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( 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

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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.,

20., Trapp, Von W., and L. Pungs, Einfluss von Temperature und

Feuchte r-.uf das dielectritche Verhalten von Natirholz im

grossen Frequienzbereich," HolzforsrhnU• Bellin, Voý. 10

No. 5, pp. 144-150, 195b.

21. Hartshorn, L.., ard J. A. Saxton "Di spf.rsion and Absorption of

Waves," Handbuchlder Pb• ik (edited bý S. Flugge), Vol,. It),

pp., 706-710, 1959,

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.

23. Kock, W., E., "Metal Lens A,,,ennas,' Proce-. ._i', f e IRE,

Vol. 34, pp. 828-836, lq4b.,

24,, Kock W. E ,• 'M tallic dela, lones," 3ý,1_•kj _n,__!-.3i

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

Publishing Company, pp. 159-167, 1952.

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

2nd ed., John Wiley and Sons, New York, pp. 305-307, 1953.

2S. LaGrone, Alfred H., "Forecasting Television Servi..e Fields,"

Proceedings of the IRE, Vol. 48 (6), pp. )009-1015, 1960.

30. Ball, Billie J. - Conversation with the author at Texas A & M

College, College Station, Texas, during Spring of 1962.

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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