/1-79-025
August 1979
ABSORPTION CHARACTERISTICS OF PROLATE SPHEROIDAL MODEL
OF MAN AND ANIMALS AT AND NEAR RESONANCE FREQUENCY
by
Vijai K. Tripathi and Hyuckjae Lee
The Department of Electrical and Computer Engineering
Oregon State University
Corvallis, Oregon 97331
R-804697-01-1
Project Officer
Claude M. Weil
Health Effects Research Laboratory
Research Triangle Park, North Carolina 27711
Health Effects Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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DISCLAIMER
This report has been reviewed by the Health Effects Research Laboratory,
U.S. Environmental Protection Agency, and approved for publication. Approval
does not signify that the contents necessarily reflect the views and policies
of the U.S. Environmental Protection Agency, nor does mention of trade names or
commercial products constitute endorsement or recommendation for use.
11
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FOREWORD
The Health Effects Research Laboratory (HERL), a part of EPA's Office of
Research and Development, is responsible for performing research into the
health impacts on mankind of many environmental pollutants, as well as unde^
sirable physical factors within the environment. One of the latter factors
with which HERL concerns itself is nonionizing electromagnetic (radiofrequency
and microwave) radiation. The basic objective of EPA's research program into
the health implications of nonionizing radiation is to develop scientific
criteria upon which the Agency can develop an environmental exposure standard
for radiofrequency (RF) radiation.
Much of the in-house research performed to date at HERL's Research
Triangle Park, NC facility has involved chronic exposures of experimental
animals to radiation of various wavelength within the RF spectrum. In these
experiments, animals have generally been assessed in a number of different
areas such as development, teratology, immunology, hematology, behavior,
neurochemistry, etc., in order to determine whether any possible adverse
effects exist. Accurate characterization of these experiments requires a
knowledge of the RF energy (or dose) which the animal is absorbing as well as
the internal energy distribution. Furthermore, there is an obvious need to
try to extrapolate animal findings to humans, a problem that is particularly
complex and difficult for nonionizing radiation. The data needed for such
extrapolations cannot be readily derived by experimental means.
For these reasons, many investigators active in this field of research
have made use of simplified mathematical models which have yielded much data
that is useful for predicting approximately the energy absorption in both
animals and man under various conditions of RF irradiation. Because of their
obvious similarity to the external shape of both man and experimental animals,
models of prolate spheroidal and ellipsoidal shape have been investigated by
several workers. These are composed of dissipative (lossy) dielectric and
the exposure is generally to a plane wave. Previous workers had succeeded in
obtaining a quasi-static solution valid in the low frequency region where inci-
dent wavelength is greater than the physical dimensions of the prolate spheroid
and ellipsoidal model. However, no solutions were available in the resonant
and near-resonant region where the incident wavelength is comparable to the
object dimensions. This region is of considerable importance because maximum
coupling of electromagnetic energy into the exposed object occurs in the
resonant region. Solutions to the problem involving a prolate spheroid of
dimensions similar to that of man in the resonant frequency region would pro-
vide realistic estimates of the maximum thermal burden which man would likely
be subjected to under these worst-case conditions. Such data have obvious
applicability to the setting of RF protection standards.
iii
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Consequently, a study was sponsored through a research grant to Oregon
State University, whose purpose was to investigate possible solutions to the
electromagnetic problem involving the lossy prolate spheroid model in the
resonant region. All of the existing techniques were to be reviewed and the
limitations and restrictions on the applicability of such techniques to this
particular problem were to be detailed. In particular, the feasibility of
applying the separation of variables technique to this problem was to be
examined since this technique does yield highly accurate solutions for the
closely related problem involving lossy sphere models in the resonant region.
This report details the results of this study. In attempting to summarize the
conclusions of the report, it may be stated that accurate solutions are pre-
sently available in the resonant region for prolate spheroid models of low to
medium eccentricity applicable to experimental animals, but that no solutions
are as yet available in the resonant region for highly eccentric (i.e., elon-
gated) models similar to man owing to mathematical difficulties encountered in
the solution of this problem. Further work is needed to overcome these diffi-
culties before a solution to this important problem becomes available.
Readers1 comments on any aspect of this report are solicited by the HEEL
project officer.
F.G. Hueter, Ph.D., Director
Health Effects Research Laboratory
iv
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ABSTRACT
This report deals with the evaluation of the absorption characteristics of
prolate spheroidal models of man and animals consisting of homogeneous biologi-
cal tissues irradiated with plane waves at an arbitrary angle of incidence at
and near the resonance frequency. Stevenson's series expansion method in the
third approximation and the extended boundary condition method are presented in
detail since they were found to be most amenable to the solution of the problem.
It is shown that Stevenson's method in the third approximation can be ap-
plied to solve for the absorption cross sections of prolate spheroidal and el-
lipsoidal models at lower frequencies. The method is quite general as compared
to other known long wavelength analysis methods which are applicable to high
conductivity models at nose-on and broadside incidences only. The general for-
mulation presented in the report can be applied to an arbitrary angle of inci-
dence and leads to an improvement in frequency range of validity and accuracy
as compared to the first order solutions. Numerical results for prolate sphe-
roidal models are presented and are compared with known results for first order
solutions and the general time-dependent solutions obtained by the extended
boundary condition method.
It is also shown that the problem of the absorption by lossy prolate sphe-
roidal objects can be formulated in general by utilizing other methods such as
the separation of variables and the point matching techniques. However, the ex-
tended boundary condition method (EBCM) was found to be implementable for real-
istic models over a wide range of frequencies. Even though the theory of EBCM
is well known, this report emphasizes the application of the method, and con-
tains some modifications for improving the accuracy and efficiency of the com-
putations. Numerical results obtained by utilizing this method are also pre-
sented.
This report was submitted in fulfillment of Grant No. R-804697-01-1 by
Oregon State University under the sponsorship of U.S. Environmental Protection
Agency. This report covers a period of September 15, 1976 to March 15, 1978.
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CONTENTS
Foreward iii
Abstract v
1. Introduction 1
2. Theoretical Formulation 3
2.1. Statement of the Problem 3
2.2. Stevenson's Method in the Third Approximation 5
2.2.1. Theoretical Formulation 6
2.2.2. Comparison with Long Wavelength Results 9
2.2.3. Prolate Spheroidal Model 10
2.3. Point Matching Method 11
2.4. Separation of Variables Method 11
2.5. Extended Boundary Condition Method 12
3. Numerical Results 13
4, Conclusions and Suggestions for Further Study 20
4.1. Conclusions 20
4.2. Suggestions for Further Work 21
References 22
Appendices
A. Scattering Coefficients 25
B. Prolate Spheroidal Model - Expressions for Coefficients 28
Case 1 - Broadside Incidence - Electric Field Along
the Major Axis 28
Case 2 - Broadside Incidence - Magnetic Field Along
the Major Axis 29
Case 3 - Nose-On Incidence 29
C. Extended Boundary Condition Method 31
C-l. Theoretical Formulation 31
C-2. Extended Boundary Condition Applied to the Surface 3€
C-3. Application to Spheroidal Problem 3<|
vi
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D. Description of Computer Program for EBCM 43
D-l. Introduction 43
D-2. Glossary of Subroutines 43
D-3. Computation of Special Functions 44
D-4. Computation of Geometrical Factors 46
D-5. Computation of Scattering, Absorption and Extinction
Characteristics 46
D-6. Integration Schemes 47
D-7. Description of Variables Used in the Program 4?
D-8. Inputs to the Program 49
D-9, Outline of the General Flow of the Program 50
D-10. Program Listings 51
vii
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SECTION 1
INTRODUCTION
A great deal of work has been done in recent years on the biological ef-
fects of electromagnetic radiation. One important aspect of this work in-
volves the determination of internal distribution of power absorbed at any
specified biological tissue in the body and the total power absorbed by the
body subjected to electromagnetic radiation. Power is absorbed by the tissues
as a function of body shape, tissue properties, frequency and incident radia-
tion.
As a first step in this direction, uniform and layered tissue spherical
models have been analysed by a number of workers. Most of the early work had
been confined to a spherical model due to the relative ease of analysis.
Early work on the uniform sphere model has been done by Anne et al. [1],
Shapiro et al. [2], Kritikos and Schwan [3] and Johnson and Guy [4]. The anal-
yses of multilayered tissue model have been conducted by Joines and Spiegel [5]
and Weil [6], These analyses are applicable to all frequency ranges. Based
on these models, in which energy deposition has been confirmed for the uniform
sphere case by thermographic camera photographs of irradiated phantom models
[4,7], it is expected that the power deposition pattern in man and animals var-
ies greatly with the properties of incident radiation including frequency,
body configuration and orientation. This leads to a need for analytical tools
to describe power absorption patterns in more realistic models of man and ani-
mals as a function of incident radiation. A model more realistic than a
sphere for the human body and many animal bodies is the prolate spheroid.
A low frequency analysis of a uniform ellipsoid and prolate spheroid mod-
el has been conducted by Durney, Johnson and Massoudi [8-11] using Stevenson's
method [12,13] as given in Van Bladel [14]. Their results, particularly the
effect of object orientation with respect to the incident plane wave vectors,
indicate a significant difference from results for the spherical models at low
frequencies and underline the need for a solution of the prolate spheroidal
model problem at higher frequencies at and near the resonance frequencies. At
higher frequencies, as characterized loosely by 0.3 <_ kb <_ 300, where b is the
largest semi-axis, the solution cannot be accurately determined using this me-
thod unless higher order (third and higher) solutions are included. Several
other known techniques have also been tried recently by these authors and
others at the higher frequencies with varying degrees of success . These in-
clude the point matching technique [15,16], the spheroidal function expansion
method [17] and the extended boundary condition method [18,19]. At higher fre-
quencies when skin-surface heating effects predominate, kb » 1, and the
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so-called slab model, consisting of planar layers of skin, fat and tissue lay-
ers has been used to evaluate the electromagnetic power absorption [20,21],
The gap of information regarding the analysis and numerical results for
electromagnetic power absorption in prolate spheroidal model at and near the
resonance frequency needs to be filled since at resonance the absorption effi-
ciency is greatest. These results are essential in order that researchers can
obtain tissue absorbed power estimates in terms of incident electromagnetic ra-
diation parameters in order to establish a sound electromagnetic safety stan-
dard over this important frequency band.
The primary objectives of this research program were to evaluate the ab-
sorption characteristics of the prolate spheroidal model of man and animals
consisting of homogeneous tissue characterized by known values of £(co), cr(o)
and y at and near the resonance frequency when radiated with electromagnetic
waves. The electromagnetic radiation is in the form of a plane wave propagat-
ing in an arbitrary direction polarized either in the plane of the wave vector
and the major axis of the spheroid or normal to it.
A comprehensive investigation of several techniques was conducted in an
attempt to arrive at the most convenient methods to solve for the absorption
characteristics. These include:
Stevenson's Method
Separation of Variables Method in Prolate Spheroidal Coordinates
Point Matching Technique
Extended Boundary Condition Method (EBCM).
Even though the problem of electromagnetic scattering and absorption by lossy
prolate spheroidal objects can be formulated by utilizing all of these methods,
Stevenson's method in the third approximation and the EBCM were found to be
the most efficient from a computational point of view. In addition, these me-
thods were most successful in evaluating the absorption characteristics of re-
alistic prolate spheroidal models within their respective range of validity.
For these reasons these two methods and the results obtained by utilizing them
have been emphasized in this report. Section 2 deals with the theoretical
formulation of the problem with a particular emphasis on Stevenson's method in
the third approximation as well as the EBCM. The numerical results obtained
by utilizing these methods are presented in Section 3. A summary of results
and suggestions for further work are presented in Section 4.
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SECTION 2
THEORETICAL FORMULATION
2-1. STATEMENT OF THE PROBLEM
The prolate spheroid model is shown in Figure 1. The object is assumed
to be homogeneous having conductivity a, dielectric constant £ = £Q£r and per-
meability u0. The incident radiation is assumed to be a plane wave polarized
in an arbitrary direction propagating along a^. Note that other wave fronts
may be constructed by a super position of plane waves. Then the electric
field of both regions (inside and outside the object) are a solution of the
Helmholz wave equation as given by:
V2E
k2E = 0
for harmonically oscillating fields (e-1 variation), where
2
and
22
k = k =
222
k = k = o> u £ +
» outside
(1)
(2)
0" inside the spheroid.
(3)
The boundary conditions which the solution must satisfy at the interface
are:
and
n x (E „ - E. J = 0
ext int
n- [e E . - (e- ~ ). ] = o
L o ext *• jco J intj
(4)
v J
(5)
v J
where_, Eext, the external field is the sum of incident and scattered fields
and Eint is the internal field in the spheroid: e and a are known constants
[4]-
Given an incident radiation which is assumed to be a plane wave,
can be determined by solving the equations (1) - (5) given above.
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O.-H
Incident Wave
i ^-o
FIGURE 1. The Prolate Spheroidal Model
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Knowing the internal field for a given incident radiation, the absorption
pattern, total power absorbed and the absorption cross section can be easily
calculated.
Numerous analytical techniques have been used to solve the problem of
scattering by spheroidal and ellipsoidal objects including the classical sepa-
ration of variables method, the series expansion approach of Stevenson, the
point matching method using spherical wave functions and other methods derived
from these. The degree of success of these methods, in terms of accuracy and
economy, is dependent on the dielectric and conducting properties of the spher-
oid and its geometrical properties, e.g., eccentricity. We have applied the
following techniques in an attempt to evaluate the absorption characteristics
for realistic models of man and animals:
Stevenson's Method in the Third Approximation
Spheroidal Function Expansion Method
Point Matching Technique
Extended Boundary Condition Method (EBCM).
Of these, the Stevenson's method in the third approximation and EBCM were
found to be most successful for the problem at hand. These methods are pre-
sented in detail in the following sections.
2-2. STEVENSON'S METHOD IN THE THIRD APPROXIMATION
As stated before, a low frequency analysis of power absorption by pro-
late spheroidal [8,9] and ellipsoidal [10,11] models has been conducted for
nose-on and broadside-incidence only by utilizing Stevenson's method in the
first approximation, and the results are known.
In this report it is shown that the electromagnetic power absorbed by the
ellipsoidal model of man and animals, when radiated by a plane wave at an ar-
bitrary angle of incidence, can be evaluated by applying Stevenson's analysis
[12,13] in the third approximation. The frequency range of validity and the
data accuracy can be improved by utilizing this third order solution for the
model being considered. In addition, Stevenson's general formulation is ap-
plicable for any arbitrary angle of incidence.
According to this method, the problem was formulated by showing that the
general solution of the electromagnetic problem can be expressed formally as a
power series in the ratio of dimension to wavelength. Each term in the series
then requires the solution of a. standard problem in potential theory, the
first solution being the quasi-static solution of Lord Rayleigh. Thus, if the
geometry of the object is such that Laplace's equation can be solved in the ap-
propriate coordinate system, the series can be continued arbitrarily although
the terms become increasingly complicated. The method is completely general
subject only to the range of validity over which the particular order of expan-
sion applies. The method can be easily applied to the ellipsoidal models
since the theory of ellipsoidal harmonics has been fairly extensively de-
veloped. Stevenson has applied this method in the third approximation to el-
lipsoids, and his results are used in this report to evaluate the power
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absorption in uniform tissue models by utilizing the forward scattering theorem
given by Van De Hulst [22]. The results obtained agree well with known results
within the range of validity, which is approximately kc < 1, where c is the
major axis.
2-2-1. Theoretical Formulation
Following Stevenson, the incident field is assumed to be:'
ci , c (9 „ m x ikC&c+my+nz)
,, ,
(6a)
= E
where (£,m,n), (£j,m^,n^
satisfying the relation:
\
and
ik(£x+my+nz)
'
fe, -,
(6bj
are three sets of direction cosines
,n) X (£,m,n)
(£2,m2,n2) = (£,m,n) X (^.rn^np
The expression for far-zone electric field components of the scattered field
are [13]:
t7a)
Here (R,9,) are the spherical polar coordinates of the field point and P and
"F are functions of the surface harmonics sCm) defined by:
p -
p -
P =
j=l JCJ'
s-1 , .
s=2
(8b)
For a three term series the expansion for P is reduced to:
* The time convention e~1(0 rather than
is used in this section to
represent the harmonically oscillating fields.
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Forward
4 Scattered
/ Wave
b-H
Incident Wave
_ _ ik.r
*-y
FIGURE I1. The Uniform Tissue Ellipsoidal Model
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P = k
/t
K2sin<(>)sine + K3cos6] + k { (I^
L2sin)sin8
N3sin<|>cos4>sin2e - ~
22
N2cos4>)sin9cos9
[sin2e(a2cos2
22 22
+ b sin 4>) + c cos 9]} (+ higher order terms).
and P" is obtained from P by making the substitution:
(9)
(£2,m2,n2) •* -(£1,m1,n1) and e -^ y.
(10)
That is, P" is given by the same equation as (9) involving "barred" constants 1(,
etc. The scattering constants Kj , Lj, "Kj, TTj , etc. with j = 1, 2 and 3 are
ipsoidal dielectric constant and permeability,
functions of the semi-axes, ellipsoi
ant^ certain integrals.
direction cosines (£,m,n), (£.,111.,^) and (£2'm2>n2^ ant^ ce
The defining expressions for these are given in Appendix A.
The power absorbed by the ellipsoidal model can be evaluated by utilizing
the forward scattering theorem [22] . For a wave incident at an angle a making
an angle 5 with respect to the x-z plane, as shown in Figure 1', the forward
scattering amplitude is given as:
S(o)
[_
-ikRe"
along the direction corresponding
to £,m,n; i.e., along 9=a and 4>=T
Then the extinction cross section [22] is given by:
The absorption cross section of the ellipsoid can then be determined in terms
of the extinction cross section and the scattering cross section and is given
by:
C . - C - C
abs ext sea
(13)
Where Csca, the scattering cross section is defined as the ratio of power
scattered by the ellipsoid to the power incident per unit area and is given by:
IT
sca
2 ' 9^0 **0 I Re[E X H ] '
(14)
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Substitution of equation (7) leads to [2] :
sea
=-|^k4 ([K.J2+ |K.|2) +
3 3 = 1 M j> l j1 o
J * *
.E. (K.L. -t- K.L.)
3=1 J J 3 J
- [(3a
b2 H- c2)(|K1[?
Kj2) + (3b2
|K2|2)
(3c2 + a2 + b2)(|K
|f3[2)]} + higher order terms.
(15)
The power absorbed by uniform lossy spheroidal and ellipsoidal models can
then be easily calculated by evaluating the coefficients given in Appendix A
for specified e and u and using equations (12), (13) and (15). For uniform
tissue models considered here u=l and £=e'+ie"=£' -i (a/coeo) . Dielectric val-
ues for high water content tissues are given by Johnson and Guy [4] .
2-2-2. Comparison With Long Wavelength Results [8-11]
Examination of the expressions obtained for the absorption cross section
reveals that for nose-on (n=l, incident radiation along z-axis) and broadside
(£.= 1 or m=l; i.e., incident radiation along x- or y-axis) incidence, the de-
rived expressions can be reduced to those obtained for spheroidal and ellip-
soidal models [8-11] if the higher order terms are neglected and if it is as-
sumed that for the tissue model, -£"=(a/oj£0) » £' » 1. For example, for the
ellipsoidal model with E K H incidence £i=m=n2=l, substituting into the expres-
sions for scattering coefficients (Appendix A) and retaining only the dominant
terms in the expressions for CL leads to:
where
abs
IT
K 3
(17a)
and
15L3 , -f3
- f(u)e
f (a2
b2)
with
- Di
abc
I"1;
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- [(e - l)(a2 + b2)Iab + JL j-1 and ^ . ^ .
Utilizing the property of_the tissue model that e > > 1 and retaining only the
dominant terms in K.. and LI gives:
Im K. - - - — = — (I8a)
3 abc I e"
a
Tc"
lb
l?
a
Substituting -e" = a/0)e and equations (13a-b) into (11) and using:
1 2
Pav = ^ 2n~ Eo C hs/volume of the ellipsoid) leads to:
'av'T'V^V - a~r-3 cw)
„,,-,._ .2 , 2 2 ,abc,2T 2,-l , _ b2
where AX - [a no C-y-) Ia 1 and BZ = - -3—J
a * b
which is identical to the result obtained by Massoudi et al [10] for E K H in-
cidence. Similarity, for other cases of both nose-on and broadside incidence
it is seen that the expression for absorbed power reduces to the ones for pro-
late spheroidal [8] and ellipsoidal models [10] if the higher order terms are
neglected and it is assumed that cr/coeo > > e' > > 1. This amounts to retain-
ing only the dominant terms in the extinction cross section and neglecting the
scattered energy which is implicitly assumed in the long-wavelength analysis.
2-2-3. Prolate Spheroidal Model
For the prolate spheroidal uniform tissue model a * b < c. Then,
! f 2 21/2
In [c, " %-*>, (20)
2 212 2 l2
(c2 - aV7 c - (c - aV/2
and the remaining integrals can be calculated in terms of I by using the rela-
tions connecting the elliptic integrals as given in Appendix A. The scatter-
ing constants for the three cases of polarization (a) nose-on with n=l and b)
broadside with m=l and either t,*l or n *1) are given in Appendix B. The
10
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tissue is characterized by U=l and e=(e'-ia/oj£Q). The values of £' and a used
in the calculations are the same as those given by Johnson and Guy [4], It
can be easily shown that scattering constants in Appendix B lead to the same
expression for power absorbed by the_ model as that derived by Durney et al [8]
if the higher order terms in P and P are neglected for the case of >
£'>>!. That is, the expressions derived in [8] approximately correspond to:
Art —
Cabs a Cext * - F- Im[Kl ' 41
for m => £.. = -n_ = 1 (or magnetic polarization case)
Cabs * Cext * - IT
for n = L. = m~ = 1 (or cross polarization case)
Cabs * Cext * - F- ImtK3 + M
for m = nl = L. = 1 (or electric polarization case).
2-3. POINT MATCHING METHOD
The problem of the scattering and absorption characteristics of rain
drops has been formulated by Oguchi [15] and Morrison and Cross [16] . The
technique can be extended with certain limitations to the case of prolate
spheroidal models of man and animals. For the point matching technique, the
incident, scattered and internal fields are expanded in terms of the spherical
vector wave functions with known coefficients of expansion for the incident
wave. The boundary conditions are then applied at a number of points (larger
than the number of unknown coefficients of expansion to be evaluated) on the
surface of the prolate spheroid (r=ab/(a2cos26+b^sin^6)l/2j with a the major
and b the minor axis) and the coefficients are then computed by using the
least square fitting procedure [23]. This leads to better accuracy as com-
pared to the case where the boundary conditions are applied to a number of
points equal to the number of unknown coefficients. A computer subroutine to
compute a least square fit for a linear system with complex coefficients was
written and attempts were made to solve for the coefficients of expansion us-
ing this technique. The accuracy of the results obtained can, of course, be
checked by using the solutions obtained to compute the boundary conditions fit
at points other than those used to obtain the coefficients of expansion.
2-4. SEPARATION OF VARIABLES METHOD
This method follows the analytical technique developed by Asano and
Yamamoto [17] who have formulated the problem of light scattering by small par
ticles in terms of vector spheroidal wave functions. The fields in the three
regions are expanded in terms of the spheroidal vector wave functions [24].
The boundary conditions are then applied at the surface by defining auxilliary
11
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spheroidal angle functions [17] which enables one to apply the boundary condi-
tions to the tangential components of the electric and magnetic fields at £=£0
resulting in a system of coupled equations for the expansion coefficients.
The coefficients are then calculated by truncating the series. A computer sub-
routine to calculate the spheroidal angle functions with complex arguments has
been written which together with known results for spheroidal functions with
real arguments leads to the evaluation of the coefficients of expansion.
2-S. EXTENDED BOUNDARY CONDITION METHOD
The extended boundary condition method (EBCM) was formulated by Barber
and Yeh [18] to evaluate the scattering by arbitrarily shaped dielectric ob-
jects by utilizing Schelkunoff's equivalence theorem. The method has been ap-
plied by Barber [19] to evaluate the absorption characteristics of prolate
spheroidal models in the resonance region. According to this method, as far
as the external fields are concerned, the object can be replaced by a set of
surface currents. These surface currents are such that the fields inside are
zero, i.e., the field produced by these currents exactly cancels the internal
fields resulting in an equation relating the internal fields to the surface
currents (or the tangential components of the fields just outside the surface).
Applying the boundary conditions at the surface leads to a linear system of in-
tegral equations for the coefficients of the unknown internal field in terms
of the known coefficients of the incident field. Knowing the internal fields,
the scattered field can be determined in terms of the surface fields (or cur-
rents). The absorption cross section can then be determined by utilizing the
forward scattering theorem of Van De Hulst [22] or by integrating the incre-
mental internal power distribution CTE^2 in the object.
The analytical technique is outlined in Appendix C for convenience. It
is basically the same as that used by Barber [19] except for some minor modifi-
cations, such as the utilization of complex linear combinations of even- and
odd-vector wave functions for computational convenience and efficiency.
The program flow chart and program listing is also appended, Appendix D.
12
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SECTION 3
NUMERICAL RESULTS
The absorption cross section of various models of man and animals was
evaluated as a function of frequency and the angle of incidence by utilizing
various techniques outlined in the previous section. Each technique was found
to be implementable only over a limited range of frequencies and object proper-
ties and dimensions. For example, numerical results using point matching and
spheroidal wave function methods converged only for a class of unrealistic mo-
dels having low eccentricities, i.e., b/a < 2 and/or having conductivities
much less than the high water content tissue medium. The cylindrical functions
with complex argument (dependent on the object dimensions and complex dielec-
tric constant) undergo an enormous variation in magnitude as one moves on the
surface from a point on the minor axis to one on the major axis, resulting in
a rather ill-conditioned set of linear equations and nonconverging numerical
solutions. We should add that the point matching method has been successfully
formulated and applied by us to evaluate the absorption characteristics of ob-
jects with relatively low eccentricities where the variation in the coeffi-
cients of the system of linear equations for unknown coefficients of expansion
is relatively small. The results obtained using the point matching techniques
were the same as those subsequently published by Ruppin [27] for near spheri-
cal objects.
The absorption cross sections for some typical cases of man and animal
prolate spheroidal models was calculated by utilizing Stevenson's method in
the third approximation as given in Section 2-2 and the extended boundary me-
thod for which the computer program is described in Appendix D. The results
obtained for a rhesus monkey model in the third approximation are shown in Fig-
ure 2 for the three standard cases of incident wave polarization. The absorp-
tion cross section calculated by utilizing the extended boundary condition me-
thod (EBCM) is shown in Figure 3 and as seen, the results obtained by utiliz-
ing Stevenson's formulation in the third approximation are in excellent agree-
ment with the ones obtained by EBCM within the range of validity of the third
order solution. The EBCM results agree well with those of Barber [19] except
that they do indicate a well defined resonance for E incidence at a slightly
lower frequency for the rhesus monkey model.
Figure 4 shows the variation in the absorption cross section near the re-
sonance frequency of the rhesus monkey model as a function of the angle of in-
cidence of the plane wave for the two possible polarization cases (magnetic or
electric field parallel to the minor axis). These results were computed by
EBCM and as expected, they indicate that the absorption cross section increases
13
-------
ISI
c
o
tr. .
3 ro
II
O J3
O (M
—1 O
O II
•a in
O •--
Z x
3
*-M
C3 -H
-o e
.-I O
o •/>
-= O
d,—.
« o -
_ +-> u
LU
O
UJ
en
o o
^ —•
— +j
C3
n e
•H
U-i X
o o
!-i
C C.
O £*
•H C3
•M
O -3
O V<
00 — i
O O
fc J=
U *J
c c
O -n
O -3
-------
a
o o
z*' NOI103S SSOdO
15
-------
o
o
O)
o
in
o
o
CD
o
in
T3
O —
O O
on c
o
in
o
o
c;
^:
a:
o: c
tc
o e
t. —
CO
o
CO
o
CM
o
UOU09S SSOJQ
16
-------
as the component of electric field along the major axis increases. The results
obtained for other prolate spheroid models and the ones obtained by using
Stevenson's method exhibit similar behavior.
The absorption cross section for average man model is shown in Figure 5.
For these models representing average man as a prolate spheroidal object con-
sisting of high water content tissue material, the EBCM numerical technique
did not result in converging solutions beyond about 80 MHz for the two polari-
zation cases in which the E field is parallel to the minor axis and not beyond
about 30 MHz for the case in which the E field lies along the major axis of
the spheroid, as shown in Figure 5. This is due to the fact that the matrices
became ill-conditioned for larger values of conductivity and/or spheroid eccen-
tricity. However, the absorption characteristics can be estimated by means of
an extrapolation method, as used by Barber [19] who seemed to have had similar
difficulty "regarding convergence of solutions for uniform tissue prolate spher-
oidal models of man. The solutions in the third approximation for the average
man model for the case of incident wave polarized along the major axis do not
converge beyond 50 MHz. However, the solution does converge for lower values
of dielectric constants. For example, the absorption cross section for normal-
ized dielectric constant = (0.1) x (^tissue) ^s snown i-n Figure 6. It is in
excellent agreement with the results obtained by Barber [19] who has calculated
the absorption cross section for spheroidal model with normalized dielectric
constants and used them to extrapolate and estimate absorption beyond 30 MHz
for E parallel to major axis incidence. The dashed curve for the E parallel
incidence case in Figure 5 is an estimated solution in the third approximation
obtained by the same extrapolation procedure. Examination of these results in-
dicates that, in general, the solution in the third approximation is valid up
to approximately kc = 1 for most cases. It should be noted that there is no
significant improvement in accuracy at low frequencies for high water content
uniform tissue prolate spheroidal models in the third approximation as compared
to the results obtained by using the long wave length, first order analysis for
the three cases of wave polarization. This should reinforce the accuracy of
the results presented for ellipsoidal models by Massoudi et al [10,11]. How-
ever, at higher frequencies or with objects where the condition a/coe > > e'
> > 1 is no longer valid (e.g., Figure 6); the analysis in the third approxima-
tion leads to more accurate results.
17
-------
z io-'
o
U
UJ
en
en
en
O
rr
o
CL
cr
o
en
ffl
<
to-
E-ln
20 40 SO 80 100
FREQUENCY, MHz.
130
• Absorption cross section of a prolate spheroidal model
of an average nan. c-0.875 «. c/b • 6.3^.
•—• results in the third approximation.
— — •results vising SBCM
estimated values in the third approximation
18
-------
10 r
CM
E
CJ>
LU
CO
CO
CO
O
cr
o
CL
(T
O
CO
CQ
20 40 60 80
FREQUENCY, MHz.
100
Fig.6. Absorption Cross Section of a Prolate Sheroidal Model with
£ =0.1 £ . , c=0.875m, c/b=6.34; E-parallel incidence.
L 15 5LIG
results in the third approximation
results using the long wavelength theory.
19
-------
SECTION 4
CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY
4-1. CONCLUSIONS
In an attempt to evaluate the absorption characteristics of prolate sphe-
roidal models of man and animals at and near the resonance frequencies, several
techniques, including Stevenson's method in the third approximation and the ex-
tended boundary condition method, have been utilized. Even though the absorp-
tion cross sections for many animal models with low eccentricities could be
evaluated at and beyond the resonance frequency, application of these methods
to high eccentricity models lead to nonconverging solutions and ill-conditioned
matrices above certain frequencies. This includes the important case of the
prolate spheroidal model of average man consisting of high water content tis-
sues. The results obtained agree well with known results within the range of
validity of the two methods.
Stevenson's method in the third approximation developed in this report in
Section 2-2 provides a simple direct approach to the evaluation of absorption
characteristics of both prolate spheroid and ellipsoid models radiated by plane
waves at an arbitrary angle of incidence up to approximately kc = 1, where c is
the major axis. Earlier work to date on the low frequency analysis [8-11] has
been applicable in the first approximation to the problem of the absorption by
prolate spheroidal and ellipsoidal models and other objects with a > > we and
for nose-on and broadside-incidence only. The technique presented in the re-
port is more general and accurate and can be applied to objects having arbi-
trary electrical properties, e.g., raindrops and other particles subjected to
radiation at arbitrary angle of incidence. The accuracy and frequency range
of validity for low frequency solutions has been extended through this work by
considering higher order terms.
The extended boundary condition method of Barber was also utilized to
evaluate the absorption characteristics of prolate spheroidal models. After a
careful investigation of other techniques, it was concluded that this method
was the most amenable one for obtaining solutions at higher frequencies. Even
though the analytical procedure is basically the same as formulated by Barber,
we have formulated the computational technique with modifications, such as the
utilization of even- and odd-vector wave functions and have utilized improved
subroutines in an attempt to find more accurate solutions with improved compu-
tational efficiency. For example, our results showed that for some animal mod-
els, resonance occurs at a slightly lower frequency than that obtained by
Barber (e.g., Figure 3 for rhesus monkey). In addition, this work represents
20
-------
an independent source of confirmation, and the results can be used to evaluate
the absorption cross sections of any prolate spheroidal model not available in
the literature.
Only typical solutions obtained for various models have been presented in
the report. No attempt was made to compile the results for various models of
man and animals since the results obtained exhibit similar behavior as those
given in the report and can be easily evaluated by utilizing the third order
approximation analysis given in Section 2-2, and the EBCM computer program de-
scribed in Appendix D.
4-2. SUGGESTIONS FOR FURTHER WORK
A. A direct extension of the work would be the inclusion of even higher
order terms in the analysis using Stevenson's method. This approach, though
time-consuming and complicated, will extend the frequency range over which the
solutions are valid.
B. Attempts should be made to formulate a computational technique by
which the so-called ill-conditioned matrix obtained at resonant frequency using
the EBCM method for high eccentricity models can be avoided. One such approach
might be to improve precision by solving the equations after separating them
into real and imaginary parts. The single word accuracy of the Cyber machine
used in our computations is 15 digits, and the double precision for complex
variables is not provided. It is believed that by providing such a double pre-
cision for complex variables, the frequency range over which the solutions can
be found can be increased for a given model.
C. Attempts should be made to formulate analytical techniques which lead
to converging solutions near resonance for high eccentricity high water content
tissue models. One possible approach would be to formulate Schellkuneff's
equivalence principle, which is the foundation of the EBCM method in terms of
spheroidal vector wave functions. Such an approach might lead to converging
solutions.
D. Attempts should be made to incorporate various realistic boundary con-
ditions in analyses involving prolate spheroid or ellipsoidal models. One use-
ful problem would be to evaluate the effect of the earth's surface on the ab-
sorption characteristics of various models. Another important problem is the
evaluation of the absorption characteristics of the models placed in guided
wave structures, such as the rectangular coaxial structures. Such structures
have been used to measure the absorption characteristics of various models as
well as experimental animals.
21
-------
REFERENCES
1. Anne, A., et al., "Relative Microwave Absorption Cross Section of Biologi-
cal Significance," in Biological Effects of Microwave Radiation, Vol. 1,
pp 153-176, New York, Plenum; 1960.
2. Shapiro, A. R., et al., "Induced Fields and Heating Within a Cranial
Structure Irradiated by an Electromagnetic Plane Wave," IEEE Trans. Micro-
wave Theory Tech., Vol. MTT-19, pp 187-196; February 1971.
3. Kritikos, H. N. and Schwan, H. P., "Hot Spots Generated in Conducting
Spheres by Electromagnetic Waves and Biological Implications," IEEE Trans.
Bio-Med. Eng., Vol. BME-19, pp 53-58; January 1972.
4. Johnson, C. C. and Guy, A. W., "Nonionizing Electromagnetic Wave Effects
in Biological Materials and Systems," Proc. IEEE, Vol. 60, pp 692-718;
June 1972.
5. Joines, W. T. and Spiegel, R. J., "Resonance Absorption of Microwaves with
Human Skull," IEEE Trans. Bio-Med. Eng., Vol. BME-21, pp 46-48; January
1974.
6. Weil, C. M., "Absorption Characteristics of Multi-Layered Sphere Models
Exposed to UHF/Microwave Radiation," IEEE Trans. Bio-Med. Eng., Vol. BME-
22, pp 468-476; November 1975.
7. Lin, J. C., et al., "Power Deposition in a Spherical Model of Man Exposed
to 1-20 MHz Electro-magnetic Fields," IEEE Trans. Microwave Theory Tech.,
Vol. MTT-21, pp 791-797; December 1973.
8. Durney, C. H., et al., "Long-Wavelength Analysis of Plane Wave Irradiation
of a Prolate Spheroid Model of Man," IEEE Trans. Microwave Theory Tech.,
Vol. MTT-23, pp 246-253; February 1975.
9. Johnson, C. C., et al., "Long-Wavelength Electromagnetic Power Absorption
in Prolate Spheroidal Models of Man and Animals," IEEE Trans. Microwave
Theory Tech., Vol. MTT-23, pp 739-747; September 1975.
10. Massoudi, H., et al., "Long-Wavelength Analysis of Plane Wave Irradiation
of an Ellipsoidal Model of Man," IEEE Trans. Microwave Theory Techn., Vol.
MTT-25, pp 41-46; January 1977.
22
-------
11. Massoudi, H. , et al., "Long-Wavelength Electromagnetic Power Absorption in
Ellipsoidal Models of Man and Animals/' IEEE Trans. Microwave Theory Tech.,
Vol. MTT-25, pp 47-52; January 1977.
12. Stevenson, A. F., "Solution of Electromagnetic Scattering Problems as
Power Series in the Ratio: Dimension of Scatterer/Wavelength," Jour. Appl.
Phys., Vol. 24, pp 1134-1142; September 1953.
13. Stevenson, A. F., "Electromagnetic Scattering by an Ellipsoid in the Third
Approximation," Jour. Appl. Phys., Vol. 24, pp 1143-1151; September 1953.
14. Van Bladel, J., "Electromagnetic Fields," McGraw-Hill; 1964.
15. Oguchi, T., "Scattering Properties of Oblate Raindrops and Cross Polariza-
tion of Radio Waves Due to Rain: Calculations at 19.3 and 34.8 GHz," Jour.
Radio Res. Labs., Vol. 20, pp 79-118; 1973.
16. Morrison, J. A. and M. J. Cross, "Scattering of a Plane Electromagnetic
Wave by Axisymmetric Raindrop," Bell Syst. Techn. J., pp 955-1019; July-
August 1974.
17. Asano, S. and G. Yamamoto, "Light Scattering by a Spheroidal Particle,"
Appl. Optics, pp 29-49; January 1975.
18. Baber, P. and C. Yeh, "Scattering of Electromagnetic Waves by Arbitrary
Shaped Dielectric Bodies," Applied Optics, Vol. 14, pp 2864-2872, December
1975.
19. Barber, P. W., "Resonance Electromagnetic Absorption by Nonspherical Di-
electric Objects," IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp
373-381; May 1977.
20. Schwan, H. P. and K. Li, "Hazards Due to Total Body Radiation," Proc. IRE,
Vol. 44, pp 1572-1581; November 1956.
21. Livesay, D. E. and K. M. Chen, "Electromagnetic Fields, Induced Inside of
Biological Bodies," IEEE-S-MTT, Intl. Microwave Symp. Digest, pp 35-37;
1974.
22. Van De Hulst, H. C., "Light Scattering by Small Particles," Wiley; 1957.
23. Businger, P. and G. H. Golub, "Linear Least Squares Solutions by House-
holder Transformations," Numer. Math., 7, pp 269-276; September 1965.
24. Flammer, C., "Spheroidal Wave Functions," Stanford Univ. Press; 1957.
25. Strattan, J. A., "Electromagnetic Theory," McGraw-Hill; 1974.
26. Morse, P. M. and H. Feshbach, "Methods of Theoretical Physics," McGraw-
Hill; 1963.
23
-------
27. Rupin, R., "Calculation of Electromagnetic Energy Absorption in Prolate
Spheroids by Point Matching Method," IEEE Trans. Microwave Theory and Tech.,
Vol. MTT-26, pp 87-90; February 1978.
24
-------
APPENDIX A
SCATTERING COEFFICIENTS
The expressions for the scattering coefficients in equation (4) are given
by:
(Al)
15L1 = f^eHCe - D^[ y (6a2 - b2 - c2) - (a2£2 + b2m2 * c2n2)]
+ e(b2mn2 - c2nm2); + f^Ce)^ {(e - !)[(£ - 2)1
+ ea2Ia - g. ] + £2u(b2 + c2)/(a b c)} + £1(e)g1
x{(I, - IG) [ j (b + c ) (mn2 + ran,,) - mn2b - nm2c ]
"7
b c)} + f1
x{(I - I)[(e - DftO + £U(£U - 2)(b2 - c2)/(a b c)]} (A2)
b
15Mi -
:a"b )
x(2a2ttl - b2mm1 - c2nn1)] (A3)
15N. = - \ (y - l)f.(u)(b2 - c2)£? + gl(£)[(£/2)(b2 + c2)
1. O X ^ i,
xCmnj^ + nmp - b nn^ - c nm1] + f^vOg^e)
x£2[eu(b2 - c2)/(a b c) - (yi - ^k^e)] . (A4)
25
-------
In the above expressions, the undefined functions and integrals are:
I »/" dU
a ° (a2 * u)R(u)
R(u) = [(a2 + u)(b2 f u)(c2 •
I.u - f ^
ab 0-2 ,. ,, 2 \ ,, /• i
(a + u) (b + u)R(u)
= ECU-
b2Ib- c2Ic-f (b2
4e
2a b
2
2. 2 2_ 2.2.
t) c (la b )
T /» du ,r _ ^ ,, 2 ,2^
J = -C - 3 = (Iab " ^c^^ " b }
[R(u)]3
j. . y^ -JLJ" , T . C2j etc.
° [R(u)]3 ab
-2 2.2 2 _ 2. 2 2. 2 ,2 2 22
2a =a + b +c Zab =ab +bc +ca.
The remaining scattering coefficients K2, £3, L2> 1*$, M2, MS and N2, Nj in
equation (4) are obtained by cyclic permutations of a, b, c; L, m, n; LI, mj,
nj; and ^2> m2> n2-
The scattering coefficients referring to magnetic field Kj , Lj , Mj and Nj
(j - \, 2, 3) are obtained from the corresponding Kj , Lj , Mj and Nj by making
the following substitution:
26
-------
The interrelationships between various elliptic integrals given above en-
able us to calculate all of the integrals I, Ia, ... in terms of only two of
them for the case of an ellipsoid, and only one of them for the case of a sphe-
roid. These relations are [2]:
a b c abc
2T ,2, ' 2T T
al + b I, + c I =1
abc
3Iaa + 'a* * !ac = ~
a be
3a2I + b2! . + c2! = 31 .
aa ab ac a
27
-------
APPENDIX B
PROLATE SPHEROIDAL MODEL -
EXPRESSIONS FOR COEFFICIENTS
CASE 1
Broadside Incidence — Electric Field Along the Major Axis
In this case, the direction cosines are n, = £ = m = 1 and £, = m, = nu
ri2 = £ = n = 0. Then the coefficients are:
K3 - | (e - I)f3(e)
2 2
15L, = £3(e)[(e - 1) 6c " 5b - cb2] + f3(e)2((e - l)[(e - 2)1
2T 4c , 2e2 ,
+ EC I -- +--)
22 22
15N = g,(e)[e b * c - b2] * f1(u)g1(e)[e b ' c ]
b c
22 22
+ f:2Cu)e b * c * f1(u)g1(e)[(Ib - IG) (b2 - e b * c )
b c
,2 2 ,2 2
e " c ] + £ C^gCeDd - I)[e(e - 2) " °
•i J •*• 1 V.H./61 Vs-J *.-*•}, •*•/»•' L v "J 9 J
be i 1 b c b^c
K = ir = L. * L- = M. * NL = M. = N0 * N, = 0
121^12523
and
28
-------
CASE 2
Broadside Incidence — Magnetic Field Along the Major Axis
In this case the direction cosines are: £ = -n~ = m = 1 and m, = n..
£9=m_ = £ = n = 0; and the expression coefficients are:
£, £
Kx = | (e -
2 2
- eb2} + ffcHCe - l)[(e - 2)1
be
2 .'2
I, - I J[(e - 1)K GO - e(e - 2) " ].
b c
15N3 = g3(e)b2Ce - 1).
15L3 = b2f3(y) - 2f32(y) | .
22 22 22
15N = f (e)(l - e) c " b ° " b c " b
- - l 1
b c
+ (e - 1)1 00].
K2 = K, - L2 = L3 = MI - M2 = M3 = Nl - ^ - 0.
\ =¥2 = S « ^ = L2 = Mx = M2 = M3 = N2 = N3 = 0.
CASE 3
Nose-On Incidence
For this case, t^ = m2 = n = 1 and m. = n. = £ = n2 = £ = m = 0. Then
the coefficients are given by:
(e -
29
-------
15L1 =
2 2
5b " 4° ) - £c2}
- 2)1
-
1 1 D C
1
(y) - e(e - 2)
22
15l2 = -f2(y)c
b \ C
b c
b c
2 b2
'~^T
- 2)
e2 - b2
b2c
22
2 7
15N = £1(£)d - £)
be
(£ -
K2 ' K3 = L2 = L3 = Ml = M2 " M3 ' Nl
0.
' K3 = Ll = L3 = Ml = M2 - M3 " N2 ' N3 = °'
30
-------
APPENDIX C
EXTENDED BOUNDARY CONDITION METHOD
C-l. THEORETICAL FORMULATION
Following the procedure given by Barber and Yeh [18] as outlined earlier
in the text, the surface currents for external fields (Figure C-l) are given
as:
-»• " -»-i -*s ~ -»•
J = n X (H + H ) = n X H
+ ^ ' +
• /*\ /\
and M+ = (E1 + ES) X n = E+ X n .
->• -*•
The scattered fields^ everywhere due to J+ and M+ can be determined from
the vector potentials of A and F.
r+exp(jk|r-r'i)
A = -=— I -
4TT •'s
and
A = -i-J ds, J = n X H (C-la)
ATT —J o-^-^i * + ^. ^^
r-r1
„
. M expCJk|r-r'|) ^ ^
- 4?JS - |-T- - ds- M+ = E> X n
(C-lb)
-- -
where r and r' are position vectors from an interior origin to the field and
source points, respectively, and eJwt time variation has been assumed. Then,
ES = -VxF - -•- (Vx7x A), (C-2a)
NS = VxA - -^ — (VxVx F). (C-2b)
Substituting equations (C-la) and (C-lb) into equations (C-2a) gives:
ES(r) = Vx Js (nXE+)g(kr)ds -VxVxJs ^— (nXH^)g(kr)ds, (C-3)
o
31
-------
E ,ir
A
71
TfT|
V
(a)
i -»=; ' *>•
+ s r//
x+ IP 7/ ^
^ .'
i^^T/:
— '" i 1Q-
» !^i i
Zero
Field
\
Figure C-l Summing of Sources and Fields to obtain the external
problem."
32
-------
where g(kr) is the free space Green's function as given by:
g(kr) = exp(jkr)/4irr with r = |r-r| , and k = 2ir/X =
-------
and \ = - VXM, M = - 7XX.
"
Here, zn denotes a spherical Bessel function of order n and P;n' denotes the
associated Legendre functions of the first kind of degree n and order jm| where
m is a positive or negative integer, and n is an integer with n > imj and n ^ 0.
For solutions of the wave equation that must be finite at r = 0, zn(kr) =
jn(kr), and the resulting vector spherical functions M and N are known as solu-
tions of the first kind. The other vector spherical functions we will be using
are called spherical Hankel functions, zn(kr) = hn^ ^r) = Jn^1"-* "*" J^nC^r) .
This solution represents outgoing waves and is called the solution of the third
kind.
Using these spherical vector functions, the incident fields are given by:
E^r) = Z E D [A M1. (kr) + B N1 fkr)] (C-7a)
^ ! mnL mn mn ; mn mn ^ J
m= -°° n> i m |
n/0
H1^) = iL- E E D [A N'1 (kr) + B M1 fkr) ] , (C-7b)
J oju ^ • • mnL mn mnv mn mn - J
o m= -00 n>_| m |
n>0
where Dmn is a normalization constant, and the expansion coefficients Amn and
8^ are known for a specific incident field. Note that,
D (2n+l)(n-
mn n(n+l)(n+jm|)!
The terms in the integrals of equation (C-5) are expanded as follows,
with the help of the free space Green's dyadic given by Morse and Feshbaach
[26].
(nXE+)g(kr) = (nXE+) • f (C-8a)
(nXH+)g(kr) = (nXH+) • G" (C-8b)
The free space Green's dyadic is:
m=-c°n>m
(k7 )N (kr ; + irrotational terms] C-9a)
'
-ran ' mn
34
-------
or = - S I D [jr (kr 3M (krj
4ff . | I mnL mn^ >J -mn <
tn=-°° n>J m|
n>0
••• N3 (krjN1 (kr.,) + irrotational terms], (C-9b)
mn 7 mn **
* * ct-> -v
where r.^, rc are, respectively, the greater and lesser^r, r'. The irrotational
terms present in equation (C-9) are washed out due to the curl operation.
Equation (C-5) is required to hold throughout the entire interior volume,
i.e., ^nside s, (analytic continuation property). Then, restricting the field
point r to lie within s,
VX J (nx£+)g(kr)ds = VX J (nXE+) • G ds
i ? l< ,DmnUs tHMM3m(kr').VXM^(kr)
m=-°° n> m
l£~ ? I D [[ (nXE )'(M3 (kr')-N1 (kr)
4ir I i mn'-Js +J ^ -mnv ' mn
m=-°° n>_| raj
n>0
f N3 (kr')-M1 (kr))ds],
-ran mn JJ J
Similarly,
A
VXVX/s jST
J o / — m=-°° n>|m|
£o
Equation (C-5) becomes:
i^izo
4Tr mn
-------
-H ->
= -E^kr)
= +£ID [A M1 (kr) + B N1 (kr) ] . (C-10)
mn L ran mn mn mn -^ J ^ '
This gives the following set of equations, because of the orthogonality of the
functions over a spherical surface about the origin (or just from the fact that
left side and right side are equal if and only if the coefficients correspond-
ing to the same M^ or M^ are the same) .
urn 'Nran
2 ,-
4^- [N3
4ir JsL -
mn(kr')-(nXE+) + jjiT M^mn(kr')-(nXH+)]ds = A^ (C-lla)
J~
o
ik2 f +3 -»• " -> /— +3
4^- [M (kr')'(nXE ) + j fiT N (kr')-(nXH )]ds = B (C-llb)
4ir J s -ran + I p_ -mn •*•'J mn
N £
o
The solution of this equation guarantees that the total field will be
zero within the inscribed sphere, and hence by using the concept of analytic
continuation within the entire interior volume under consideration.
C-2. EXTENDED BOUNDARY CONDITION APPLIED TO THE SURFACE
Now the field inside the lossy dielectric can be approximated by:
[c M1 (k'r) -i- d N1 (k'r)] (C-l2a)
i l L mn mnv ' mn mnv 'J
i> m
o m=-oo
where k1 =
The boundary condition at the surface requires that the tangential compo-
nents of the fields be continuous, i.e.,
nXH = nXH (C-13a)
nX£+ = nXE_ (C-13b)
36
-------
The (+) and (-) sign subscripts on the surface fields indicate that these fields
are the external and internal fields, respectively, at the surface. From equa-
tion (C-13), the tangential components of the internal fields at the surface
are:
nXE = E I [(c nXM (k'r1) + d nXN (k'r')] (C-14a)
no v. i I mn mn ' mn mn ^ ' J *• '
m=-°° n> m|
nXH = j ~ I I [c nXN (k'r1) + d nXM (k'r')] (CM4b)
J coy | i L mn mn v ' mn mn ' J *• ;
o m=-°° n>| m|
Now, due to the equality of the tangential surface fields in equations (C-13) ,
equations (C-14) can be substituted into equations (C-ll). Then equation
(C-lla) leads to:
7
/£
r f -*-^ -*• -*•! •*• / T I -*"$ -*• -*•! -*•
I {[! N (kr')XM , .(k'r^-nds W ^ J M (kr')XM , fk'r')
n'>im| S ~mn m" ^ Wr S "mn mn
Us
•ndsld , , = A (C-15a)
J m'n1 mn v '
Similarly,
n, (k'r'
'n' = Bmn ^C
m = _<» to °° n ^_ Imi , n i 0
n1 = -°° to ro n1 ^_ |m , n t 0
Using the notations:
37
-------
,. a K I -+•$
mm'nn' ~ 4iF j n"' -j
(k'r')ds
nM'nn'
, C*'?')ds
Kmm'nn' = '
leads to:
m''n'
7^ J]cn,n,
I]dm,n, -
(C-l6a)
L]c H- [L
m* ,
(C-16b)
These are infinite sets of linear equations and can be approximately solved for
c's and d's by truncating and utilizing suitable numerical methods.
Subsequent similar analysis provides a relationship between these coeffi-
cients and the expansion coefficients of the scattered field. Scattered fields
are assumed to be expanded as follows.
Es(kr) = - Z Z D [a M° (kr) + b N J (kr) 1 ,
1 ' t mnL mn mn } mn mn^ ; J '
m=-°°
and
o m=-°° n> |m
b NT (kr)'
mn mn
Following as in previous case leads to:
m'
(C-17a)
v J
(C-17b)
38
-------
mn
I E {[I- + /e L']c + [J' * /S~K']d ,
m'n' hr1 Ir^ m n
v/ rr -J yr
(C-18b)
where
''.-an- = "-
;lT1,(k'r'-)ds
•)ds
K
mm'nn'
= T-
4ir
^kr'DXMj^.Ck'r'Dds
Wan-
llT,,(k'r')ds
From (C-16) and (C-18), a and b can be expressed in terms of A and B by
. . mn mn c mn mn
using matrix notation.
-i
mn
mn
wr
J'+/e~K'
1 r
J Pr
i+ /r; L
mn
mn
CC-19)
C-3. APPLICATION TO SPHEROIDAL PROBLEM
Due to the axial symmetry property, the integration from $ = 0 to = 2ir
gives the complete orthogonality property, i.e., the terms with m1 / m vanish.
Then since,
nds
e)r2 sin6d6d
mnnf
(cos9) + -.
39
, (cos9)]
(C-20a)
-------
Similarly,
J
*
mnn' i
V
, (cose)
j,(k'r)
CcosB)
(C-20b)
dP™ (cose) dP™, (cose)
h1 (kr)
dPm, (cose)
(c.20c)
mnn
dPm (cos0)
Pn' (cos6:)-d¥ } + Pn (cose) P^, (cose) [n
™
J_,(k'r) dP, (cos6)
-T- M, (k'r)] (P
h1 (kr)
h'(kr)
(_n_
J,(k'r)
And using the mirror symmetry property of spheroid, r(u - 8) = r(9), and with
aids o£ the properties as:
cos (IT - 6) = -cos6, sin(ir - 9) = sin6
P^(cos(Tr-9)) =
= (-I)m+V(cos9)
40
-------
I <*r (a2-b2)sin6cos9 1_ dr
r d9 = 2 2Q , 2 . 2a " r d6
a cos o + b sin o
and
1 dr
= -j [(n-m+DCn+m)?^"1 (cos(ir-9)) - P1^1 (cos (T-9))]
leads to:
I . = L . = 0 for n + nf even (C-21a)
mnn' mnn1
J , = K , = 0 for n + n' odd (C-21b)
mnn' mnn1 v '
and integration is performed from 9 = 0 to 9= ir/2. Equation (C-21) reduces the
matrix size further by a factor of two. In addition, our definition of complex
spherical wave functions leads to:
I I.I.I
a = -a . b = b
-mn mn -mn mn
For polarization I,
(H field parallel to I I I I (C-22a)
minor axis) z = -c , d = d
-mn mn -mn mn
all = all II = II
For polarization II, -mn " mn ' -mn mn
(E field parallel to (C-22b)
minor axis) II _ II .II _ II
-mn mn ' -mn mn
These lead to the following expressions for I, J, K and L.
(kr) d9; for n + n' odd (C-23a)
41
-------
r.r/2 , jn,(k'r) , m2 m m
J , =1 {h (kr)(-~-; -- j , (k'r)) [-^-7- P (cose) P, (cos8 )
mnn ' J 0 n v ' v k'r Jn'^ ^Ls in6 n n ' J
*
dP.-.e) dP(unfcO , j (k'r) dp^e-j
11 11 -i i/*ii-\i/"i*\l* **
+ — r- — -3 — j + n'(n'+l)h (kr) - =-; -- 77?-
d8 d6 rr ' k'r d8
)> (kr)2sin6de; for n + n' even (C-23b)
TT/2 h^kr) , 2
, =-| {[Jl - + h1 (kr)]j ,(k'r)[-Ar- P (cos6)P , (cosG)
mnn' JOLL kr n^ ^JJn' 'L.2Q n n1
sin o
de de J "<.«--j kr Jnl ^ ^.
JDm / " ,i\
dP , i_c«r>t?) j ~
—3T- (^- )}(kr) sinede; for n + n1 even (C-23c)
dQ rdQ
12 hX(kr) , j ,(k'r) ,
{[JL + h1 (kr)] [ n — + j (k'r)][P^(cos8)
,v .- dr
- * ~ p' (COS6)1 *
h'(kr) j ,(k'r) , h1 (kr)
j, (k'r)) + n'(n'*l)C ~-
j , (k'r) 2
n }(kr) ae.
k.r
The expressions for I' , J'^,, K'^, and L'mnnl are the same if we re-
place h^(kr) by ]n(kr) .
42
-------
APPENDIX D
DESCRIPTION OF COMPUTER PROGRAM FOR EBCM
D-l. INTRODUCTION
This program has been written in FORTRAN IV for CDC 6400-CYBER computer.
Several factors were given consideration in the design of the program.
Efficient coding was emphasized to reduce the computer run time, and re-
dundant expressions were avoided as much as possible. The complicated expres-
sions are degrouped for the use of other inter-related expressions. This pro-
gram can be run on any other standard computer with FORTRAN compilers support-
ing complex variables, with one addition of subroutine for the solution of
complex linear equations. In our program, the math-subroutine from IMSL was
used for that purpose.
Full single word accuracy of CYBER machine is 14-15 digits, and the double
precision for complex variables are not provided, even though for some situa-
tion, it was desirable. The memory occupation is not very significant to this
program because several (20 x 10) matrices are sufficient to give accurate re-
sults as long as the matrix doesn't become ill-conditioned. It was observed
that once the matrix becomes ill-conditioned, the increase in matrix size does
not contribute to any significant improvement.
D-2. GLOSSARY OF THE SUBROUTINES
The program operated via a main routine and 10 subroutines including one
IMSL math routine, which are briefly described and listed below. Standard I/O
and mathematical routines; e.g., sin, cos, etc., are assumed to be available.
The main routine, called EBCD, controls overall run processing, and com-
putes [I], [J], [K], and [L] matrices, and the final results of the scattering
efficiency, the absorption efficiency and the extinction efficiency.
Subroutine Scatter: computes the contribution to the overall scattering
and the extinction of each azimuthal index m.
Subroutine Sphere: computes the scattering and extinction coefficients
for the known ideal case of the sphere for compari-
son and for the program testing for the early stage
of this work.
43
-------
Subroutine Shape: computes the geometrical variables for our model of
spheroid.
Subroutine Hankel: computes the spherical Hankel functions of 3rd kind
for real arguments .
Subroutine SPHBESS: computes the spherical Bessel functions for complex
arguments .
Subroutine Legendr: computes the associated Legendre functions.
Subroutine LEQT2C: (IMSL math-routine) performs the L-U decomposition
of complex matrix, and then computes the solution of
complex linear equations by iteration method,
Minor Routines
Subroutine MULTRX: computes the multiplication of square matrix with
column matrix.
Function DFACT: computes the double factorial defined as (2n) ! !
Function FACT: computes the factorial of integer n.
D-3. COMPUTATION OF SPECIAL FUNCTIONS
A. Associated Legendre Functions
Subroutine LEGENDR generates the associated Legendre function P^(cos6),
its derivatives (dP^) / (dQ ) , and the frequently used (mP^)/(sin9) for a given
angle 9, a given value of the azimuthal index m, and for all values of degree
n from m to specified nmax.
The following recursion relation and formula were used.
C2n + l)cos9P™ = (n + m)P™ l + (n - m + l)PjJ+i
(2n-l)(cos9Pm-Pm J
nm _ anm nm n n-1 ,n , ,
or P , = 2cos6P - P . + - = - (D-l)
n+1 n n-1 n-m+1 v '
and for the derivatives ,
dPm
-m - ^9 u» - m + ^Ci - (n + 1
Beginning with:
_.m ., Q, (2m)! sin 6 A ro , . . . mQ
P (cos9) = - — - - = (2m)!! sin 9
2mm!
44
-------
~ (2m 1- I)cos9pjj(cos9). (D-4)
Special note for ra = 0, and 6=0°. n n
dPu mPu
i) Form* 0, 9 = 0"; P° = 1. -£ . 0. ^ - 0.
dPm mPm , .,
•••> T* i a no nm n n n n(n+l)
ii) For • - 1. 6 - 0 ; Pn - 0. -35- -
dPm mPm
Hi) For«> 2. 6 .O'; Pj.-ajj. - ^ . 0.
B. Hankel Fimctions
Subroutine HANKEL generates the Hankel functions of the third kind, and
its derivatives, for a specified real argument x, for all values of order n
from 0 to njjgjf specified.
i) Spherical Bessel Functions
The backward recursive relation is used as:
beginning with sufficiently large N > > n^x, JN+I = JN = °- Then normal-
ized with respect to JQ(X) = (sinx)/x.
ii) Spherical Neumann Functions
The forward recursive relation is used as:
beginning with HO (x) = (-cosx)/x, andnjCx) » (cosx)/(x ) - (sinx)/(x)
iii) Derivatives in either cases are computed by:
nf ,(x) - (n+l)f (x)
" ' - CP-6)
C. Spherical Bessel Functions for Complex Arguments
Subroutine SPHBESS generates the spherical Bessel functions and its deri-
vatives for complex arguments, for a specified argument x and for all the val-
ues of n from 0 to nmax specified, by the same method as for the real argument.
45
-------
D-4. COMPUTATION OF GEOMETRICAL FACTORS
Subroutine SHAPE computes r and (dr)/(rd6) for a given 6 and for a given
dimension of a and b, as:
(a cos 9+b sin 8)
(a2 - b2)cos9sine/(a2cos29 + b2sin2e)1/2
(D-7)
(D-8)
D-5. COMPUTATION OF SCATTERING, ABSORPTION AND EXTINCTION CHARACTERISTICS
n
Qt - "
(2n+l)(n-
m
A*
nmAmn
mnmn
. I I
= -Re[S S (-i)n'1{[a -?— Plm| (cosa)
L v J L mn sina n ^
m n
for electric polarization, and
Qt . -Re[Z E (-
m
m n
dP' (cosa)
-^_ -
for magnetic polarization.
m n
m
mn
mn
Qa = Qt - Qs •
Then the cross- sections are computed by:
"1
d? (cosa)
mn da
]}
J
(cosa)] (D.9)
and the efficiencies are computed by:
= a /ira(a cos a + b sin a)
(D-13)
, , 2 2 ,2.2 ,1/2
= a /ira(a cos a + b sin a)
22 22 1/2
= a /fra(a cos a + b sin a)
cl Si
46
-------
D-6. INTEGRATION SCHEMES
There are various numerical schemes of integration available. Our choice
was the simple Trapezoidal rule against other methods. For the periodic func-
tion with a continuous ktn order derivatives, if the integration is taken over
period, then this scheme gives better results than other methods. Though
Bode's 6th order scheme was also tested, and the results were the same.
x £_ f
Trapezoidal rule; /" f(x)dx = h[ ~ + f. + . . . + f . + ~ ] (D-14)
u z j. n™" A /
x. ,
Bode's 6th order; f° f(x)dx = -™r [41fQ + 216f + 27f + 272f
X0
+ 216fc + 41fJ.
o o
D-7. DESCRIPTION OF VARIABLES USED IN THE PROGRAM
Special care was taken to have the variables named as close to its repre-
sentation in mathematical expressions.
A. Special Function
i) Legendre function:
. dPm(cose) mPm(cos9)
P(n) = P°(cos6), DP(n) = —^ , PM(n) = —r
ii) Hankel function:
SPHANK(n) = h (x) = j (x) + jn (x)
DSPHANK(n) = hn'(x) = j^(x) - jnn(x),
iii) Spherical Bessel functions for complex argument:
SPB(n) = j (x)
; x complex.
DSPB(n) = j'(x)
B. Matrices for Surface Integration
A v f ~
IMMl(i,k) = JjpJ n • M.(kr») x M (k^^ds, for i+j even
47
-------
JMN1
2
(i,k) ^ ~ J n
, for i+j odd
KMNl(i,k) = ^1' n • N^.(kr')
, for i+j odd
LMNl(i,k) = jp n '
and IMN2, JMN2, KMN2, LMN2 correspond to NT
C. Matrices for Linear Equation
.(k,r')ds, for i+j even
(1) M(3)
M1
.
-m,i
N1
i) PMTRXl(n,n) =
i) QMTRXl(n.n) =
-
mm
TL
m+l,m
KJ _
m+2,m
~JK
m,m
LT .
m+l,m
JK 0
m+2,m
LI
m,m+l
m+1 , m+1
LT
m+2,m+l'
TL
m,m+l
KJ . .
m+1 ,m+l
ZL
KJ
TL
• •
JK
LTi
* *
m,m+2
,m+2
m, m+ 2
i
m+1,m+2'
where KJi j = KMNl(i,k) + /sTJT JMNI(i,k) and so on. PMTRX2(n,n) and
QMTRX2(n,nj correspond to IMN2, JMN2, KMN2, LMN2. (Note: for m = 0,
PMTRX1 and QMTRXl(n,n) interchanges its role.)
iii) WAI (N) [, The first (nxn) locations contain the L-U decomposition of
< = a row-wise permutations of linear matrix. The last n lo-
WA2 (N) j cations contain the pivot indices. The remainder is used
as a temporary work space.
WAI, WA2 correspond to PMTRX1, QMTRX1, respectively.
iv) The right-hand side of linear equation:
(cosa)
AT ii i
COEFI(n) for PMTRX1 4 in+1 JL- PH (cosa) or in+1
COEF2(n) for QMTRX1
Other are like: X(n), Y(n) are used for temporary input and output of
linear equations.
48
-------
QA(10,2) = Absorption characteristics
QS(10,2) = Scattering characteristics
QT(10,2)'= Extinction characteristics
where the first dimension represents the incident angle, and the second
dimension represents the polarization mode.
COA(IO) = the cosines of incident angle a
SIA(IO) = the sines of incident angle a .
D-8. INPUTS TO THE PROGRAM
Card 1; A,B
where A ? the dimension of the long axis of the spheroid in cm.
B = the dimension of the short axis of the spheroid in cm.
Card 2; FREQ, EPS
where FREQ.= the frequency of the input wave in MHz.
EPS = the corresponding relative dielectric coefficients of the
model (complex).
Card 3; NMAX, MMAX, KMAX, NSECT, KEY
where NMAX | njnax
MMAX | "hnax
KMAX f the number of incident angles considered
NSECT ? the number of integration steps from 0 to ir/2
KEY * the parameter to be used for printing control of (I,J,K,
L) matrix. If K = 0, printing suppressed, if K = 1,
print the matrices.
49
-------
D-9 OUTLINE OF THE GENERAL FLOW OF THE PROGRAM
_
Initiailzatlor
Raad Input Data
Compute the appropriate constant variables
Determine the Integration step sire69 and the Incident aju?l» increment
___ Set overall snaracterstjes QT - Q3 - 0
i Sat aztnuthaJ Index m - Qv H-; in the prcgrajn) 1
l>t I.'j ii.L i
LStart Irtecration ]
Set v - C
Compute geometrical factors
Compute Legendre functions.Hankel functions and Bessel functions of
complex arguments.
Coapute the step-size Integration for each element of
. I.J.K.L Mtrices,then add to them correspondingly
IChecK for integration ending (9 -TT/2)| .If no'SBt °"*tte "d fi° '
If yea, integration ends for anazimuthal index i
Perform BOM modifications to l.J.K.L.e.g.. changing signa. multiplying
aoB* couon factors. For ••O.th* role of J.K Bat^ice* ar* changed for
future consistent trsataent aa for other »'a.
i Check for KEY
I If KEY IB t print the I.J,K,L matrices otherwise skip printing
Computation of PHTRX.QMTRX matrices for linear equation*, j
For n-O.half of this process is skipped because the |
corresponding matrix elements are tero . j
_ of the matrix operation i
Decompose the PHTRX1 and 4HTRX1 through LU decomposition by
IKSL lubroutln* called LEQT2C.
Save the decompoasd aartix in MAI and MA2 for future use.
I Set incident
=-
Computations of corresponding incident coefficients
First compute th« Legendr* function for oc then compute the
coefficient. For «-0. the role of COEF1 and CDEF2 are
changed for future consistent use.
For each polarisation and for each PHTRXl and QKTHX1 chose the
appropriate right hand side coefficients. Then find the solution |
by iteration method via subroutine L£QT2C.The solutions are the j
absorption coefficients. j
Compute the scattering coefficients by simple
multiplication of the absorption coefficients and
the corresponding PHTHX2 and 3KTBX2 matricea^^
If the model is a perfect sphere .compare th«s«i
results with the known analytical solutions by j
calling subroutine SPHERE
' Canpute trie contribution of these coefficients (
1 to the overall character-sties by calling <
I subroutine SCATTER
[Add thVse contributions to the overall charaeterstlcs gf Q3 and QTj
~\ If no's° to(^ )
Did all four
If yes.flriiBhed for one incidjant^angle
for the Incident
If n^.set^«"<
Check for n "
If yee, finished for one azimuthal index •
~\ If no. set IB •»*i and go tofT)
If yes flnishxl for all n's
Compute the overall cross sections for the
absorption. jsatt*ring and «jrti ittlon and print
Compute the overall efficiencies of the absorption,
scattering and extinction of the mcxlel to the incident
electromagnetic wave and print th«m .
50
-------
D-10 PROGRAM LISTING
"3G~AX E2CDC I.\'=U7* OLTPV)
"EAL K3,KE,KPSC,P(43), DP C 43 ) , ?M (40 )* COA( ! 3), SI AC 12 )
"AL CAC 13*2)* ^SC 1 3*2)*C7( 13*2)
COMPLEX IMG, IMS, EPS* PATIO, XI *?.X! , V~X ! ., EE3 IJ , VH, '.'HD, S?K, 33PH, 31 J
COMPLEX ~OIo',KI,HJl*HJ2,HKl * HX2 , HL 1 , HL2, HL2 , X ( 23 ) , V ( 23 ) *S?HANKC40 )
COMPLEX DSPHAN'K(43)*3P3( 1 03 ) , DSP3 C ^3 ) , CO EF I C23), C3EF2 ( 23 ) , VA 1
COMPLEX VTEM,'.'A2(^63), IMN'l C20* I 3 ) * JXN 1 (23* 1 3 ) > KM!; 1 (23* 13 )
COMPLEX L.MN1 (23, 13 ), IMN2(22, 13 ) ,u'MN2(23, 1 3 ) , X:1N2 < 23 , 13 )
COMPLEX LMM2(23, 13),PMTHX1 C23,23) , PMTP.X2 <23* 22 ) * CMTP.X 1 (23,23)
COMPLEX '?"'1""0X2 (23, 23 )
COMMON/3LXl/P,Da,?M/3LX2/S?HANX, DS?HA:JK/3LX3/ SPS , DSPS
sa. 1^1 5926535?9«
ECTa? . ?*n I
IMG=(3 ., 1 . )
IMS=(3 ., -1 . )
OM=(PI+PI )/2.993E13
'IAD 1333, A, 3
~C°.MATC2F'2.4)
133* -?E
~Oc.MAT(3ri3.A)
. E6
1233, .«JMAX*MXAX,XMAX,NSECT,XEV
1233 FOT3.MA'T1(3I2, 13* 12)
NHALF=MMAX/2
NMAX=2*NHAL~
••'PITE 233^, A*3,Fr>.EQ,E?S,XH*N'MAX,MMAX,X!1AX,MSECT
2333 FOP.MATC IH!, //////, 3X, *A**F6 .2, 3X,*E*,~?.2* 3X* *FPE:** E! 1 .4, 3X*-
! «EPS*,2F8.2,3X,*K3«,F?.3,//, 3X,«MMAX«, 13, 3X**MMAX«, I3,3X,
2 *KMAX«, 13, 3X,*INTZG?.AL 3ECTI CW* , I 3, /// )
J=KMAX-1
= DTH*P.ADIAn
~HETA=3.
DO 13 !=2*J
THETA=THETAfDTH
COACI )=COS(THE7A)
31 A( I )=SUJ(THETA)
CONTINUE
COA( ! )=SIA(XMAX)=1
SIAC I )=COA(KMAX)=3
- 51
-------
DO ^ l-\,1ft
?S( I ) =OT( I )=0 .
2(7 CONTINUE
D'HaREcr/FLOAT < ::SECT >
DO 40 1=1, NMAX
DO 4^ J=I,MMA:'
=>M7".Xl CJ, I >sPMTP.X2CJ, I )=<3.,1. )
Q.MTP.rt »<0 . >3. )
4^ CONTINUE
00 77!^ M=l,:iMAX
00 12? I=1,NHALF
oo 12^ j=:,rn-iAX
IMNl CJ^ ! )=JMN! C J, ! )=KMN1 C J , I >=LMNl CJ>!)=(3.J3.)
IMN2CJ. ! )sJMN2(Jj ! )=KMN2(u> I )=LMN2( J, 1 ) = C '1 . , 3 . )
12 "I CONTIMUI
COS~H=1 .
DO 4 = 3 ISZCT=U3
!F< ISEC~-2) 125/133,125
125 S ?.MUL = i3 • 5 * 0~H
GO TO 15*
S?MUL*OTV.
IMAX=NSEC~-1
DO 4^^ 1 = 1, I MAX
CALL SHA?ECCOSTH, S ! NTH , ?.K3 , ?.K 1 , ?.D* VP.X3 . VP.K 1 / KHSQ )
CALL HANKEL ( ?.K3 / JIN ^ NMAX )
CALL S?HBESS(°K1 ,MN,NMAX)
CALL LEGENDP.CCOSTH, S INTH, M/ MN, NMAX)
I~LAG=1
Naf-JN
DO 350 MR 0 T.v = 1 , MM AX
IF C! FLAG .EC. 1) GO TO 160
1
GO ~0 2
M!=MN+1
S!Ca=CON*SIM~H
HI=COM*SPH
HJ1=SICO*S?H
-------
HJ2s(?.D*D°MN>*HJl
HK1»SICO*WHD
HK2* ( RD*PMN*FN*S I CO ) *VH
HL1»CON*WHD
CON»CON*P.D*PMN
HL2=(CON*FN)*'.TH
DO 333 MCOL* 1 , NHALF
IF CM .£0. 1) GO "0 252
PI J«P (Ml )
BESIJ=SPB(NI )
CONsPMN*DPIJ+DPMN*PIJ
DIJ*CON*BESIJ
TDIJ=P.EAL(H1)*DIJ
DIJaHI*DIJ
IMNl
WH»BESIJ*WP.K1
DIJ»SPH*HLi+DSPH*HL2 + V
TO I J=P.EAL ( HL 1 ) *S?H+R£AL C HL2 ) *DSPH+R£AL C HL3 ) * WTEM
LMN1 (MP.OV,MCOL)»LMNl (MROW,MCOL) + 01 J
LMH2(MP.OT.;,MCOL)»LMN2CMP.OW,MCOL)fTDIJ
250 PIJaP(NJ)
D?IJ*DP(NJ)
3ESIJ=SPB(NJ)
CON»P»1MN*PMI J+DPMN* DP I J
WH«9ESIJ*VRX1
WHD»WH*DSPQ(NJ)
S?H*CON*'JHD
DSPH»+OI J
JMN2 (MP.OW, MCOL ) » JMN2 CMROW, MCOL ) +TDI J
SPH=CON*BES!J
DSPH»DPIJ*BESIJ
DIJ»HK1*SPH+HK2*DSPH
TDIJ»P.EAL(HK1 >*SPH+REAL(KK2)*DSPH
KMM1 (MP.Ot'7,MCOL>»KMNl CMRO-'jMCOL) + 01 J
XMN2»KMN2CMP.OV,MCOL)+TDIJ
53
-------
CONTINUE
IFLAG=-IFLAG
350 CONTINUE
THETA=THETA+DTH
COSTH=C03MCOL)=-KMN1 CMP.O'.S MCOL)
KMN2 C MP-OV, MCOL ) =-KMN2 CIRO1^ MCOL )
455 CONTINUE
IFCM .ME. 1 ) GO TO 464
DO 466 MCOL=UNHALF
DO 466 !1POW=UNMAX
VH = KMNl (MP.OV/MCOL)
KMN1
JMNl
T-;H«KMN2< MHO W^ MCOL)
KMN2 CMP.OU* MCOL ) =»JMN2 (MP.OWj MCOL )
JMNS CMP.OW, MCOL ) =WH
466 CONTINUE
464 IFCKEY .NE. I) GO TO 475
y^ITS 3503, C < IMN1 < I , J ) > J= I ^ NHALF )
VP.ITE 3500> ( CJMNI C I, J) , J» I /NHALF)
VP.ITE 3500* C (KMNl C I* J)*J=l * NHALF )/ I=1*NMAX)
VHITE 3503* C (LMNI < I * J ) * J= I * NHALF ) * I = l * NMAX )
WRITE 3500* < (IMN2CI* J)* J= I* NHALF)* 1 = I* NMAX)
VP.ITE 3500* C
-------
PMTRX1 CMP.OV-H*MCOL) = IMN1 CMP.OV+1, I)+RATIO*LMN1
QMTP.X1 «LMN2CMROV+ 1 , I ) + RAT I 0* IMN2CMROV+ I , I >
470 CONTINUE
477 IsPI
DO 480 MCOLaUNMAX, 2
1 = 1+1
DO 480 MP.OW»1«NMAX,2
PMTP.X1 I )
PMT?.X2(M?.OW,MCOL)*KMN2(MROV, I ) +PATIO*JMN2 (MRO* , 1 )
CMTRX2CMP.OV^MCOL)»JMN2CMROV, I >+RATIO*KMN2CMSOV, I )
^MTP.Xl CMP.O'J+1 jMCOL+I )*JMN1 (MROV+U I ) +RATI 0*KT1N 1 CMP.OW+1 , I )
QMT°.X1 (MP-OM+UMCOL+l )»KMN1 CMROV+1 , I )+RAT!0*JMNl CMROWf \, I )
SKTHX2CMROU+UMCOL+1 )*KMN2CMP.OV+ I , I ) +RATIO*JMt»2 (MROV+ \, I )
480 CONTINUE
C
C
C *** DECOMPOSE THE MATPIX ****
r
C
CALL LSCT2CC«MTP.X1,NMAX^23*Y, 1 ,20, 1 , WU , IE?.)
IFCIEP. .EC. 129) GO TO 760
CALL L£QT2CCQMTSri,NMAX,2C,Y, 1,20, 1,^A2, IEP.)
irclEP. .ES. 12 ,M,MN,NMAX>
IFCM .EO. 1) GO TO 510
DO 505 J=UNMAX,2
COEF1 (J)*PM(K)*SPH
COET2(J)aDP(K)*SPK
SPH»S»H*IMG
K»K+1
COEF1 CJ+I )»D?(K)*SPH
COEF2CJ+1 )*PMCK)»S?H
SPH*SPH*IMG ^
K»Kt- 1
505 CONTINUE
GO TO 5213
510 DO 515 J=1,NMAX,2
COEF1
-------
COEF1CJ+1)=PMCK)*SPH
COEF2CJ+1 )=DP(K)*SPH
SPH=S?H*IMG
K=K+1
515 CONTINUE
520 IPQ = 0
525 IPQ=IPQ.+ 1
IPOL=*CI?0+1 )/2
GO TO (530., 540, 550, 563/600) IPS
533 DO 535 J=1,NMAX
VCJ)=COEF1CJ)
535 CONTINUE
GO TO 580
540 IF(I .EC. KMAX) GO TO 525
DO 545 J=i,NMAX
YCJ)=COEF2CJ)
545 CONTINUE
GO TO 573
550 IF (I .EG. KMAX) GO TO 525
IFCI .EC.1) GO TO 600
DO 555 J=1,NMAX
Y(J)=IMS*COEF2(J)
555 CONTINUE
560 DO 565 J=1,NMAX
YCJ)=IMS*COEF1CJ)
565 CONTINUE
573 CALL LECT2CCSMTF.X1 /NMAX/20/Y/ 1,20, 2, WA2, IE?.)
WRITE 2250,CYCJ>,J=1/NMAX)
2250 FORMAT(//, <4X, 8E13.6))
IFCIEP. .EQ. 130) GO TO 770
CALL MULTP.X C CM~P.X2, X/ Y, 20 , NMAX)
GO TO 590
580 CALL LEGT2CCPMTP.X1 JNMAX>20,Y^ 1 ,20,2>UA1., IEF.)
VP.ITE 2250. CYCJ),J=1 ,NMAX)
IFCIEP. -EQ. 1305GO TO 770
CALL MULTP.XC=MTF.X2, X, Y, 23 >NMAX)
590 WRITE 2200, M,IPOL/ CI- 1)*DAL, CXCJ),J=1,NMAX)
2200 FORMAT(///,2X, *M=*/12,2X,*?OL*,12/2X,* INCIDENCE ANGLE*,~1.2*//^
1 C2X,8E15.8))
CALL SCATTEP.CM/MN/NMAX, IPOL/ IPO, SCAT,TOT)
IFCA .EQ. 3 .AND. IPQ .EQ.l) CALL SPHEHECM/MN,NMAX,EPS/HK0/RK1)
IFCM .NE. 1) GO TO 595
SCAT=0.5*SCAT
TOT=0.5*TOT
595 QSCI, IPOL)=QSCI, IPOD-t-SCAT
QTC I ,1POL)=QTCI,IPOL)+TOT
GO TO 525
600 CONTINUE
700 CONTINUE
»
56
-------
R0»8.*PI/CK0*K0)
QT(1*2)»QTC1* 1 )
GSU*2)»QSU., 1 )
DO 750 I«1*KMAX
DO 750 J*l*2
CTCI*J)=CT(I*J)*R0
CS(I*J)*QSCI*J)*R0
QAU* J)*GTU* J)-GSCI* J)
750 CONTINUE
WHITE 2000, A*S*FP.ES*EPS*KB*NMAX*MMAX*KMAX*NSECT
VF.ITE 2530
253IZ FOBMAT(//,40X**CP.OSS SECTIONS**//)
WP.ITE 3600* CCI-1 )*DAL* C QAC 1 * J ) * J= 1 * 2) * *J=1,2),I=1*KMAX)
3600 FORMAT (2X**I NCI DENCE ANGLE** 10X*
1 *AESO?.PTION**20X**SCATTEF.ING**20X,*EXTINCTION**///*
2 (6X*F6.2*5X*6E15.7*/))
P.0=l ./CPI*A')
DO 755 I=1*KMAX
H2=P0/SQ!?TCESQ*SIACI )**2-»-ASO*COAC I )**2)
DO 755 J=l*2
Q£(I*J)*CSCI,J)»F.2
3A ( I * J ) *QA CI * J ) *F.2
755 CONTINUE
'JHITE 2553
2553 FORMAT C///, 40X* *EFFI CI ENCIES** // )
V'P.ITE 3600 , C*DAL* ( QA ( I * J ) * J= 1 * 2) * (QS Cl * J ) * J= 1 * 2) *
1 (CT(I*J)*J»1*2)*I»1*KMAX)
GO TO S00
760 VF.ITE 2600*M
2600 FOP.MATC2X**M»**I3** MATP.IX IS SINGULAR*)
GO TO 800
770 WP.ITE 2730 *M
2700 ro?.MATC2X**M»**I3** MATP.IX IS ILL-CONDITIONED*)
800 STOP
END
3FM
57
-------
SUE-ROUTINE SCATTER CM, MN, NMAX, IPOL, IPO, SCAT, TOT)
COMPLEX XC2e),SP, IMS,VASP
REAL PC 42), DP (40 >,PMC40>
COMMQN/3LK 1 /? j DP >PM/3LK4/X
IMS=C0..,-1 .)
COM=1 ./"ACTCMi-M- 1 )
-=2 .
L=MN- 1
~L=~LOA~
-------
SU3P.OUTI NE HANKEL < X* MN, N )
COMPLEX SPHANKC40),DSPHANK<40>
REAL S?B(72!>,SPN<40),DSPNC40>
COMMON/3LK2/SPHANK., DSPHANK
ETA=COSCX>
STA=SIN(X>
Ysl./X
3PN(I)sCl=-ETA*Y
OSPNC1 )=-C2
D« I .
S=»0.
DO 10 I»3*NL
3*5+1 .
SPNCI)=C3-D*C2*Y-C1
DSPN(I-l )=CS*Cl-CS-t-l .)*C3)/D
CONTINUE
K=IFIXaCMPLX<-C2/DSPNC I ) )
F»l .
00 30 I-3..NL
C3»SPB»CMPLX
-------
SUBROUT I NE SPH3ESS C X* MM, N )
COMPLEX X, Y, FACTOR* Cl , C2* C3* SP3 ( 103)* DS?B<
COMMON/3LK3/SPS* DSPB
V=( 1 .*0.)/X
IFfCASS(X) .GT. 13.) M=NL*53
IFCM .GT. 100) M=130
CI = (3.*3 . )
C2=( 1 .* 1 . )
K=M
M=M-1
00 50 1=1 *M
K = K-1
S?3CK)=C3=F*C2*Y-C1
C1=C2
C2 = C3
50 CONTINUE
SP3C I )=C1=CSIN(X)*Y
FACTOR=C1/C2
SPSC2)=C2=FACTOR*SP3<2)
DS?3(1)=-C2
F=l .
DO 60 I=3*NL
SP3CI)=C3=SP3CI)*FACTOR
DSPBCI-1)=
60
-------
SUBROUTINE LEGENDS C ETA, ST,M, MN, N)
REAL P<49>,DP<40).,PMC40>
COMMON/BLXl/P,D?,PM
NMsN+MN+1
ITCETA .EG. 1 .) GO TO 40
r "FLOAT CM)
CM-FLOAT
SM»CF-1 . >/ST
PCM)»PA1=X*DFACTCM-1 )
P(M>1 )-PA2»ETA*PAl*CM
V? CM) » »
-------
SUBROUTINE SPHERE OP^ PM/SLXS/SPHAIJK, DSPHANK/BLK3/ SP5, DSPS
VP. 1TE 500 /M
~OP.MATC//,2X,*EXACT SOLUTION ~OP. THE SPHEP.E^ :•*. = *, 13, /)
00 130 I=MN,N'MAX
31=SPBC I )
V3D=DSP3 < I )
SP=SPHAMK(! )
SM= DSPHANK ( I )
Al =°EALCSP)
VAD=PEALCS?/!)
°.l=°.K0*WAD+Ai
)/C31*FAC-S?*VP. 1 )
'.'3 = DP AE20 . 12)
CONTINUE
RETURN
END
SUBROUTINE MULTRXCA, X,Y,M,N:>
COMPLEX A
-------
SUBROUTINE SHAPE! ( ETA, 2 ETA, RX3, RX I , RD, VP.X0 , WRX 1 , XRSQ. )
REAL K0,KRSQ
COMPLEX KUVRKURKl
FACTOR=ASQ*ETA*ETA+BSQ*ZETA=*ZETA
R=AB/ SORT ("ACTOR)
RX0=R*K0
VRX0»1 ./RX3
RD=ABQ*STA*ZETA/FACTOR
RETURN
END
FUNCTION DFACT(N)
DFACT=1.
IFC N . EQ. 0) RETURN
DO 10 1=1,N
DFACT = DFACT*FLOAT CI + I-t)
10 CONTINUE
RETURN
END
FUNCTION FACT(N)
FACT=l.
IF(N .EQ. 1) RETURN
DO 10 I»2>N
FACT»FACT*FLOAT
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing/
1. REPORT NO.
EPA-600/1-79-025
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
Absorption Characteristics of Prolate Spheroidal Model
of Man and Animals At and Near Resonance Frequency
5. REPORT DATE
August 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
V.K. Tripathi- and H. Lee
18. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Electrical and Computer Engineering Department
Oregon State University
Corvallis, Oregon 97331
10. PROGRAM ELEMENT NO.
1FA628
11. CONTRACT/GRANT NO.
R-804697-01-1
12. SPONSORING AGENCY NAME AND ADDRESS
Health Effects Research Laboratory
Office of Research and Development
US Environmental Protection Agency
Research Triangle Park, NC 27711
RTP, NC
13. TYPE OF REPORT AND PERIOD COVERED
Final. 9-15-76 to 3-15-78
14. SPONSORING AGENCY CODE
EPA 600/11
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This report deals with the evaluation of the absorption characteristics of prolate
spheroidal models of man and animals consisting of homogeneous biological tissues
irradiated with plane waves at an arbitrary angle of incidence at and near the
resonance frequency. Stevenson's series expansion method in the third approximation
and the extended boundary condition method are presented in detail since they were
found to be most amenable to the solution of the problem.
A general formulation of Stevenson's method in the third approximation is
presented which can be applied to an arbitrary angle of incidence and leads to an
improvement in frequency range of validity and accuracy as compared with first order
solutions.
The report also deals in detail with the extended boundary condition method
(EBCM), since it was found to be inplementable for realistic models at frequencies
approaching resonance. Though the EBCM theory is well known, the application of the
method has been stressed and certain improvements in the accuracy and efficiency of
the computations are incorporated. Numerical results are presented and compared with
known data published by others.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSAT! Field/Group
Electromagnetic radiation, radio frequency,
microwaves, electromagnetic absorption,
resonance, numerical methods, mathematical
models, spheres, muscle tissue, humans,
laboratory animals, standards, protection.
06F
18, DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (ThisReport)
Unclassified
21. NO. OF PAGES
71
20. SECURITY CLASS {Thispage)
Unclassified
27. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION is OBSOLETE
64
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