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UNG.sLASSIf FIED
410836
DEFENSE DOCUMENTATION CENTERFOR
o•,C luilFiC AND TECHNICAL INFORMATION
CAMERON STATION, ALEXANDRIA, VIRGINIA
UNCLkSHIFIED,
BestAvai~lable
copy
NC:.:... WIen government or other drawings, speci-fications or other data are used for any purposeothe.- than in connection with a definitely relatedgoverniment procurement operaLiuri, the U. S.Government tLhereby incurs no responsibility, nor anyobligation whatsoever; and ihe fact that the Govern-merit may have formulated, fwnished, or in any waysur-' Led the said da.awings, speciftc.tions, or otheriJla is not to be regarded by implication or other-wise Ls in any manner licenojing the holder or anyother person or o;orporation, or conveying any rightsor permission to manufacture, use or sell anypatented invontion that may in any way be relatedthereto.
( AFCRL-63-483
Electrical Enr'*:e rig Research LaboratoryThe University nf Texas
mmSI Austin, Toxas
Report No. 6-53 31 May 1963
/C...) CONSIDERING FOREST VEGETATION AS AN
IMPERFECT DIELECTRIC SLAB
D. J Pounds40% A. H. LaGrone
Contract AF 19(604)-8038
Project 4603
AIR FORCE CAMBRIDGE RE'Z-ARCH L,,.JRA1ORIEQ
Office of Aerospace IResearck
United States Air Force
Bedford, Massac' ,Cotst
Ig
Requests for additior2al copier by Agencies of tOi Departmn6nt ofDefense, their contractors, and rother Government Agencies should bedirected to
Defense Documentation CenterArlington Hall StationArlington 1Z, Virginia
Dcpartmdnt of Def'ense contractors must be established for DDCservices or have their "need-to-know" certified by the cognizanttnilitary agency of their project or contract.
All other persons and organizations should apply to the
11, v Department of CommerceOffice of Technic.;- ServicesWa±shington 25, D.C.
AFCRL .- 483
ELECTRICAL Et4C,•;.• .. RESEARCH LABORATORYTHE UNIVERSITY OF TEXAS
A*ustin, Texas
R,.port No. 6-53 31 May 1963
CONSIDERING FOREST VEGETATION AS AN
IMPERFECT DIELECTRIC SLAB
by
D. J. PoundsA. H, LaGrone
Contract AF ;9(604)-8039
Project 4603
AIL FOR('E CAMBRIDGE RESEARCH LABORATORIrSOf;ice o" Aerospace Research
United States Air ForceBedrord, Massachusetts
€°."
TAll,.. G.. , NTEN TS p
Page
ABSTRkCT v
I. INTRODUCTION
II. GENERAL REMARKS ON FORESTS
A. Conifers 4
B. Hardwoods 5
Ill. TYPES OF FORESTS 10
IV., STATISTICAL DESCRIPTION OF FOREST I ?
V. DIELECTRI(., ý'•,OFERTIES OF WOOD 18
VI. DIELECTRIC PROPERTIES OF BARK 30
VII. DIELECTRIC CONSTANT OF A MIXTURE 33
VIII. ELECTRICAL PROPERTIES OF LEAVES 39
IX., MFTHOD FOR SYNTHESIS OF THE DIELECTRIC
SLAB - CALCULATION OF c 50
X., AN EXAMPLE OF ,HE COMPUTATION OF THE
PROPOSE D DIELECTRIC 6LAB ý3
Xi. CONCLUSION
REFERENCES
' :OF I GURES
Page
1. Cellular Structure of a Sample of Conifer Wuod 6
2. Cellular Structure of a Sample of Hardwood Wood 9
3. Forest Vegetation of tie United States II
4. Dielectric Coi.stant of Buckeye Wood Z _
5. Dielectric Constait of Oak ý d WycL Elm Z3
6. Loss Taagent of Oak innl Wyzh Elm ?4
7, Dielectric Constant of Fr Wood as a Function o0 25
Moisture Context
8., Dielectric Con'-•nt rf Fir V ood as a Function of Frequency 26
9. Loss Tangent of Fir Wood as a Function of Moistuie Content 27
10. Loss Ta.-igent of Fir Wood as a Function of Frequency 28
11. Dielectric Constant of Loblolly Pine Bark 31
12. Los. Tangent of Loblolly Pine Bark 32
13a. Pteallel Plate Capacitor with Diel-ctrics in Parallel 34
13b. Parallel Plate Capa,",tor with Dielectrics in Series 34
lJ. Orientation of a Leaf wiLh Resp, -t to the E and H Fields 46
15a. Artist'. Con.Lptpon of M-tz's Pc-rest Q
Db. Dielectric Slab Repr-sentation of Metz's Fores,t
IM. T' OF T. BLES
Page
Table I. Sta.tistical Description of a One-Acre Fully 16
Stocked Stdnd of Loblolly Pine
Table II., Values of U for Use in tae Dielectric Mixtureb 38
Equation 12
Table III. Polarizabilities of Conducting Bodies 49
IV
A method for representin.g a grove of trees by an imperfect
dielectric slab is developed and illustrated., The tnethod considers
leaves as conducting bodies and bark and wood as dielectric bodies
using artificial dielectric and dielectric mixture theory to compute
the r,•al part of the dielectric constant of the proposed slab. The
imaginary part of the dielectri, co.istaat of the sl~b is computed
from measured attenuation data. The method is illustrated using
data front a real forest to compute the characteristics of an imperfect
dielectric slab.,
1. INTRODUCTION
It is well known that tre.•s surrounding a recciving antenna have
a decided effect on received signal strength at frequencies abnov*
approxirmately 30 megacycles per second. These effecty are known
qualitatively, but lVttle has been don,', to give them a quantitative v'.luc,
Quartitative information on the signal loss duc to tiees near the antenna
would be valuable for selectinL an optimum rtcel'.',sr site, computing the
effects of a large forest on wavy., propagation, finding the optimum
antenna height at a given locp.dion, the effect of a forest on the coverage
area of television stations, and other problems. Thus, the problem in
a, Y study is to develop an appro,.itnate maLhe,'staLicat eitrof. .1 (ja!rove
of trees for use in theoretical calculations of field dtrength in the
vicinity ,f tie grove of trees.
One approach to the problem is to represent the grove of trees by
a random distribution of short conductors and dielectric b.dies from
which the dielectric .,unr.stant is computed. The attenuation is then
determined from loss measurements. This is the approach adopted
for this study.
First, the two principal tree types and some of fh. "i:'r;ipal
forcst types it the United States are considerea. TJ'hey As :r,,• ted
and described in termr. Df physical characteristics such as the names
and description of major species including size and appearanc,,. A
general d 9c, iption of the J ,.luding ,uch thiczs as shapt, sizc
and some variations irom species to speciks is 1resentea,
Second, a statistical description of a fully stocked (moderately
dense) even-aged forest stand is presented. These statistics hich•de
such things as the number of trees per acre, average diameter at
bre-ist height, basal area per acre, average heigh, s, c-stlmated ot
measured volumes of wood av.i bark per acre as a function of age and
perhap% thi. cl-aracteristics of chL. site. These data are used in the
syithesis of the dielectric si-jb model.
Third, the electri:al properties of wood, )brrk, a d leaves are
eenortcd. Chey are measured ou, estimate, el(romn the phv i.,.i, zt-ucture
and moisture content of the subject. Cons'deration is givctj to qu~h
factot r as frequency dependence of th. dielectric proPLrtie, a.d
%ariationh in the moisture content of the wood,
Fourth, methods for synthesizing the electrical properties of
wood, bark, leaves. And air into the dielectric properties of an in-
Verf 'ct dielectric slab are derived.
Fifth, the method of calculaticn is illust,'ted by an example
followed by a summary.
The dielectric :lab concept is adopted he:atase of ts':, ,...p•'~ty
of application and the avajiability of a suitable program for calc !at,')
by means of a digital computer.
II. G'.INERAL REMARKS ON FORF3rS
A forest is a large ar,;n. of land cuvered b,., a imiude ratt to dvnft"
growth of trees. A forest may or may not have underbrush. Some
forests, for example, do not even have grass growing under the tle('.
A tree ib a "woody perennial, seod-bearing plant' A"'hit, ,t
maturity is 20 feet or more in height, with a zingl, trunk, unbranched
for nt I,', ,L •everal feel ahnvv 0n" grotin'i an ha.v'ig a rnore. or lebs
definite crown." This definition cannot bv taken Ab absoltv but only
as a guide. For example, some willow trees have multiple stems
while many trees do not reach 20 feet in height under advroc conditions.
In this study, a tree is con.Adered a.4 made up oi th,.'- vri..cijpal
componentl, - wood, bark, and leaves. A number of componwits arc
n,:glected by such a division, These includo, the roots below the ground,
the buds, flowers, fruit, and seed normally found above ground, and the
cambium or growing layer between thL wood and bark. It io also
necessary to neglh.. r'ne variations within the wood, bark, and ltlav0s.
Foresters generalIV divide forest trees into two group:;
(Pi i,.'ters or softwnoods, and (2) bruadlehf tre, , or hdrdcrodq. There.
are some minor exccptions to the above clasqif'cai,'or 1-,i, 6 ;h-,y are
insignificant to this study. A survey of the litaratu,? r-, fo, (xa i.'ple,
shows that a few conife-5ý -xhibit some characteristics like broadleaf
trees and vice versa. Trhest two groups of trees generally dif."- in
ceown form, Irat'hing habits. a,:d wood strurct-rc as well as ],,.bf :hape.
4
A. Conifers
Conifer is a common name forestors use insttad of the
scientific name gymnosperms. Conifer literally means cone bearing.
The terms soft'voods and evergreens are also used but these terms arc
misleading since some conifers are not evergreen and some softWod
Jumb,'r is barder than the lumber of some ha-'dwoodf..3
Most conifers tend t nave conicallbr shaped crowns. In the
forest, however, competition forces the crowns to be more cylindrical
in shapc, smaller in horizonttl dimensions,and the base of the crown to
be located at a greater height above the ground.
Tlhe conifers generally 1,ave small, relatively !,,',t, nearly
horizontal branches. These branchee are usually of small diameter com-
pared to the trunk and do not significantly affect the trunk's diameter;
i. e., the diameter above and below a given branch is approximately
the same. This structural form means that almost all the wood volume5
is present in the stem Probably more than 80% of the wood volume is
cogitained in the stem of large trees.
ConiferLt are resinous trees with needle- or scale-like leaves.
The leaves are evergreen on most conifers but certain cyp-,., tamarack,
and larch shed their leaves in the Fall. The Icaves may bu ;,.c: • singl+,3
or in clusters.
The seed of tnust onifers are, as the name implies, borne
in) conts, but the ju:-.ipers be.r a berrylike seed, and the yew a lieshy
scarlet disk. The conifers inc:'de cyp'esses, pines, hemlcks, spruces,
firs, ccdar, tamaracks or larchies, pinyons, vews, ijinipers, the giant
-equoias and redwoods.3
The cellular structure of a conifer wood sample is sh6wn iti
Figure 1. Up to 90% of the volume may be occupied by "verticAl1 rienited,
thick-,-walled dead cells (of r.-lulose) varyi-, in. length from 0. 5 to 1a
millimeters." f) Transversely, ,. iented wood rays are also present.
Note from the figure that large, vacant volumes arc prebent wnich are
6filled in the tree by gases and water. All of these have an important
effect in dete~mining the declectric properties of the conifer wood.
B. Hardwoods
Foresters use the term "hardwood" instead of the scientific
name angiosperms even though palms and yuccas are angiosperms hut
not hardwoods. Hardwoods are also called broadleaf trees and deciduous
trees but the term decidsious can lead to confusion because a fcw hard-
woods have evergreit lt;-ves. 3
Generally, hardwood trees tcnd toward spherical, ellipsoidal
4and cylindrical crowns. Their crowns tend tr. bc proportionally larger
in horizontal dimensions than the conifers. In the forcst, *. .,cti~iot
causes the crowns to be narrower than shade tree hardWooL pf...'iles
indicate.
The different crown shape covers a different branchina,
structure in the h-Adwood. The branches of hardwood trees are relatively
TT - CROSS SECTIORR RPMDAL SECTIN
T'-TANGENTIAL SECTION '
iR - TRACHEIDSML - MIDDLE LAMELLAS - SPIRINGWOUD SRE IRVPvLTLA TCk
SM - SUMMER WOOD U ~RS EVEFNSN SLB-~TG
fP,- ANNUAL RING3MR - WOOD RAY
H..R~ - FLISJFORM WOOD RAYHRD- SIMPLE PIT
BP-BORDERED PIT
CELLULALR STRUCTLP11 OF AA SAMPLE IF r_-,.FEIý ACýOD
,7 1
7
large cr.-nparcd to the trunk. Thus, often the stem is conviderably sniallhr
aot-ve a branch or circle of branclhes than below the br'inch, Near the
top, heavy branching causes the stem to lose .n Eize rapidly. In many
species, the branches form an acute angle with the trunk going upwjatd
as well as outward. These branching habits add up to a smaller percentage
.5of the total wood being found in the sten of h;ardwoods,
Hardwood trees are • on- -,',nous and have brosd leaves. The
leaven may Le .3imple or compound, i. e., made up uf leafP-t•. Most
sp...cies shed their leaves in the fall, i, e, , are deciduous, However, live
oaks, magnolias, American holly, laurel oaks, redbay, laurel cherrv,
"-nny small trodp.ti. and rubtropica' t:?eb, and posaibly a few other species
have green leaves throughout the winter. The leaves of hardwood trees
art,, net veined, the seeds are enclosed in a fruit, and the bark is distinct
from the wood which has annual rings. The hardwoods include catalpa,
dogwn,(.d, maples, ashes, box elders, buckeyes, walnuts, butternuts,
pecans, hickoriet,, mahogany, locusts, sassafras, mulberry, osage-orange,
9ijins, sycamores, magitolias, bays, tupelos, persimmon. holly basswood,
ehn, hKckberries, cottonwoods, poplars, birches, willows, cherries,
beecheschestnuts, oaks, 'aluer, buckthorn, madrone chinquan-ti, and
3perhaps others.
Wood structure of a hardwooti is illustrated in, Figure 2. Its
structure is mno. , complex t1- .;i softwood struct,,re. Most hardwoods
: .a!.-. ,. ' ! .- '"nt:ed *:, - and .'erti fib .
rhe c! -~f fun'ction of the -,mbes is to conduct ýý,ater, but bome of the uldei
t'ihos become blockcd, Ohcc aga:.i, a Luri~iderabie volume of thie wood
is OCCLIP.Ad b> vai -ýr and gases.,
k" iA16 SETONMTG TANGEN'IAL SECTION
SC -PERFORATION PLATE RRhAT VESSEL END
-WOOD FIBERK - PIT
Me~ - WOC-jD PAY UiS FID'EST SERVICE,, "2'IND4UCTS -A MTOPAlAR - ANNUAL RINGS - SPRINGWOOD
SM - SUMMER WOODM L - MIDDLE LAMELLA
LC L' 1 Af r- C r~ M10 - r I I I tI I -, L_dLAM 41 I t:.L. I ) rI k c M . OF m1&-LJVVUU WUUD
FIG 2
III, I ;PES OF FORESTS
The original forests cuvered ,abou 48'% ot hthe total area •f thvt
United States7 but this amoint has been reduced considerablv by
lumbering and clearing for agricultural and other purpose.. There
8are 1, 182 different kinds of forest trCes in the o1,g ,al 4H ,tatk . 8 Of
',hi:-h, over 100 are of cursunercial importance. ViiVy other species
can be found in Hawaii and Ala -a..
The forest classifications and locations used throughout this
7report are from Shantz and Lan, who divide the natural forest vegeta-
tiun into 12 regions tnd 18 utubregions. (The areas specilied as regions
f. t i, report were called subret .its by Sha".'7 at.r. Z,'i
The western forest rei'.ons ate spruc, -fir, western whitu pine -
western latch, Douglas-fir, redwood, pinyon-juniper., chaparral,
ponderosa pine - sugar pine - incense-ctdar, ponderoua pine -
Douglas-fir, and lodgepule ptne.: The eastern forest regi( ns are *Spruce -
fir, .orthvastern t,.e'oithero hardwood, chestnut-chestnut oak -
puplar, oak-hickory, jak-pine, cypress-tupelo - sweetý,,n, souther,,
7pin-es, and nidgrove.
These regions may be located by use of the nias, F :r 3) witn
more detailed data in Shantz and Zon. Thic iap is u., itation
Shantz tnd ZLon's 5 or igiol ,i)und in Forestry tHandbook 9
-10.
IIL
I- P I
IV.. S:- 'TISr±CAL DESCRJP-tION OF'FORES"
In order to approxir.-.at., a forest b\ a diLlectric slab itr S
necessary to know within reasonable limits how initch of the o-erali
volume repi-esented by a forest is occ upDld b\, wood dud bark and thel
number of leaves,: This is necessary since the dielektri" L-- operties
of wood,, bark, leaves and a r dif(,,
Information ot this t pe o- ota1t,,, C1 yl, (I tables which l pre--
dict the wood ,.txailable fo urn an acre of land fullh ',o.al,_J ", J1, tr•Ls,
I icbe tables normally give thy,' "total'' or botnet tncs merchanlt oh,)
ciibic feet ff wooo in the stern, i , t r i:'e s on an acrt,' ut land iii is
,,eiid Is uSuPlly 6 I 1 as a func-c-, (f ,kc 4 no site quality for a particular
species of treL,
The vield tables serve the rteeds of the forester well b it must b_
modilied to include the wood and bark in the branches, tree top, and
st uni., since these are important in this ht-dý . An e stiimate Of tltc
number of leavL. ;,st also be ready scnce this data is not normnally
ija býl1 ,1e,.
It is boyvoid the scope ot this paper to present volutne s of bark,
wood and leaf quantities fo, all 1niportanit SPUCCt S, but al eC%,lePh v'".1
be presenAte to ill -,tr,,te' the Mtethodýt Ube(] 11, 011lyl!'. , ,rv-" io
"FT, clard ,,v the data to be prt sciat a a ew i ,t', ,,1i "H . d1, t ( iid
alnd disLUsseiU
S.... . . ...
13
"'eld taoles,which Form the basis of the data used in this stud,
ar- prepared for even-aged stand., of onc spcciCs "If groups of SpcCies d
that are fully or n.,rmalh, stocked (fairly dense), Even-aged means
that the members of th, stand are of about equal ag,'; that is, all
started growing within the no-rmyii iiteo for natural replanting. This
may be one or more years,
The age of a tree is expressed ini ,,,,ears Age is ubually determined
by i.8 a sample out of a tree at breai; height and counting the nimber
,of annual rings. The forestcr then adds the number of years that heý
eStImates at took the tree to grow to oreast height.,
The diametp" at breast height of a tree (d. b. h.) in inches as
measured outside the bark at a point 4-1/2 feet above thc` ground., For
elliptically shaped stems, the major and minor axis values are averaged.
The average diameter at breast height in the tables is the d.,b. h. of the
trce of average basal area.
The number of trees per acre (N) is a count of all trees with a
d. o.,h., greater than a certain value selected by the i',vestigator
The basal arca (Ab) per a- r in square feel b the sum of the,
ar-as of all tree stems at breast hoght (larger tha. a C ..... . i ,. )
on an average acre of land fulle stocked with trees, 1-ahi t, , stemn is
considered as a circle for purposes of z!a catcoL,:ns.
14
Dominant Liveb are those with crowns exteiding
above the general le, A of thc irest canopy°, receiv,ng full
lhghi from abd ve anu partlt from the bide. they are larger
than thre a' ,-ragv' tr- n the bt,•nd and have cro,"" that are
well du\:eloped though the%- may bc somewhat cro-ded on the'
sides.,
C-odominant trees are tosve W, tb crowo,• viichI form the
I-, le el of tle forest canopy and recen, (, ull lipht from
above but comparatively 1,ttle from the sides.: th' are u,,uallv m
trees with m,,dium-sized crowns that are more or less crowded
on the sides
Intermediate trees are those with cruwns bciow, but s ill
extending iato the general level of the forest canopy, receiving
a little direct light from above, but none from the sides, these
trees usually have small crowns that are shaded on all sides
Overtopped trees are thos, with crowns that arf entircly
below thegeneral forest canopy and receic no direct l1ght
either from above or :rnm t,. sides.
The average height of the dominant and codominant trees i1d) i.
the average of the heights (ground to tip) of only the d -'ii .Ad LU
dominant trees, The avera-e' height ot a,. the trees (H. ) ',wclhdes all
tour classes ot tiu same appl ximate, age. fl. II , for example' \.wilda
0111"' a t ii year -ei' seedlin,, ii, a fift, sear L, stad.d, F ur .it hu-igft (IL)
IS - ter'1m1 Coifnld iuo use, •i thiýs p'ipti" ilnd is all Zkragr- 0: tt !, , L\ erdy.-'
height. -)f thr dominan! and codomninrint treL' andi thf A'~urigt. i.Qo' v
7.1 the trees. Forest height. he. is an average of the two average
),V~ t'g L, dcii.oitv aoovt-, For,..t height, a:- 'ýOl rat. poin'ea out later,
is aitci assumed to be Aie' height of the imperfc cr J2 'r('tr'.c SlaD bvii,L.
dev.eloped lur use in signal strength caIlcla ionIs.
The volume of wood in cubic feet (V wIis taken from t, ywild !abie
In somec cases, the orikinal values shu-ild be miult~plivd 1)% -Ir -*Qtian'±ttd
cut ,~t~fa ictor in order to obt~ar Lhe total wood "olamre on an Acrp.
This correction facter ib based on the structure of the tree Loo0 LIA- pa)t IN
of the tree rot, ink-Alided in the volume reported in the yield table.
Vie bark vc"'- me iv cubic feet (V b) per Lce.e is taken from the
yielk' table (with estimated corictions as abovc) or cestimated 1141lg
experience gaiied fromn other Species and various percentage values
found in the literature.,
Annual leaf fall (W L) in pounds and number o" leaveF (N L) per
cubic meter ;tre approximations mad-! from meager data. These leaf
estimrates ure based o"l a few reports on oven dry wt-ight (it lie Alnnidl
fail and the average live i,,*e of I. "af.
Per unit volumes of wood (' ) and bark (v b) are ci.t. Lakou,
the base volume to b-2 that of a rectanigular solid of ý, art ft-,t
* se area -id a height equal to the fore ,height..
A statistaj.ao dtesc-ipt~on for lobl-)ll pine is gix an in ,e 1.Au
rA cI nN I o mr -
0 4
.0' a, 00r- r- o- r- aoiA-1~ If) m E
'4 4 4N u 0 a4) u - > -
514 co
'Cl~~ ~ ~ M n' D o' 0 'D4
0 p
In co -- -- v .0 0D .i: N wd
> J 0 1ý Z)
11 11 4 0 4)
vn 00 v~~ 00 N~ d. 'o 0 o 0 N E 'o -1.m~~~ ~ ~ ~ v Ln 'D r-r-c -,a ,0 a0 z k >oý
0~ 0
In' f'4 in OOO OC 00 0 0,Cn00 .-
-4 P 4
o
:4~~U %DmCY
ot q4 ,
a4 In r- w M. o 7 0 Ino UýN~ r ' o
L) 0V~~~. ) N. I- 4)nr l nI
M~ ~~~~. '0' D' Nr - ~ .4
0*
4-4 z4)-I V f
v f- N ID P t- a, 0
S. -j * ) 0 r
>1 U -rq -r -t u) I(0o r
should be renebered that the figures gi" en in Table 1 arc for
fIly stocked, even-aged pure staocs with little or no 11lderbrLIshI Thlis
may g sorc ie 1 for .iderstocked areas, however.,
At the present time, it is not clear how to gcý *11e best estimates
on understo':ked areas, One might assume that woos and bark per ui.!
volumes are directly proportional to basal area or can perhc. , be
expressed as a function of basal area.
V. DT .LECTRIC PROPER'IES OF' WOOD
Wood has long been used as a di(e-,ctr,, : i-aterial, ,. L dr,.
wood or wood witi, a low moisture Lontent. The rnoi-ture content is
normally reported as a per cent of the ou en dry wv:ght of wood.
Expressed mafhematically,
W et W Nd r%-
mc :; 100 ! .td' (I)IN dr
xk, re MC rmoisture cont et in per Lent ••> weight)
INwet wveq ht of ti", ooud wher 'w ,t', i e, , Ii for.e,
uvt n drying
tdry weight of w, .,I when "ov! n dr'dry
"'he wood (i.e. , seasoned lumber) used to build houst.b, turi itur',
etc. has a moisture content of approximately o to 24 per evni depending
on the ilet:: od of seasoning and the relative humidity where it I. ,,,,, boiei14
and used. However, this studv is ( mctrned %%ItIh lvIng i wo•d
Wood which has ,i 't been Lut is refer red to as grtten wood G ret.-I
wuud imitialll hais the same, we i,.t physI.al st ri,. re, ani mInoi.vt ire
content as liviaib wood., Oniyt" - '. 01Sti.e l gN.'t-A" pron . . .'., iiS'.
the lItving wood. The.reture, it is reasotnable to asd ., t. ti I, ,t t '* . wi(I
and livtng Wood ar- electricallý ident icr
MoistUr', L, itt ni•I1 " gli 1. 1 woo,0 d are a\xailLlt• in. Womt j iai,(ibn , ,k 14
It ".1 1iim ,diatelx oO\'lci-b , t Sr u Stud> 1,8 this 1,f,. Inati-h, tha' ge
q !Antitieb. oLI ý t .'Lr a rte p"- - t , lt III 11i 2 V , o d t1' a 11%'.11g' t:','u
I-,-
T kcd"I- a:,d Saarm,'m nave suggested that the water present
,0 tlmbcr bc grouped in thret aJdCL-
(1) strongly bound water (moisture Lorntent roughly
J to 5 per cent)
(2) weakly bound water or capillary zo.orbcd water
(moisture corteat roighly 5 to 39 per cent)
(3) nearly free water ;moi.'.ui cortoi ;e , .tv- ttar,
30 per 'tnt).
Reported dielectric cun:'tant curves seem to indicate that
these classes change gradually from one to another.
Bound water is reported to '.a- much 'ougex relaxation time
15than free water leacing to a lower dielectric constant for wooc at
higher frequencies.
Wood definitely has anisotropic dielectric properties. The
dielectric consta:ýr in a directon paraliel to the long wood vessels
(longitudinal) is considerably greater than along a radius of the tree
L.'unk (radial), or tantgent to the wood I'ngs (oangential), ,r aný other'
direLtion prpenc Lcular to the woo'_' vessels (trans'.'ersc)..
18Skaar explained the anisotropy as follows:.
The parllel-tu -grain constaut ol wuOoa ,-
nificantlv grcdter 'han the corrt sponding pcr:I.,dLcular-
to, ra* ,'I 0nbt-11it . , acreason for t thin ditlre cie • 1.ai y
b(- re! idt nt in thuv .1, A itedu htrlot u, i't the *'-l Y'11"
20
The '•ilulose componeut of c"11 walls consists of chair
molecules which aze in pa AIlel at intervals throughout
their i.gti and I.;nce form crystallites. The long axes
of the bulk of tbesr crystallites are essent~allv parallel
to th- long axes of the cells of which they are a part, and
hence they are parallel to the grain of the wood.. On '..
sides of the chain molecules, hydroxyl groups and water
r-_olccules are so arranged that rotation or vibration may
occur more readily in an electric field which is parallel
to the crystaiiitee tnan "n one which is at right angles to
them. The degree of the rotatiuu of the mrole-ule in an
electric field contributes to its dielectric constant.
This ma.y explain the directional differences in the
dielectrical constants as determined.
Since the media is anisotropic, the dielectric constant of wood
to be used can be chosen only after consideration of the E field direction
,. the forest, Tree trunks can be considered (in most cases) to be
verLi,.al and perpk ndicular to the g~ ound.: Thus, for veitical polari-
zation, the longitutdinal dielectric constant is used.. SImIl, , fo,-
horizontal polarization, a Lransverse dielectric f. nstant r ,d.,
Substitutions dand estimat, ; will be used where tCe rejuired information
is not available,
1ý'tar18 reports that the dielectric constant of bucke',ýe wood
increases with moisture ccaLitent (F eure 4). The measurements were
made at 2 Mcpj.m
19Ilearmon and Bur,-ha'-n report measurement- nf the dielectric
properties of oak and Wych elm. Figure 5 is a plot of dielectric constv.,t
versus frequency, and Figure 6 gives the corresponding resul'c for the
loss tangent. Hearmon and Burcham report dielCe.rIc u.unstait-s greater
fb1n ,Ion for low frequencies, These values seem sonrewhat high and out
of line with those reported by other workers. They attribute their high
values to 'polarization effects caused by electrolytic conduction in the
material. ",19
20Trapp and Pungs reported on the dielectric propcrties of a species
of fir wood which is probably common in Germany.. Figure 7 shows that the
dielectric constant increases with moisture LoLIterIt. Figure 8 shows that
the dielectric constant decreases with increasing frequency Note tlat at
high moisture contents and low freque-ic~es, the dielectric, constan, of fir
wood approaches the dielectric constant of water. Figures 9 and 10 give the
corresponding loss tangent data ,•:,ch shows that g.'een fir wood is a very
lossy dielectric. In fact, at the l_,wer frequencies the c ,. rnt
exceeds the displacement current so that it acts more 'iL,- iductor than
a dielectric,.
Taeda 1 Jle inasurements on basswood win.0-i are uf •imited
\awue to this study and are ,eadijy available in the literature.
36
"32"---
28-...=_ - -i_ -
24
I-20 - -
00
FI-
UD
wi
28/
* 0A /
Sf =2~ IO@.,os
0 20 40 60 80 100 '20PERCENT OF MOISTURE
DI ELECTRIC CONSrANT OF BUCKEYE ;:,•,'
FIG -
23
1= 1z I
0
0<200
LL J 00 1DLL z
z
I-
.LNVIS NO)DIW!313ia
s.1
24
_______-. I I 2
1 r L
w In
N j a.
N ~J 0
zzID
z
NVI-
100 , I
E FIELD RADIAL 460 I00 I,
- . - .. ...f...........K 8 0---.- ..-.
40 z z:_K 2o~j~
-0 "
-0
•,u 4 - ''' / ' 1
0 02 0406 1 2 4 6 10 20 40,
MOISTUk CONTENT- PERCENT
DIELECTRIC CONSTANT OF FIR WvOn AS A
FUNCTION OF MOIS'URE CONTENT
FIG
zI
40 *1
U)
00
-0 0I 00~C) ) Ll --
W H
c I N V I S N M'4
0-101z,
4,4
E FIELD RADIAL cp-A L
6 -H I
:4- i ----,.tga -0.- -
4 7- -
' 1I
6---- ---
2
6
0002 --
0,1 02 04 I 5! 4 10 20 4)0 100%MOISTURE CONTENT-PERCENT
LOSS TANGENT OF FIR WOOD AS A
FUNCTION OF MOISTURE CONTENT
FIG Q
.M\
44
8
Lr01 .
8-"•
60 - ----- ---001
4
E FIELD RADIAL
0 0 1 i EIL -LI I
10 10 I10 104 10 5 10 107 ,0' 109 1010
FREQUENCY -CPS
LOSS TAG,,3ENT OF FIR WOOD Ab A
FUNC"iON OF FREQUENCY
FiG 10
Ro K
Th- data presented in detail in this section were not taken on 'he
sper'es of greatest interest ,n this r.dy; however,, the 1ata inidicate that E
the dielect ic piope 'ties o! green wood, whatever the ýpccies are primarily
dLpendent on moisture content. The wood structure of conifers and hard-
woods dififers considerably, however, so it seems reasonable to use coniter
data to estimate the dielectic properties where possibLe, of conier woods
rather than hardwood data., rhe wood dielectric da,a pro ýontly a'allable
A,-,u ., :rccMplete that it cannot be said whether spec eh to species varia-
t.ons among the conifers or amoig the hardwoods art significant.,
A second possibility exists for the treatment of the stem and branches
of trees in a forest in determining the electrical properties of the forest.
The cambium layer,, growth layer, contains a great deal ot mnoistuie In6
which there are many dissolved substances; therefire, it probably has a
relatively high condactivity. In this case,, it is possible that the cambium
Lyer eCf: Ltively forms a conducting sheath aroind the wood parts of thQ,
trunk and branches. li such a case, the wood parts of thL tree hoiald bL
represented by conducting bodies of the same shapLs. These shapes would
be approximated by conducttig so'As of bi-,rpler g•,,,ney sLih ab (-tise,
cylinders or para,.loids., This approac, to the problem i J ,, , linsldrud
Ii this paper,
S .4€
Vi. DIELECTRIC PROPERIMiS OF BARK 4
Diele•LriL mniasurements on loblolly pine bark were made and
are reported in 'igure I! The method of measurement is describcd
in Appendix C. The results show a decrease, in diulectrtc c)unstant
with increasing frequency. The loss tangent curve of the ,ark
(Figure 12 ) appears to be very similar in shapt, to a portion of thet
loss tangent curve for fir woofH ()6% MC.) given by Trapp ai'd Ptng 20
, j'l. The measirements on loblolly pine uark ,. .c' made tn a
radial direction.. It is very probable that results would be different in
a longitudinal or tangential direzt ion due to the layer structure of pine
b.L rk.
The sample measured was taken from a large loblolly pine in
Bastrop State Park, Texas, The sample extended from a tangent to
the ccmbium layer outward about 4/10 of an inch.. The measurements
were made in February, 1963. It is proba .A" that somewLat different
values would be obtained at other seasons, relative humidities, physio-
logical condition of th. ttee, and weather conditions. Fu)r vxamph.,
measuretnentb on bark immediate,, after a rain would protjao! git a
higher dielectric constant.
io-
31
00
I- z
0 >
U-IC, 0
U0
U
Z IL
ILAZ
INVISNOH
- A
wI32
0 0 cc
ww
0 0z
o LWa
NVID
VII. P1ELE(,TRIC CON,'STA'Nj W A MIXTURE
Strictly speaking, the term diclectri ( ontant (an be applied
only to hlumugeneous substances. However, tht concept of A dielectric
constant rar bt. succesfally applied to mixtures cOihiposed ot tvwo or
more substance. The dielectric constant of a nmixture u. several
things tan be expressed as a function of the uicictri-' propi -es of
each material in the mixture, tO-e georrietric shape' ot the particlos o1
each mdter:al and the proportion of th,, total vot. .. ,. ,',.d by ,,( Ii
material. Two limiting cases of a paiallel plat'! Lapacitor with mUltiplh'
layers will b'- considered,
First, sup"uL,.-, tha planee o" d 4,'Iectric layers are pa allel to the
applied E 11( ld aS shown in Figure 13 a. I'his can be. considered as a
group of capacitors in parallel.. Therefore
CT = C + C + C3 + C4 + + C , (2)
But for parall-I '1 ,t, capacitorS
c' z E% ( Ad
where C ,'"ap.citance
K = diele,.tri,, constant
t : permittivity of free sp :e
d - epa ... n bt-tweeii plate .
A arez ,f ul, tS
34
L
PARALLEL. PLATE CAPACITOR WITHDIELECTRICS IN PARALLEL
FIG
E 2
PAR,ALLEL PLATE CAPACITOR WITHDIEL.EC'.RICS IN SERIES
FIG 1Bu
MEW
The above equation can now oe written
Km.o AT KIcoA IK zo A
d d d
Kc A
+ n o n (4)
d2
Multiplying both sides by - results in
0
K VT = V1 K + V2 K +., + V K (5)
where V = volume
. ing both sides by VT and let ing vi pc" d-lt voi~n ,
gives
v 1 K 1 + v 2 K 2 + - + v iKn (6)
Km v.K (6b)
For dielectrit' layers perpvuLa cular to the fAld hitics, the'
effective condenser may be ."oniuv, •d a, made up of a scries .1
parallel plate condensers (Figure For this L:.±a'
I I" 1.F + --7-, +(7
T 1 U2
as bef"-e
K-: oIC -- 0 8d
thus
1 I 1 iK cA K KIA1 Kz A.2 K 0 Amo lo 1 o n o n
dt d1 d2 dn
Multiplying both sides by * 01. ard SImphIfYmg , 'el,0
VT VI V VN
Km K 1 ' K K n
Dividing through by VT results in
m 1 v 2 VnS= m + +. . + u (-L
orn
V ( i b-)
In tht- examples considered iL. this study, the distributions of
particles in the medium art much more complicateid ,ha, li," two smnplc
cases illustratea. However, Wiener22 has shw,: that f.- . .,n v
the actual dielectric constant lies between the extrenit values g.vvi'
above.,
37
Wiener derided t general equ.•tion for mixturcs. r-is se of
general e.uations may be ,,., IL:"
K -K n- K.- Km n I: / v. -- -(12) m
K ÷U i RK+U.m II I I
where
K = dielcctric constant of the mixt'r,m
K = dielectric c, .istant of the rredi"n
K. dielectrik constant of the different materialsI
scattered throughout the media
v. per unit volume of the scattered materials
n-1 K-K
V U K--i i K.+U
u I =1(13)m n-i K- K
- i nvi K. + U.
i=1
and U. is a paramer'-r determined by particle geometry and oriuntation
f,,. simple cases. U has been given for several commor bl-dP• , by
Wiener22 aild Hartshorn and Saxtoa.21 See Table Ii.
It will be noted that equation (12) reduces to eq, Atio,, I([, whei,
Ll = ao and to equativ.,(1l) when u = 0. For purposes of t',,. -:.idy,
equation (12) is valid it'
38
TABLE iI
Values of K ':se x the Dielectri d
Mixtures - Equation (12)
geometrical shape orientation U
sphere any n
circular cylinder perpendicular to £
field lines
thin discs rand ,n disk
needles random 1/2(f needle + 3cn)
layers or laminates planes of laminateparallel to field
layers or laminates planes of laminate 0perpend;, u.:.r to field
1. K <K < <K <K . K, (14)n Kn-I Kn-2 - Kn-3-1
n-r2and ) vKi+ v K
2. K i= r (15)"n - 1 - v n -1
Condition 1 can usually be satisfied by careful choice oi Lhe or,, r
of K 's.: It will be shown later that this condition is aiso ,led' ,
example given. Therefore, the rnathcn,,itical rest-icto.- are niet and
formula (12) n, ) 'e used,
VIII, .LECTRICAL PROPEKTIES OF LEAVES
Si• ce no rn(-aurementr)t, on in,. dit IC LL'-( 11opC'ty, q Of 1( -'I CS
were found in the literature, approximate measutireents were made
by the method described for the bark of the tree in ,-L.tion VI.
The high conductivity of the leaves and metal to ledf ,-ontact
probleris made the measurements of que;tonlth-. v-aIAL', % , le tht"
dielectriL measurements were viot cornpleLely SUiCessiul, they di•(
indicdte th.Lt for frequencies up to 40 megacyclce ,n.i pr..bd i higher,
green leaves act as conductors rather than dicletrics. This con-
clusion is further supported by the composition of leaves themsel-,s.,
Water, in which , .;ny ch-mical subcIances z.re dissolved, makes up
52% to 78% of the total weight of a green leaf.0 Apparently, sotnt ot
these dissolvec chemicals radically increase the conductivity of water.
Since the ltaves are to be L0o1bit-red as conducting bodies rather
than ,'ielectric bodies, the effect of the leaves must be ha'idled by Lneans
of artificial dielectritc theory. W. L. Koek23, 24 has developed two
t 'pes of artificial dic. ctric which have some of the radio fr-,qncoc•
properties of ordinary dielectri, *,)aterials.
The first type has a ,efra,.tivc inuex o1 less than onu i.. - 1)
where
I Z5 (171
RAW-'
,101The see. ad type of artifivýiai dielectrics is referred to as metallic delay
d~e~octrics and has a refrtactive- itluex greatcr than l(in > 1), Kock 24
describes the basic idea 01 delay dielectrics as follows:.-
The artificial dielectric rnat--rial which constitutes the
delay lens was arrived at by reproducing, onl a much Larger
scale, those processes occurrin~g in the molecules of .
true dielectric which pro(! ce the obsei-ved dclay 01 electro-
magnetic waves in such dit±Lctrirc,; Thi- IIIVOIivc a! lanlging
metallic elements in a tbýree -dimensional array or lattice
structure to simiulate the crystalline lattices of the dielectric
material,. such an array ie,,ponls to rz.dio waves Just as a
molecular lattice responds to light waves; the free electror.s
in thc. metal elements flow back and forth under the action of
the alternating electric field, causing the elements to become
escillating dipoles similar to the oscillating molecular
dipoles ot the dielectric. In both cases, the relation between
the effective dielectriC constant C Of the medium, the density
ýjf thL .leme~ it b N (numbe r pc, unit volumne) and the dipole
strength (polarizability a of each element) is approxinntely
given by
c + Na 2(l'4 In thl,:, study)
where E : he dIieletric cot' stant )f tree bpaLC.
41
There are two requirements which are imposed on
the lattice structure, First, the bpdAtIkg uf thik el,_nmcnts
must be somewhat less than one wavelength of the shcrtcst
radio wavelength to be transmtted, otherwise C .fra'tion
effects will occur as in ordinary dielectrics when thc
wavelength is shorter than the lattice sraciig !X-ray
aiffraction by crystalline substances). Secondly, the size
of the elements must be small relative to the v•,inim,,rn
wavelength so that resonance effects are avoided. The
first resonance occurs when the element size is approximately
one-half wai- -.,gth. and for freO, ucn2ies in the viciiiitý of
this resonance frequency the polarizability LL O0 the
element is not independent of frequency., If the element
size is made equal to or less than one quarter wavelength
a, the smallest operating wavelength,, it is found that r. and
hence c in equation 2 (18 in this study) is substantially constant
for all longer 'aav, lengths.:
Since lenses of this t p, will effect an equal amount of
wave delay at all waveengthb ,.hich are long compa i .' Lilu
size and spacing (. the objects, ttocy can De
operate oLer any, desired wavelengt band ..... . . .. ,
Aticth, wa, uf h )king at the wave delay prduted UL
lattices tf bndli c0ndO•ttorb is to ceh.~der them as capactiativ,'
e, menLs which "load" free space. just as parallel capacitors
on a transmission line act aý ioading elements t"., reduce tht"
wave velocit,, Consider a charged parallel plate air condenser
with its electric lihne of forc. perpendicular tki !,e" plates,.
Its cap-city can be increased either by the insertion of
dielectric material or by the insertion of insulated cond ting
objects between the plates if the objects ha, c some length in
Lhv direction of the electric lines of force. Tlis 4s boause
such objects will cause a rearrangement of the lines of force
(with a consequent ncrea se in their number) similar to that
produced b-' shift, due to -n applied field. of the opposiely
charged particles comprising the molecules of t1ke dielectric
material. The conducting elements in the lerns may thus be
considered either as rjrtions of individual condensers, or as
objects which, under the action of the applied field, a'zt as
dipoles and produce a dielectric polarization similar to that
formed by the re•i"rangement of the charged particles com-
prising a non-polar dLelectr, Either viv%,.puint leads to the
delay metchanism obse-ved t,, 1he fucusing action of L1•L
artificial dielectrc lenses to be descr o,;ed.
Thus, Kcck describe, the basic pr:,aciples if artiticial dielctrILcs.
EqCatio, (18) toU- omputing perniittivit y of dan art:ficial ieler( LIc
I:
iq a b, opilfication oi dw' 'ý 'ir . thLorý jt vill be used in tl,
CS."dv and nkaý be altcriidte1ý writieri aD
K I t -r - L719
00
where P poldrization per unit. Volume,
An artificial dielectriL affects the rnagncti- field of a propagating
NA,;ve as well as the electric field. The magneticf field induces
currents in the conduct tog hndi - ý th flit re-ilit thaL ti~e conducting
bodies act as magnetic di;ýuics as Aell as eleCtriL dipolvs, 25 1het
effective pe rmeabilit V cf Othe *rtific. tal ,ILecI v t ri( iý u tiio t he rfag -
nutIc equivalent of equation (18)
p p 0 (20)
Fh~s may also be, written
N air 110
wvhe 1ii pe rmeaý-i. of a vacuLum
N i,imvher of knydoi ting object pe" ¼ '.
a mngnct ii. polar iZa Ility, 01 1 bm1gl CmnC
and ýLr CffCctive r elatiý pe rrea bi ltv of the
a."' 'fic al d11ieLetric.
44
lince, in this study, the leaves have been assumed to randoml!y
H49tributed and all orieotations 1,.,\ e been absumed to be equally likcly.
an averdge cuntr'bution &o the total dielectric constant by each leaf
must be determined.
The average contribution of a single leaf is determined by the
following analysis which is a modification of a derivation gi, -i byz6
Bottcher,26 Assume that all leaf oriontations are equally probable,.
,Ti the number of leaves oriented ir. a given s.lid angl- ijq dir-ctly
proportional to the solid angh. enclosed. Thu expression for a differ-
ential solid angle dil i:•
dfi = sine 0 ',) dO '22)
Then, the number of leaves (dN) oriented in a differential solid
angle will be given by
dN= C sine 0 do dO (23)
where u a constant to be evaluated,:
If equation(22 is integratur -)ver a solid angle of 4Tr
,-r 2 T
N = SdN - S ý C sine 0 o d•, (24)
0 0
Evaluating( .4) aL.4 soving for C gives
C = N-- (25)4i-
4 51
There -.re,
dY" - d,,i dO 2 o)
Tht lea,,-,, t oweecr,; will bt. polarized by an ext,.rnal E or H field
and will act as dipoles. Since these dipoles will all be set up by an
external field rather than being pola- molecules to be rotated bk ine
external field, they will be ', btricte, in t h 11 dr -ion to thc range
rI < 0 < rather than being al jwerd the fall 0 rage from 0 to Tr.
faking this factor into accoant, cquation (26) must ot modified to
N idN -:a sine 0 d0 dO for 0 < 0 <
0 for <0 < 1, (27)
Equation (27) gives the distribution of leaf orientations to which
the polarizing field is applied. The polarizing E or H field may be
Urokei, up into t v€o components for dIbc - bliaped and ne.dle - sha pod
lea Ne; For leaves shaped like discb, one component is in the plaie
o! the disc and the second component is perpendicular to 'he plane of
thty disc. Fr a ncedle-shaped leafvne component is aluag the axis of
the needle and the second 's p, rp-ndiLuiar to tb, a,<is f t0... -: le
(FFigur -1.).
cross section ofueedle or disc.
0
E cosO i
/ , sin 0
Orientation of a leaf with respect to the E and H fields
F gcirt 14
The neL electrit- po.la-i'dtiWI (dli ul leaves in a dift..rential solid
angle may be written a~
JP C (N )a (E cosO 0)a l (E sinle 0) (8
w he r,ý E is the a pplied elvC.tril. field ",tensit y
e lec~t ro. pOla rizabilit y 'or 1, t- tc ~ . i he-
ia, r -as (or pdrallel toc the, luc-igt!,) ocf tlhe ccup'.c
a. - lec-tl-.. pil.±nizcdaoitý ton E field pvrpeCldlIc.. r t) theC
r iaj L)r ji. ro i io r t naIt t !. I2 v 'gt h) ci the 1.1 t
47
'ý'nce dN is zero over Lhe range - < 0 5 Tr, the integration is done.
over a solid angle of Tr ir•.sLead e. irr, After substituting for tIN and
setting up the i.itA jratio,.
iT
P - dP E [ a cos 0 + a sine 0 sine 0 do dO. (2Q)ep en
0 0
After performing the 0 integratin and removing constant terms
LLf. t.hC integrals
P = ENep cos o sin,? 0 d 0 + N,n, sine 0 d 0, (30)
But
It/2Ssine 0 cos 0 d 0 ( 31)
0
iijZ2 i
sine 0 d 0 (. 32)0
Therelore
aI Tr a
P ý EN e. + C .• ,3\4 4
From dielectric theocy
P ý (K- 1) F.. (34)
S 'Cuiv ig to, K,
K - . ... (35)
bstituting equation (33) into equation (35) gives
N . ,n TE- IK = 1 + + L 1 (36)
f0 L4 1
By a similar derivation, it can be shown that
A-= 1 + N Trm (37)I + 4 I+ (37)40 ýI0 ; -4-
where
"a magnetic polarizability for H field parallelmp
to the major axis (or parallel to tle length)
of the object
"a = magnetic polarizability for H field pcrpcndicu~arm n
to the major axis (or normal to the length) ut
the object
I =effective permeability of the artificial dielectric,.
Schelkuntjff2 ai.d Cohn26 have given the electriL a, d magnetic
polarizabilities of several conducting ob)ects of several shapes. For
o-,nvenicnce, some of these have been compiled ii, Table I1. Cautious
use of thizo table s ,necessarv since the ,ularizabtl1tt, of in F-
is different tor different orientations of the objeCt wit ,e to the'
E and H fields.
TABLE III
Polarizal-Q ic- of onducting Bodcici
Shape Eler~tric MagnLtiC
4r3
,Thin circular rod a. 3 lu(a Lm is very small
2a n 4irca I a zi -4Tr~.a I
8 312 I CL -- ýL
en n
IX, Mf_ :HOD FOR SYNTI-TLSIS OF THE DIELECTRIC SLAB -
CALCULATION OF c"
The imaginary part of the dielectric const,•it is determined by
the use of measured attenuation data. This data musL oe changed
from the form of experimental attenuation data to the imagi.z ry part
of the dielectric constant. In order to make this c,.v,.rsion c,.,taitr
relations between a, P, c' and f" must i.- known.
"ThLa study is conIcelaed with plane waves.: t,,C .,u p. oposcd
blab is imperfect, the use of Maxwell's equations ior conducting
media is indicated., As a result of Maxwell's equations, Ramo and
28,Vahinery 2 'ivc .ie following forn-,uliss lor the propagation constant
for plane waves in a ccnducting media:
= C + 7 (3b)
and
o+ jA JWB-(c (39)
where
y= propagation conbtant
u attenuotion L•onstant
P phase cor,stant
cc = complex permittivit,' of Lil ,iater,. I
C C'-i permittivity of the materiai
LT und,,ctiV'ty Of the m.iterial
w =r.1 -- angular frcqucncy
4-
The complex permittivity is more commonly repre ~ited by
.1 -l el (40J)C
Combining equations (38) and (40) and separating real and
lznagnr,, p., r' -r, ,Jts In
S- ,(41)
and
"- (42)
Equation (39) may now be rew.,ritten using *" as
y z a + j1P = j , l-.c' - j(1) (43)
Squarviv hoti- •ideE- a'id equaLLng real terms and imarinary
termb gves
2 2 2 (44)P - Cl = 4a1 t( 4
41 2 , (45)
k % ing t44) and (45) f,)r f" in te rm at " ',d t . g- s
a / 2 2•' = - , , r '+ a (46)
22el -s determined by" usiirg the methods deb~ribed in the last two
chapters. a is determined from an equation given bii LaGrone.2
Attenuation do/meter - 1. 29 x 10 (fMC)0.77 (47)
or
- 4 .771.49 x 10 4f mc ).-7nepers/meter. (48)
This completes the theoretical development.
X. AN :,XAMPLE OF THF COMPUTATION OF THE PROPOSED
DIELECTRIC SLAB
In order to illustrate the method a complex dielectric constant will1 •, 16
be computed for a slab to represent Metz'b plot No.- 1. This plot
is described as being 75% loblolly pine and 25% shortleaf pu., 25 years
of age. The basal area is .,iven as 102. 6 square feeL total with ±02.0
square feet oeing pine.ý The rerr-'inder is hardwood but will be considered
as pine.,
The basal area of each species is -omputed assuming the 75% and
25% to apply.
751--0 x 102. 6 = 76.9 scuare feet bajal area o± wuujliv pine
,!5Y"-5 x 102. 6 = 25,: 7 sqiare feet basal area of shortleaf pine,
Ihe volume oi wood and bark present on the acre will be considered
to be directly proportionval to the basal area. From Table I at !b years
of .ge, the basal area of a fully stocked acre is found to bc 144 square
feet for loblolly pineý and 158 square i,,et for shortledf pine (from a
similar table).
Thus, fur wood volume
7109 6 5 ft loblolly ptiw- \vo.d144 310 1r,5f
3Z5.7 2430 39Q ft shortle,,f piune wood158 3
1941'- iL ,.yoo
Foj bark volumes
76.9• x 800 z 425 loblolly pine bark144 x 1
25.,_ x 620 = 101 shortleaf pine barkl8 526 total bark ft 3
The forest height will be taken to be the woighted average ior tke
two species. Thus,
H -(5 -5 (40) = 52 feet.,f 100 100
The volime of the slab is then
V = (52)(43, 560) = 2. .65 x 106 cubic feet..
The per unit wood and per unit bark for this forest is determined
by
w= 2.0 4 5 x 10 9.U3x0-4
w Vs 2, 265 x 106
Vb_5. _x_2 --I- -(5.26x1 0 = 2.32x!9
s 2.265 x 10
The number of leaves per cubic meter will also -a s pro-
portional to the basal area, thus
76. •74"- 1250 L6t loblolly pvic nee,iles per meter
25. 7. \ 2410 430 bhort/,caf pine nutdles per meter144
Itarlow and lat rar' -,ve thc' approxi niatc 1 ngth of the mwe
needle as 6 to q inhe- for in,,ljll, pine and 3 to ' inches for shortleaf
pine. Averagc values of 7. 5 inches and 4 inches will be used here.
The cross sectio, o' a plt needle i ' er) ciO,. to a 120 svgil rtt
of a circle. For simphcty', ýh! crob-ý section will be approximated
b% c ircile of equal area. Measurmrnent s n the laborator\ •-v .cate
the diameter of this equivalent circle to be approximately 022 inches,
Tbhi completes the required statistical data, The computation
of the dielectric constant will fe,)llowv th( procedure of considering the
dieleu(ric constant resulting from the polarization of the leaveq to be
the dielectric constant oL the media for ubt it hhe mixtare formula.
By use of the leaf data given above and ti~o formula,, of rFable Im
the electric and magnetic P, zabilities are calculated., When the
calculations are corns ted ,w both Species. it ,q obv'oi,•, timt nmp
a ,M 1 nd a can be neglected,*r , en
Evaluating ae for lobloiy pineep
I = .0932 ineterb
a = 2.:8 x !0"4 mcters
o. E 0 1583 1), 4
ep 2I3 log (L) - I
e a
Deliintions ar given on pages - nd !
56
Co -idering only the effects of a i , equation (36) simplifies to
Na
K = _e0
Since tw, kinds of needles are considered and interaction is
neglected.4(668)(5, 83 x 10 - ) 430(1, 8r x 10 ) 0*
K 1 + + 0
0 0
K = 1 + .1947 + .040 1. 2347
Before the example can proceed further, a frequency of operation.'"-Arization must be chosen, frequencv ,) 1 3 e1acvc1 .. and
horizontal polarization is chosen, It ;s well l'nown that Ios,,,t-, will be
greater for \,ertically polarized signals.
Using the dielectric mixture formulas of WienerZ2 the complete
dielectric constant can be calculated,. From Figures 8 and 11 the
dielectric constants ,_- 34 for wood and 2.42 for bark. I'hese values
and the per unit volume-, calculated are used in the Wienc formulas
presented in the meý.hods chapters. First the wood and bark will be
considered as cylinders with axis perpe.dit.ular to the fi1,1 ,ivuing
U K - 1.2347..n
SeC page ,'
-. 1
V I K+
U (49)m n-I
I n
2lvi K i K1=1
Substitutins In equation (49)
n 3
v, 9.03 x 10 per unit volume
-4v2 2. 32 x 10 per unit volume
K 1 34
K2 2,42
U U : K3 - 1.:2347
and evaluating gives U 1. 2347
Another condition on the use of .quation (12) irn',t be checked,
>•vK( + v3 K3ii
K > i (50)
Substitution in this inequality (50) g'ves
S42 > 1, 2 '3.
Therefo .. th' cormputation ,.dii pro o.ed. Uing tl,, in:',.rt forrnii!p
K-K K K_n V, - n {
K+Uv U mU.mI i-I
Substituting the values giver' above in (51) and so]v)ng for K gives
K -1.2370.,
ýI'hUI, 0 " 1. 2370 e farads/meter.0
Now using LaGrone's equation (48) for a where f - 30 mega-
cycles
-4 )0. 77 -3,41 x PO mc -) 2 04 x 10 itepers/lneter
rhus, the complex permittivity oi the slab nmy be v/ritteIL
1-11 -2,0c- j" 1, 093 x 10- - 8,02 x 10- farads/meter
(1., •" 7 j().Ub x U) ) 9 fi rid,/, r cter
See Figure
__ a
ARTIST'S CONCEPTION OF METZ'S FOREST(PLOT NO I)
FIG "
DIELECTRIC SLAB RE ,RESENl,4IAC'N
,F METZ'S FOREST
P'LOT NO I)
FIG. t5k
X.CONcLUS;ON
An approximate mrethod lo.i rer:t.,jij a forewr bv an imperfect
dielectric slab nafs been preeent ed and illumtrated by an example. This
method is presented as i; possible approach to a rath,;& difficult pr~blemn
rather than as a proven trithod of soluition. The usefulness ,f the
method can only be det~ermioned b%, con'siderable (Npc rimnental .- ,)k
Scveivdl a-ssumpt ions hatve- beet) made. Some', of the more important
ones arc*,
i That dielectric mfl1iiurt; tihei-ory is applicable when
objects and spacirqs- are' both significant in size
'~ ~dto a waýOelrsw'h
,2' That the u-ffec? t of interac~ion between leaves, anid
.)etwe(!n lca"ct and 1h(, wood and bark are necgligible,
(3 That a sing le srisooth topped homogeneous slab
can rcprc j"tiI a forcostf
(4) Thiai l'-af placvrm,ýnt acid orictitat ion are random.
1! was nccvsa rF,- 1.. ake' Owse and othu'r assumptions in order to
circurnivc-tt probleum'i Uru'oubt abute.t, 1n~oi-nplete data in some areas
-i acid an ec's aMrOLu.It of ii'uvd matliaemat'cal nianaipa&lationn.
Some, if not all, of tht s.' pribli.-ni-4 cin be solved or rm'tig.-- 'v'
additional work -in this area,
(1 Fu t~ ~~~~S-,% thet 111 t '~ V V ~-o ~lI rg't .t~i
s houlId be con -4idierci a s ctiduct r :I, bud i s rat he r
than dielectric bod~cs Tfho would t(ce .qiate a
diffe~rent a pproa di to calIculxating thi- c- ffvrct ,t oi the~
wood
interaction between i''"-iand possikbly be-tween,
le-avesi bar]', ind '%'ood intot accc.. rit could bt:
a ppl ied,
3; More co. ndu.ct, "t% tind dice tCt ic mras'~i rerriv-I't
on leaves co;ild be rn~adt! to d-.tLrmin., #nor(*
accurately the frc-quAr-flcý range in which the
leaves ma%, b. cot;-.sdt-ri-d c-ondl*4tIf:,
(4) Balil 30 SUggL C3'((1 that 1-ii. '(!xpa~erittcc - 'rtd-icati.' tha.t
pirn m . ,.v art cn'iato tignalf' tr the region of
30. rrfqeg,, CV le pi. r -' . o'd no rc thin ha rr~vcodc
f o.-, ,;t s. E,!i'w.al work is tic-edf.- fto de~termitic
Ilii" s~grbf1 icaym oJ -1101 II qua l~!ativ&(- al)%'e r at ý)n
-)idtiakc pos~bl Lij'ort, t-Li 1r,11 apl. oi
the irnpir:is ef. chelcctric -slab concerit.
ý' I m t. ie p.'.o lo *1jpproxr mic1fl%' c"al.1l 1fI.1.i
R loss 1- a fore-it ;otc) dcit-rrnine whiether -.t isn
a significant par'• of the total signal attcnuation by
using the mrnapnary parts of the dil1-cric constants
of wood and bark determined exper.mentally and the
conduct-vit)y of the leaves.
(b) More than one dielectric slab or perhap- a dielect ic
slab whos,. propertles ,ar% w.-th 1height and Ic. ".,iort
could be used to approx-."na'e the forest. One slab,
for example. might bc used to repr,osent the litter
on the forest iloor. a socond to inlude the under-
brush, a third to include the stem below ,.he crown
; fourth to reprc •lc! the crowvn.
(7) Some means of cons'dr tg the fact that ihe top oi
the forest 1-i not 1.ýti rn'-lht be devhed.
It would be possibl,, to l-,! -.-vei ,-nor(- ibing4 bt wie ( •vll suffi(c.
to show that much , still tfo be don-
REFERE4C ES
1. Harrar, Ellwood Scot,,, "Tre, - " Encyclopedia B:littanica,
Enycilt pedia Zrittanica, Inc., Chicago,: Vol,. 22, pp. 444, 196Z,
2. Harlow, W. H., aad S, S. Harrar, Textbook of Dendrology,
McGraw-Hill Book Co.,, Inc., New York, pp., 2, 42, 1958.
3. Little, E. L. , Jr.,, "Important Forest Trees of the Unit-,, States,"
Trees, U. S. Dept., of Agriculture, pp. 763 P14 1949.
1. May, Curtis, "Keeping Shade Trees Healthy," Trees,. U.S., Dept.
of Agriculture, pp, 91-100 1949,:
5,: Jerram, M. R. K., Elementary Forest Mensuration, Thomas Murbý
and Covyiranv, London, pp. 6-8, 1939,
i. Kamer, Paul J., and T. T.: Kozlowski. Phsiology uf 1'ree,
McGraw-Hill Book Company, Inc., New York, pp., 12 .57,
342-367, 1960.
7. Shintz, H. L. S., and Raphael Zon, "Part I ý The Physical Basib of
Agriculture, Section E - Natura! Vegetatiorn,' Atlas of
American Aericulture, Bureau of Agricultural Econonic t
U. S. Dept. of Agricl, re, pp.: 1-15, 2-, 1924.
8 Bergoffen, W. W.,, "Guegtions and Answers,` Tree'. 11 S D-pt
of Agriculture, pp., 19-36, 1949..
9.. Forbes, R. D., Forestry Handbool , Ronald ejs, Ne", York,,
SeLLIU, 6, 1955.,
.'r.
0-k
10., G,'ise, Cedric H.. Tihe Management of Farm Woodlands, McGraw-
Hill Book Co., hc. , is Ad., New York, pp.: 81-82, 1939,
11. _, Volume, Yield, and Stand Tables for Second
Growth Southern Pines, misc., publ. no. 50, U.: S. Dept. of
Agriculture, Washington, 1929.
12., Metz, L. J., 'Weight and Nitrogen and Calcium Content of the
Annual Litter Fall of the South Carohna Piedmront,"
Proceedings of the Soil 6cience Society of America, Vol., 16,
No. 1, pp. 38-41, 1952.
13. Metz, L. J., "Foreit Soil-Water Relations in the 'Piedmont',"
Report of the Southeastern Forestry Experimeiit Station 1952,
pp. 19-22, 1952.
14. __, Wood Handbook, Forest Products Laboratory,
Forest Service, U.S. Dept. of Agriculture, Handbook No., 72,
pp. 322, 1955.
15. Takeda, Masatami, "Studies on Diclcctric Behavioi of Buund WateL
in Timber ,n the High Frequency Region," Bulletin of the
Chemical Societv of Jpan, Vol. 24, pp., 169-173, 1q51.
16. Stamm, AlCrcd J., "The Fiber-&Sturation Point of Wood as
Obtained from Electric Conductivity Meabur-, r., n:"
Industrial and Engineering Clrims, ta'y'L~al .Ldition,
Vol 1, p., 94-97, 1929.,
17, St. -rnm. Alfred J. and W. Karl Loughborough "Thermodinarn', s
of the Swelling uf Wood Journal of Physical ChemLstrý,
Voi.: 39, pp. ilI-132, 1935.
18. Skaar, Christian, Diclectrical Properties of Wnnd at Several
RLdio Frequencies, New York State College of Forestry
Technical Publication No., 69. 1948.
19. Hearmon, R. F. S. , and J. N. Burcham, The Dielvctric Propertieb
uf Wood, Forest Products Research, Special Report No. 8,
Her Majesty's Statonary Office, Londor, 1954.,
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Feuchte r-.uf das dielectritche Verhalten von Natirholz im
grossen Frequienzbereich," HolzforsrhnU• Bellin, Voý. 10
No. 5, pp. 144-150, 195b.
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Waves," Handbuchlder Pb• ik (edited bý S. Flugge), Vol,. It),
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22, Wiener, Otto 'D,e Theorie Des Miichkorp-rs Fr'r Das Feld Der
Stationairen Str-munL "' Abhanulungen (1 k, t, Ge-i. d Wistz
Math,. khys. C i Vol,, 1 pp, 5U9q-bit) 1q12.
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Jour__2al, Vol., 27 pp., 58-8& 1q48,
66
25. Scllkunoff, Sergei A., and Harold T. Friis, Antennas. Theory
and Practice, John wiley .d Sons, New York, pp. 576-591.
1952.
26. Bottcher, C. J. F., Theory of Electric Polarisation, Elsevier
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27. Cohn, S. B., "Artificial Dielectrics for Microwaves," Mndern
Advances in Microwave Technique, Syrposium Proceedings,
Vnl. 4, Polytechnic Institute of Brooklyn, New York,
pp. 465-492, 1954.
28. Ramo, S., and J. R. Whinnery, Fields and Waves in Modern Radiol
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2S. LaGrone, Alfred H., "Forecasting Television Servi..e Fields,"
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30. Ball, Billie J. - Conversation with the author at Texas A & M
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31. Pounds, D. J., "Considering Forest Vegetation as at, Inmperfect
Dielectric Slab," Masters Thesis in Electrical Engineering,
The University of Texas.; May 1963.,
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