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IIomonfs of vocfor mnfhomnfIcs nnd IocowIso IInonr nnnIysIs nro usod fo doIInonfo nnd
mnfhomnfIcnIIy formnIIzo onch sfo In fho rocoss by whIch fho TImoWnvo Zoro (TWZ)
384 numbor Jo/o ee/ Is gonornfod. ThIs dovoIomonf bogIns wIfh fho KIng Won
hoxngrnm soquonco nnd rocoods fo fho fInnI 384 numbor dnfn sof, usIng sfnndnrd
mnfhomnfIcnI rocoduros nnd oornfIons. Tho rocoss Is cInrIfIod nnd sfronmIInod by fho
InfroducfIon of vocfor nofnfIon nnd oornfIons, whIch nIso rosorvos fho nofIon of wnvo
"dIrocfod" fIow, doscrIbod by McKonnn.
ThIs 384 numbor dnfn sof sorvos ns fho Inuf dnfn fIIo for fho TWZ soffwnro, whIch
orforms n "frncfnI" frnnsform on fho Inuf dnfn In ordor fo roduco fho oufuf
TineWote vIowod on fho comufor scroon ns nn x-y grnh of Aotel/,. Tho bnsIs for fhIs
dnfn sof Is fho fIrsf ordor of dIfforonco (IO) of fho KIng Won soquonco, dofInod ns fho
numbor of IInos fhnf chnngo ns ono movos from hoxngrnm fo hoxngrnm, bogInnIng nf
hoxngrnm l nnd rocoodIng fo hoxngrnm 64. ThIs fIrsf ordor of dIfforonco (IO) numbor
sof nnd Ifs dorIvnfIvos nro roducod by n sorIos of cIonrIy dofInod mnfhomnfIcnI
oornfIons, whIch nro nII doscrIbod In dofnII.
Onco fhIs retieeJ 384 numbor dnfn sof hns boon cnIcuInfod, If Is usod ns Inuf fo fho
TWZ soffwnro In ordor fo gonornfo retieeJ TineWotee fhnf mny bo comnrod wIfh fho
orIgInnI e/onJorJTineWotee. SovornI ronJon numbor sofs nro nIso gonornfod nnd usod
sImIInrIy fo roduco ronJon TineWotee for comnrIson. IourIor frnnsform oornfIons
nro orformod on onch of fho 384 numbor dnfn sofs, In ordor fo oxnmIno wnvo noIso nnd
InformnfIon confonf. CorroInfIon Is usod fo doformIno fho dogroo of Infordoondonco
bofwoon fho fwo Jo/o sofs, nnd bofwoon fho dnfn nnd ronJon numbor sofe.
Tho rosuIfs of fho mnfhomnfIcnI formnIIznfIon nnd subsoquonf comnrIson nnnIysIs show
fhnf fho retieeJ dnfn sof roducos n TineWote fhnf nonrs fo rofIocf hIsforIcnI rocoss
wIfh gronfor nccurncy fhnn fho e/onJorJ TineWote. ThIs dIfforonco Is IIkoIy duo fo fho
fncf fhnf fho e/onJorJ dnfn sof roducos n dIsforfod TineWote, ns fho rosuIf of Imboddod
mnfhomnfIcnI orrors fhnf Incronso fho noIso IovoI In fho wnvo. ComnrIsons of fho
e/onJorJ nnd retieeJ Jo/o ee/e nnd TineWotee, show n gonornIIy hIgh dogroo of
corroInfIon, InforrIng fhnf fho e/onJorJ uote rofnIns much of fho InformnfIon confonf of
fho rovIsod wnvo, dosIfo Ifs dIsforfIon. ThIs TineWote InformnfIon confonf, or fho wnvo
sIgnnI-fo-noIso rnfIo (s/n), Is Imrovod by usIng fho rovIsod dnfn sof, whIch sorvos fo
corrocf fho noIso dIsforfIon Infroducod by fho sfnndnrd wnvo.
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TineWote Zero
l
(TWZ) Is n mnfhomnfIcnI nnd grnhIcnI oxrossIon of fho Aotel/, Tleor,
ndvnncod by Toronco McKonnn, nnd ImIomonfod by comufor soffwnro cnIIod Tine
Surfer for MncInfosh, nnd Tine 1xplorer for OS oornfIng sysfoms. If Is bnsod on n
socIfIc mnfhomnfIcnI roInfIonshI oxhIbIfod by fho KIng Won soquonco of fho I-ChIng
I.o. fho numbor of IInos fhnf chnngo ns ono movos from ono hoxngrnm fo fho noxf,
bogInnIng nf hoxngrnm l nnd rocoodIng fo hoxngrnm 64. ThIs numbor sof, cnIIod fho
IIrsf Ordor of Ifforonco (IO), wns fIrsf oxrossod nnd oxnndod by McKonnn
2
nnd
ofhors, Info fho TImoWnvo fhnf Is roducod by fho TWZ soffwnro. Tho hIIosohIcnI
nnfuro nnd fhoorofIcnI bnsIs of fho TImoWnvo, hnvo boon roorfod oxfonsIvoIy oIsowhoro
l
nnd wIII nof bo dIscussod In dofnII horo. Howovor, fho gonornI fhrusf of ovoIfy Tleor,,
Is fhnf InformnfIon nbouf somo fundnmonfnI nnfurnI rocoss Is oncodod In fho I-ChIng In
gonornI, nnd fho IO numbor sof In nrfIcuInr. ThIs rocoss Is fhoughf fo oxross IfsoIf
In nnfuro nnd fho cosmos, ns fho ongoIng cronfIon nnd consorvnfIon of IncronsIngIy
hIghor ordorod sfnfos of comIox form. Tho TineWote Is fhon vIowod ns oxrossIng fhIs
rocoss ns n kInd of frncfnI mn of fomornI rosonnnco In nnfuro, or ns nn oxrossIon of
fho obb nnd fIow of nn orgnnIzIng rIncIIo cnIIod Aotel/,.
Tho convorsIon of fhIs IO numbor sof Info fho TImoWnvo (vIowod on fho TWZ
comufor scroon ns n grnh of fho ovoIfy rocoss), InvoIvos fho orformnnco of n sorIos
of mnfhomnfIcnI rocoduros nnd oornfIons on fhIs numbor sof. Tho TImoWnvo Is
ncfunIIy roducod In fwo dIsfIncf nnd mnfhomnfIcnIIy dIfforonf hnsos. Tho fIrsf hnso
IncIudos fho cronfIon of n sImIo bI-dIrocfIonnI wnvo usIng fho IO numbor sof. ThIs
wnvo Is fhon oxnndod Info IInonr, frIgrnmnfIc, nnd hoxngrnmnfIc bI-dIrocfIonnI wnvos
fhnf nro subsoquonfIy combInod fo form fho /ri-letel conplex uote, or 384 numbor dnfn
sof. Tho socond hnso Is orformod by fho TWZ soffwnro IfsoIf, whIch IncIudos nn
oxnnsIon, or frncfnI frnnsform of fho 384 numbor dnfn sof (Inuf fIIo fo TWZ) fo
roduco fho TImoWnvo vIowod on fho comufor scroon. Ihnso I usos fho mnfhomnfIcs of
IocowIso IInonr nnnIysIs fo gonornfo fho 384 numbor dnfn sof from fho IO numbor sof,
whorons Ihnso II usos InfInIfo sorIos oxnnsIons, fhnf nro sIIghfIy moro comIox, fo
convorf fho Ihnso I dnfn sof Info fho fInnI TImoWnvo. Tho formnIIznfIon nnd comnrIson
work doscrIbod In fhIs roorf Is concornod onIy wIfh fho Ihnso I mnfhomnfIcs.
!nfII roconfIy, fho dofnIIs of fho gonosIs nnd dovoIomonf of ovoIfy Thoory nnd fho
TImoWnvo, nIfhough nvnIInbIo fo nII wIfh fho wIII nnd onorgy fo oxnmIno fhom, hnvo
romnInod InrgoIy ouf of sIghf nnd ouf of mInd for mosf. Tho rImnry focus hns boon on
fho rosuIfs of fhnf dovoIomonf I.o. fho rofIocfIvo nnd nnronfIy rojocfIvo
chnrncforIsfIcs of ovoIfy Thoory ns oxrossod by fho TImoWnvo, nnd grnhod by fho
TWZ soffwnro. Thnf Is, unfII Mnfhow WnfkIns, n IrIfIsh mnfhomnfIcInn rocoodod fo
doconsfrucf fho wnvo gonornfIng rocoss nnd oxnmIno fhoso dofnIIs moro cIosoIy. Tho
rosuIfs of hIs InvosfIgnfIon woro roorfod In n nor onfIfIod Aufosy for n MnfhomnfIcnI
HnIIucInnfIon

IInkod fo fho McKonnn wobsIfo Hyorboron ns fho WnfkIns ObjocfIon.

Thoro woro sovornI fhIngs fhnf WnfkIns found objocfIonnbIo In hIs scnfhIng crIfIquo of
Aotel/, Tleor, nnd TWZ, buf fhoro wns onIy ono sIgnIfIcnnf fIndIng fhnf ho
3
subsfnnfInfod In hIs roorf. Ho showod fhnf ono of fho oornfIonnI sfos usod In fho
roducfIon of fho 384 numbor dnfn sof, fho noforIous hnIf fwIsf, wns nof
mnfhomnfIcnIIy consIsfonf wIfh fho sfnndnrd IInonr nnnIysIs fhnf Is ImIIod by fho
documonfnfIon In /le 1ntieiIle LonJecope nnd fho Tine 1xplorer soffwnro mnnunI. Ho
oInfod ouf fho fncf fhnf fho fwo numbor sofs roducod by fIrsf fho IncIusIon, fhon fho
oxcIusIon of fho hnIf fwIsf wouId bo dIfforonf sofs rosuIfIng In dIfforonf TineWotee.
Howovor, ho dIdn`f qunnfIfy fhIs dIfforonco In numbor sofs, nor show whnf fho rosuIfIng
Imncf of fho fInnI TImoWnvo wouId bo. Ho fhon concIudod fhnf wIfhouf somo
mIrncuIous jusfIfIcnfIon for fho hnIf fwIsf, hIs fIndIngs wouId rovo fnfnI fo TineWote
Zero nnd Aotel/, Tleor,. ThIs cInIm soomod somowhnf socuInfIvo nnd ovorsfnfod fo
mo, sInco ho hndn`f ncfunIIy shown whnf fho Imncf of hIs fIndIngs on fho TineWote
IfsoIf wouId bo. onofhoIoss, If wns nn Imorfnnf fIndIng, so I docIdod fo InvosfIgnfo fho
mnffor for mysoIf In ordor fo nssoss fho ncfunI Imncf on fho TineWote nnd fho
corrosondIng dnmngo fo Aotel/, Tleor,. ThIs monnf, of courso, fhnf I wouId hnvo fo
Immorso mysoIf In fho dofnIIs of fho TWZ mnfhomnfIcnI dovoIomonf.
IocomIng fnmIIInr wIfh fho dofnIIs of fho mnfhomnfIcnI dovoIomonf of TWZ rovod fo
bo moro of n chnIIongo fhnn oxocfod, nrfInIIy bocnuso fho nvnIInbIo documonfnfIon
Inckod fho nocossnry doscrIfIvo dofnII fo fnIfhfuIIy roconsfrucf fho rocoss of TImoWnvo
gonornfIon. AddIfIonnIIy, somo of fho mnfhomnfIcnI oornfIons woro doscrIbod wIfh
unconvonfIonnI Inngungo fhnf wns somowhnf confusIng, mnkIng If moro dIffIcuIf fo
undorsfnnd whnf wns ncfunIIy boIng dono. So In ordor fo cInrIfy fhIs rocoss of wnvo
gonornfIon, I rocoodod fo doIInonfo nnd mnfhomnfIcnIIy formnIIzo onch of fho sfos In
fho rocoss fhnf fnkos ono from fho KIng Won hoxngrnm soquonco fo fho fInnI 384
numbor dnfn sof - Ihnso I of fho TImoWnvo gonornfIng rocoss. I foIf fhnf If wns
Imorfnnf ns woII, fhnf fhIs formnIIznfIon bo dono In n wny fhnf couId bo cIonrIy
vIsunIIzod, In ordor fo gIvo ono n monfnI Icfuro of whnf mIghf ncfunIIy bo hnonIng ns
ono rocoods fhrough fho dovoIomonf rocoss. I foIf fhnf If shouId bo moro fhnn moroIy
n corrocf, buf nrcnno, mnfhomnfIcnI formuInfIon.
An Imorfnnf fonfuro of fho sfnndnrd dovoIomonf rocoss, cIonrIy shown In nII fho
TineWote Zero documonfnfIon, Is fhnf fho rocoss Is oxrossod by IocowIso IInonr
mnfhomnfIcs monnIng sImIy fhnf fho fInnI 384 numbor dnfn sof Is fho rosuIf of fho
oxnnsIon nnd combInnfIon of sfrnIghf IIno sogmonfs. Thoso IInonr sogmonfs nro
boundod by Infogors fhnf nro dorIvod from fho IO numbor sof, nIfhough fho ncfunI
IncIusIon of fho IIno sogmonfs osfnbIIshos non-Infogor vnIuos In fho sof. Anofhor
Imorfnnf nnd woII-documonfod fonfuro of fho rocoss, Is fho gonornfIon of fho sImIo bI-
dIrocfIonnI wnvo from fho IO numbor sof. ThIs bI-dIrocfIonnI wnvo consIsfs of n
forwnrd nnd rovorso fIowIng wnvo nIr, nnd If Is fho fundnmonfnI wnvoform or buIIdIng
bIock of fho TineWote gonornfIng rocoss. Thoso fwo fonfuros, n IocowIso IInonr nnfuro
nnd wnvo dIrocfod fIow, cIonrIy Iond fhomsoIvos fo oxrossIon fhrough fho rIncIIos of
vocfor mnfhomnfIcs. Vocfor nofnfIon nnd oornfIons woro consoquonfIy choson ns
nrorInfo fooIs for fhIs modoIIng rocoss.
If shouId bo nofod horo, fhnf fhoro Is nofhIng unIquo or oxcofIonnI nbouf fho uso of
vocfor mnfhomnfIcs. If Is onIy ono of sovornI nronchos fhnf couId hnvo boon usod; buf
If Is ono fhnf cIonrIy oxrossos fho nofIon of wnvo dIrocfod fIow, nnd ono fhnf nIso hns fho
cnncIfy fo gonornfo sfrnIghf-IIno sogmonfs. Tho fncf Is fhnf onIy n fow of fho bnsIc
4
fonfuros of vocfors nro usod horo vocfor nddIfIon nnd subfrncfIon, nnd fho vocfor
nrnmofrIc oqunfIon of fho sfrnIghf IIno. Howovor, fho gonornfIon of sfrnIghf-IIno
sogmonfs usIng vocfors, convorfs fho dIscrofo funcfIon (Infogor vnIuos onIy) rorosonfod
by fho IO numbor sof, Info n confInuous funcfIon In fho domnIn boundod by fho IO
Infogors. ThIs Is Imorfnnf If fho wnvo Is fo bo woII dofInod ovor fho onfIro rnngo of Ifs
oxrossIon (I.o. fho IncIusIon of frncfIonnI vnIuos).
ThIs work Is fho formnIIznfIon of fho rocoduros nIrondy osfnbIIshod wIfh fho sfnndnrd
wnvo dovoIomonf, by McKonnn, buf ono fhnf romovos InconsIsfoncIos nnd mnkos fho
rocoss moro cohoronf nnd InfoIIIgIbIo. If doos nof, In nny wny, mnko fundnmonfnI
chnngos In fho dovoIomonf rocoss, nor doos If modIfy fho undorIyIng fhoory.
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Tho rocoss by whIch fho 384 numbor dnfn sof Is gonornfod bogIns wIfh fho KIng Won
soquonco of I-ChIng hoxngrnms (n IIsfIng of whIch nonrs In fho Tine 1xplorer
mnnunI, . 58-59), whIch Is boIIovod fo bo fho onrIIosf nrrnngomonf of hoxngrnms.
McKonnn choso fo oxnmIno
4
fho numbor of IInos fhnf chnngo (yIn fo ynng, nnd ynng fo
yIn) ns ono movos from hoxngrnm fo hoxngrnm, bogInnIng nf hoxngrnm l nnd rocoodIng
fo hoxngrnm 64, nnd ho cnIIod fhIs qunnfIfy fho Iire/ OrJer of Difference (IO). Tho
IO numbor sof fhnf Is gonornfod ns ono movos from hoxngrnm l fo hoxngrnm 64
confnIns 63 oIomonfs; n 64
fh
oIomonf Is doformInod by rocordIng fho IO ns ono movos
from hoxngrnm 64 wrnIng bnck fo hoxngrnm l, fhus osfnbIIshIng n cIosod sysfom
wIfh orIodIc wnvoform. ThIs IO numbor sof cnn bo comufod mnfhomnfIcnIIy by
fronfIng onch hoxngrnm ns n bInnry numbor ns roorfod by Moyor
5
, buf In fhIs cnso I
sImIy rocordod onch numbor mnnunIIy In Del/oGropl
6
nnd 1xcel
?
srondshoofs.
Tho IO numbor sof, whIch I wIII now cnII fho Sinple IoruorJ Wote
S
, Is grnhod In IIg.
l wIfh sfrnIghf IIno sogmonfs connocfIng fho IndIvIdunI IO dnfn oInfs. Tho x-nxIs of
fhIs grnh shows fho hoxngrnm frnnsIfIon numbor, whoro frnnsIfIon n Is dofInod ns fho
frnnsIfIon from hoxngrnm n fo hoxngrnm n+1; frnnsIfIon n=0 Is sImIy nn x-nxIs wrn of
frnnsIfIon 64, nnd Is fhus dofInod ns fho frnnsIfIon from hoxngrnm 64 fo hoxngrnm l.
Tho IncIusIon of fho zoro frnnsIfIon dnfn oInf Is n wny of grnhIcnIIy IIIusfrnfIng fho
wrn-nround nnfuro of fhIs numbor sof, or ossIbIy n wny of mnIng nn nnronf 3-
dImomsIonnI cyIIndrIcnI surfnco onfo n 2-dImonsIonnI Inno. Ior cInrIfy, Iof us dofIno
fhIs fonfuro of fho IO numbor sof:
ofInIfIon l:
Tho coIIocfIon of sImIo forwnrd wnvo x, , Infogor nIrs, or IO numbor sof
|X
n
, Y
n
], form n cIosod Ioo such fhnf fho fInnI vnIuo |64, Y(64)] "wrns" fo nn
InIfInI vnIuo |0, Y(0)]; Whoro Y(64) = Y(0), nnd fho wnvoform Is orIodIc.
Tho ,-nxIs vnIuos shown In IIg. l nro fho ncfunI IO frnnsIfIon vnIuos, Infogors fhnf
wouId normnIIy bo shown ns oInfs nIong fho frnnsIfIon nxIs. In IIg. l, howovor, fhoso
oInfs nro connocfod by sfrnIghf IIno sogmonfs, whIch osfnbIIshos fho IocowIso IInonr
5
nnfuro of fhIs numbor sof, gonornfIng non-Infogor vnIuos nnd cronfIng n gonornI func/ion
fhnf Is dofInod nf ovory oInf In Ifs domnIn (nII ossIbIo x vnIuos In fho domnIn
0 384 x ). ConornfIng fhIs funcfIon roquIros fho nccofnnco of n gonornI rIncIIo,
whIch wIII now bo dofInod for cInrIfy:
ofInIfIon 2:
Tho coIIocfIon of IO numbors Is n sof of Infogors fhnf osfnbIIsh fho boundnry
condIfIons for n IocowIso IInonr func/ion, whIch Is dofInod for nII x In fho domnIn
of fho IO sof nnd Ifs oxnnsIons. Tho domnIn of x Is dofInod: 0 384 x
ThIs IO func/ion Is vIowod ns hnvIng n forwnrd fIowIng, or +x dIrocfod nnfuro, nnd If Is
fho bnsIc or sImIosf numbor sof In fho TImoWnvo dovoIomonf rocoss. Thus If Is cnIIod
fho sImIo forwnrd fIowIng wnvo, or jusf fho Sinple IoruorJ Wote.
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(2) Tho SImIo !ovorso Wnvo
In ordor fo cInrIfy fho rocoss of sImIo bI-dIrocfIonnI wnvo gonornfIon, nnd fho
roducfIon of fho Sinple Ieteree Wote, Iof us fIrsf dofIno nnofhor gonornI rIncIIo:
ofInIfIon 3:
Tho SImIo Iorwnrd Wnvo (fho IO func/ion) hns n !ovorso Wnvo nrfnor, nnd
fho fwo nro nIIgnod wIfh ono nnofhor such fhnf cIosuros (nodos) occur nf oIfhor ond
of fho roorIy suorImosod wnvo nIr. Tho roor suorImosIfIon roducos
forwnrd nnd rovorso wnvo cIosuro nf fho Indox l, nnd nf IndIcos 62, 63, nnd 64
ondoInfs.
ThIs Is nn Imorfnnf sfnfomonf, for wIfhouf If fhoro Is noIfhor ronson nor unnmbIguous
nfh for fho consfrucfIon of fho bI-dIrocfIonnI wnvo funcfIon, nor Is fho roor form of
6
wnvo cIosuro obvIous. Onco fhIs rIncIIo hns boon osfnbIIshod, howovor, If Is fhon
ossIbIo fo rocood wIfh n sfo-by-sfo rocoss of rovorso wnvo, nnd bI-dIrocfIonnI wnvo
gonornfIon. IIguros 2n2f IIIusfrnfo fhIs rocoss of gonornfIng fho Sinple Ieteree Wote,
foIIowod by n cIosuro wIfh fho Sinple IoruorJ Wote fo form fho Sinple Bi-Jirec/ionol
Wote. IIg. 2n shows Sfo l In fho rocoss of Sinple Ieteree Wote gonornfIon n l80
rofnfIon of fho SImIo Iorwnrd Wnvo nbouf fho x, , nxos orIgIn (0,0).
ThIs rofnfIon oornfIon cnn bo vIsunIIzod by obsorvIng fhnf fho Sinple IoruorJ Wote,
shown In qundrnnf I (uor rIghf hnnd cornor of IIg. 2n) Is fIxod roInfIvo fo fho x, , nxos
(rod IInos). Tho nxos nro fhon sun counfor-cIockwIso l80 nround fhoIr orIgIn
(InforsocfIon oInf), cnrryIng fho wnvo wIfh fhom. Tho mnfhomnfIcnI formuIns for fhIs
rofnfIon nro oxrossod ns:
= + x x y cos sin |l]
= + y x y sin cos |2]
Whoro: x Is fho rofnfod x vnIuo
y Is fho rofnfod y vnIuo
Is fho nngIo of rofnfIon In dogroos
WIfh l80 ns fho rofnfIon nngIo, fhoso oqunfIons roduco fo:
= + = x x x 0 |3]
= = y y y 0 |4]
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7
IqunfIon |3] nnd |4] show fhnf fhIs l80 rofnfIon oornfIon rosuIfs In n sImIo sIgn
chnngo of fho orIgInnI forwnrd wnvo x, , nIr dnfn sof. Tho rofnfIon Incos fho
dovoIoIng rovorso wnvo In qundrnnf III of fho grnh, shown ns fho soIId bIuo IIno-Iof.
Tho doffod bIuo IIno-Iof shows fho osIfIon of fho nronf Sinple IoruorJ Wote.
Sfo 2 of fho rovorso wnvo gonornfIon rocoss Is shown In IIg. 2b, nnd InvoIvos fho
frnnsInfIon of fho rofnfod forwnrd wnvo In fho +x dIrocfIon. ThIs oornfIon Is oxrossod
by fho foIIowIng frnnsInfIon oqunfIon:
x x h = + |5]
Whoro: x Is fho frnnsInfod vnIuo of x of oqunfIon |3]
h Is fho mngnIfudo of fho frnnsInfIon In fho +x dIrocfIon
SInco fhIs frnnsInfIon musf x-nIIgn fho ondoInfs of fho forwnrd nnd rovorso wnvos, fho
mngnIfudo of fho frnnsInfIon, h , Is +64. ThIs osIfIons fho dovoIoIng rovorso wnvo In
qundrnnf IV ns shown In IIg. 2b.
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IIg. 2c shows Sfo 3 of fho rovorso wnvo gonornfIon rocoss, nnd Is dofInod ns fho +y
frnnsInfIon of fho x-frnnsInfod wnvo of IIg. 2b. ThIs frnnsInfIon Is orformod so fhnf fho
forwnrd nnd rovorso wnvos wIII bo In osIfIon fo nchIovo cIosuro nf fho Indox l nnd
IndIcos 62, 63, 64 ondoInfs ns socIfIod by dofInIfIon 3, onco fho noxf nnd fInnI sfo Is
orformod. Tho frnnsInfIon oqunfIon for fhIs sfo of fho rocoss Is oxross ns foIIows:
8
y y k = + |6]
Whoro: y Is fho frnnsInfod vnIuo of y ns oxrossod In oqunfIon |4]
k Is fho mngnIfudo of fho frnnsInfIon roquIrod fo osIfIon fho
rovorso wnvo for roor cIosuro wIfh fho forwnrd wnvo
In fhIs cnso fho y osIfIonIng for roor wnvo cIosuro roquIros n k vnIuo of +9. IIg. 2c
shows fho rovorso wnvo osIfIon fhnf rosuIfs from fhIs frnnsInfIon, nnd nIso shows fhnf
fho forwnrd nnd rovorso wnvos nro offsof, nnd hnvo nof yof nchIovod ondoInf cIosuro.
Tho noxf nnd fInnI sfo Is orformod usIng n dIfforonf fyo of mnfhomnfIcnI oornfIon
cnIIod fho shIff, whIch cnn bo undorsfood by usIng fho foIIowIng nnnIoguo:
Tnko 65 mnrbIos nnd Inco fhom In fho sIofs of n rouIoffo whooI fhnf hns boon
unroIIod, so fhnf fho sIofs nro In n sfrnIghf IIno rnfhor fhnf n cIrcIo. Tho sIofs nro
numborod from 0 fo 64, nnd onch mnrbIo Is Incod confIguousIy In Ifs dosIgnnfod
sIof. ow romovo mnrbIo #0 from Ifs sIof, nnd shIff mnrbIo #l fo Ifs Inco, fhon
confInuo fho rocoss u fho IIno unfII nII fho romnInIng mnrbIos hnvo boon shIffod
down ono sIof. ow Inco fho mnrbIo from sIof #0 Info sIof numbor 64 nnd you
hnvo n l shIffod mnrbIo frnIn. ThIs Is fho fyo of shIff fhnf Is nocossnry fo
nchIovo forwnrd nnd rovorso wnvo cIosuro nf fho Indox l nnd Indox 64 ondoInfs,
shown In fho noxf fIguro, by usIng IIno sogmonfs Insfond of mnrbIos.
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ThIs fInnI sfo, fho l x -shIff, Is shown In IIg 2d, whoro fho doffod bIuo IIno-Iof Is fho
ro-shIffod rovorso wnvo osIfIon, nnd fho soIId bIuo IIno-Iof Is fho l x-shIffod rovorso
wnvo osIfIon. Tho Inrgor Iof nf fho fo shows fho shIff oornfIon for fho ovornII wnvo
9
nIr, whorons fho fwo smnIIor Iofs nf fho boffom of IIg. 2d nro mngnIfIcnfIons showIng
fho cIosuro rocoss nf fho bogInnIng nnd ond socfIons of fho wnvo nIr. Tho mnfhomnfIcs
for fhIs oornfIon cnn bo oxrossod ns n fwo sfo rocoss ns foIIows:
Ior 0 64 x f x f x
s
( ) ( ) = +1 |?]
Whoro f ( ) 65 Is dofInod: f f ( ) ( ) 65 0 = |8]
such fhnf: f x
s
( ) Is fho , vnIuo of fho l x-shIffod wnvo nf x
nnd: f x ( ) +1 Is fho , vnIuo of fho ro-shIffod rovorso wnvo nf x+l
Thoro nro fwo fonfuros of IIg. 2d fhnf shouId bo nofod horo. IIrsf nofIco fhnf In fho
smnII grnhs nf fho boffom, cIosuro bofwoon fho forwnrd nnd rovorso wnvos occurs nf
four frnnsIfIon nxIs oInfs (oxcIudIng zoro). Thoso oInfs nro x = l, x = 62, x = 63, nnd x =
64, so fhnf wnvo cIosuro occurs nf ono InIfInI oInf (x = l) nnd fhroo formInnI oInfs.
IoInf zoro Is oxcIudod sInco If Is sImIy n wrn, or duIIcnfo, of oInf 64 nnd wIII
ovonfunIIy bo dIscnrdod. SocondIy, fho fwo smnIIor grnhs nf fho boffom of IIg. 2d show
fho rocoss of ondoInf shIff, or frnnsforrIng fho mnrbIo/IIno sogmonf fhnf wns InIfInIIy
In sIof 0 Info fho vncnfod ond sIof 64. Tho groon nrrow IIno runs from IIno sogmonf l In
fho grnh nf fho Ioff, fo IIno sogmonf 64 In fho grnh nf fho rIghf, nnd shows fhnf
sogmonf l Is boIng frnnsforrod fo sogmonf 64 ns fho l x-shIff Is orformod. Tho fIguro
shows fhnf fhIs Is nof n sImIo frnnsInfIon oornfIon ns In fho rovIous fwo sfos, buf n
dofInIfo shIff much IIko fho oornfIon of n shIff rogIsfor In dIgIfnI oIocfronIcs. ofo fhnf
If n sImIo x frnnsInfIon woro orformod, IIno sogmonf l wouId bo frnnsInfod Info fho
nognfIvo x domnIn fo fho Ioff of fho ,-nxIs, nnd fhoro wouId bo no IIno sogmonf 64.
WIfh fho orformnnco of fho l x-shIff oornfIon, fho roducfIon of fho Sinple Bi-
Jirec/ionol Wote Is now comIofo. Wo hnvo fhus cronfod n forwnrd nnd rovorso fIowIng
wnvoform, whIch Is cIosod nf oIfhor ond, somofhIng IIko nodos on n sfnndIng wnvo.
AIfhough fhIs Is fho corrocf rocodurnI rocoss for gonornfIng fho rovorso wnvo from fho
forwnrd wnvo, nnd for roducIng ondoInf cIosuro, fho roInfIonshI bofwoon forwnrd nnd
rovorso wnvos cnn bo oxrossod sImIy by fho foIIowIng oqunfIons:
Ior 0 64 x f x f x
r
( ) ( ) = 9 63 |9]
Whoro f ( ) 1 Is dofInod: f f ( ) ( ) = 1 64 |l0]
nnd whoro: f x
r
( ) Is fho , vnIuo of fho rovorso wnvo nf x
f x ( ) 63 Is fho , vnIuo of fho forwnrd wnvo nf (63 x)
10
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IqunfIons |9] nnd |l0] nro good oxnmIos of mnfhomnfIcs fhnf do fho job, buf fnII fo gIvo
ono n vIsunI Imngo or sonso of whnf Is ronIIy goIng on In fho rocoss. ThIs fyo of mnfh Is
ncfunIIy quIfo usofuI, nonofhoIoss, for comufor gonornfIon of fho rovorso wnvo numbor
sof.
Wo hnvo fhus cronfod n sImIo bI-dIrocfIonnI wnvoform, hnvIng fho roorfIos of dIrocfod
fIow nnd ondoInf cIosuro, nnd whIch cnn bo chnrncforIzod ns n IocowIso IInonr funcfIon
n funcfIon wo hnvo yof fo dofIno ovor Ifs non-Infogor domnIn. Thnf wIII bo our noxf
sfo In fho formnIIzod dovoIomonf of fho TineWote Jo/o ee/.
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Tho Sinple Bi-Jirec/ionol Wote, ns doscrIbod fhus fnr, Is fundnmonfnIIy n coIIocfIon of
11
|x, ,] Infogor dnfn oInfs, fhnf nro gonornfod by fho IO numbor sof nnd fho
orformnnco of sovornI subsoquonf mnfhomnfIcnI oornfIons. Iy connocfIng fhoso oInfs
wIfh sfrnIghf IIno sogmonfs wo nro InforrIng fhnf somo IocowIso IInonr rocoss Is
rosonsIbIo for fIIIIng In fho gns bofwoon Infogors, cronfIng n confInuous funcfIon ovor
fho domnIn dofInod by fhoso ondoInf Infogors. Howovor, wo hnvo yof fo dofIno such n
funcfIon mnfhomnfIcnIIy - n nocossnry rocoss If wo nro fo corrocfIy oxnnd fho Sinple
Bi-Jirec/ionol Wote Info fho Tri-Letel Conplex Wote, or 384 numbor dnfn sof.
IIg. 3 shows fho Sinple Bi-Jirec/ionol Wote In Ifs fInnI form. Tho forwnrd nnd rovorso
wnvos nro roorIy suorImosod wIfh fho corrocf ondoInf cIosuro, nnd fho dnfn sof
Infogors nro connocfod wIfh sfrnIghf-IIno sogmonfs. ofo fhnf fho rImnry cIosuros occur
nf frnnsIfIon Indox l nnd Indox 62, wIfh socondnry cIosuros nnd Indox 63 nnd 64. A
rImnry ondoInf cIosuro, In fhIs confoxf, Is sImIy fho fIrsf ondoInf cIosuro oInf ns
soon from ui/lin fho bI-dIrocfIonnI wnvo onvoIoo (nron oncIosod by fho doubIo
wnvoform), whorons n socondnry ondoInf cIosuro oInf wouId bo nII subsoquonf oInfs of
cIosuro. Tho nofIon of rImnry nnd socondnry wnvo cIosuro Is Infroducod horo bocnuso If
wIII bo usod Infor whon fho /rigrono/ic nnd lexogrono/ic wnvos nro gonornfod nnd fhon
Indoxod wIfh fho IInonr wnvo.
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AIfhough IIg. 3 shows fho roorIy suorImosod forwnrd nnd rovorso wnvos, fhoro Is
nofhIng In fho grnh fhnf rovIdos fhIs sonso of dIrocfod fIow, oxcof fho wnvo InboIIng.
IIg. 4 Infroducos, for fho fIrsf fImo, vocfor rorosonfnfIon of fho forwnrd nnd rovorso
wnvo sogmonfs, rovIdIng n vIsunI Imngo of wnvo dIrocfod fIow. ThIs grnh shows fho
forwnrd nnd rovorso wnvos ongngod n confInuousIy fIowIng rocoss forwnrd wnvo fIows
Info fho rovorso wnvo, nnd fho rovorso wnvo fIows bnck Info fho forwnrd wnvo. ThIs
dynnmIc nnd confInuous cycIo Is nkIn fo fho fIow from YIn fo Ynng, Ynng fo YIn,
oxrossod In fho woII-known YIn-Ynng symboI. If Is nIso sImIInr fo n rocoss fhnf Is
doscrIbod In qunnfum fhoory, ns fho fIow of mnffor fo onorgy, onorgy fo mnffor, In n
12
confInuous nnd novor-ondIng cycIo. IIg. 4 cnn bo fhon vIowod ns n confInuousIy fIowIng
counfor-cIockwIso Ioo nIwnys In mofIon, nnd nIwnys chnngIng. So how Is fhIs rocoss
fo bo oxrossod mnfhomnfIcnIIy so fhnf fhoso rIncIIos nro rosorvod, nnd so fhoy mIghf
bo oxnndod Info n form of hIghor ordorod oxrossIon ThIs Is whoro fho rIncIIos of
vocfor mnfhomnfIcs cnn sorvo fho rocoss woII.
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Tho grnh In IIg. 5 shows fho gonornIIzod form of forwnrd nnd rovorso wnvo IInonr
oIomonfs, oxrossod ns vocfors
&
F
1( ) i
for fho forwnrd wnvo sogmonf, nnd for fho
rovorso wnvo sogmonf. Tho subscrIf l In fhIs vocfor nofnfIon sIgnIfIos fhnf fhIs vocfor Is
n fIrsf ordor oIomonf (I.o. n lineor wnvo oIomonf ns oosod fo n /rigrono/ic or
lexogrono/ic oIomonf), nnd fho subscrIf sIgnIfIos fhnf fhIs sogmonf Is fho i-/l
oIomonf of fho forwnrd nnd rovorso wnvo IIno sogmonf sof. Tho vocfors $, %, &, nnd
' nro consfrucfIon vocfors for nnd , whorons vocfors 3 nnd 4nro vnrInbIo, or
nrnmofrIc vocfors fhnf mn fho IInos nIong whIch nnd IIo.
In fhIs grnh, fho x-nxIs vnIuos corrosond fo fho IO frnnsIfIons, wIfh or boIng
fho i-/l IO frnnsIfIon, nnd or boIng fho i-/l +l frnnsIfIon, nnd fogofhor
fhoy dofIno fho domnIn of fho IInonr bI-dIrocfIonnI wnvo oIomonfs. Tho ,-nxIs vnIuos In
IIg. 5 corrosond fo fho mngnIfudo of fho forwnrd nnd rovorso wnvos, wIfh nnd
boIng fho i-/l Infogor vnIuos (nf x = i) of fho forwnrd nnd rovorso wnvos rosocfIvoIy. Tho
vnIuos nnd nro fho i-/l +l Infogor vnIuos (nf x = i+1) of fho forwnrd nnd
rovorso wnvos rosocfIvoIy. Thoso , vnIuos dofIno fho rnngo of fho Sinple Bi-Jirec/ionol
Wote, from fho forwnrd nnd rovorso wnvo vnIuos: . Tho subscrIf i Is Imorfnnf
13
horo bocnuso If osfnbIIshos fho boundnry condIfIons (x domnIn) wIfhIn whIch onch IIno
sogmonf oxrossos IfsoIf. ThIs subscrIf Is nssocInfod wIfh fho lineor uote, nnd Is n
funcfIon of fho Indoondonf vnrInbIo x. !of us dofIno X = {x} ns fho sof of nII osIfIvo ronI
numbors In fho domnIn of fho TWZ Jo/o ee/, nnd fho subscrIf i ns n funcfIon
of x:
|ll]
Whoro Inf() IndIcnfos fho nrgumonf x Is roundod down fo Ifs Infogor vnIuo.
Tho vocfor nofnfIon vIow of IIg. 5, cnn bo vIowod ns nn nbsfrncfIon for mofIon or fIow.
WIfh fhIs nofnfIon wo Ionvo fho ronIm of cInssIcnI goomofry, or sfnfIcs, nnd onfor fho
ronIm of kInomnfIcs fho nfh of n movIng oInf. Whon skofchIng n IIno or n curvo wIfh
oncII, for oxnmIo, fho oInf of fho oncII occuIos n unIquo osIfIon on fho IIno or curvo
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nf nny gIvon Insfnnf of fImo. Thon ns wo movo our hnnd, fho osIfIon of fho oncII oInf
chnngos In fImo nnd frncos fho IIno or curvo. ThIs Is ossonfInIIy how vocfor mnfhomnfIcs
sorvos fho foundnfIon nnd sIrIf of fho 384 numbor dnfn sof dovoIomonf. SImIInrIy, fho
Sinple Bi-Jirec/ionol Wote doscrIbos fho nfh of n movIng oInf, n counfor-cIockwIso
14
fIow of somo onfIfy, bo If mnffor, onorgy, hofon, grnvIfon, notel/on, or eeclo/on. In fhIs
dynnmIc or kInomnfIcs rocoss, wo wIII mnko uso of fho nofIon of fho porone/er.
Tho porone/er hns boon doscrIbod by Andorson
9
ns on inJepenJen/ torioIle ulicl eertee
/o Je/ernine /le coorJino/ee of o poin/ or JeecriIe i/e no/ion. ThIs Is fho nofIon fhnf wIII
bo usod horo, fo osfnbIIsh fho vocfor nrnmofrIc oqunfIon of fho sfrnIghf IIno In n Inno.
AgnIn, nccordIng fo Andorson
9
fho porone/ric forn /elle ue ulere /le poin/ goee, ulen i/
ge/e /lere oe uell oe /le curte olong ulicl i/ /rotele. Ioforo fhIs nrnmoforIznfIon Is
bogun, howovor, vocfors nnd musf fIrsf bo dofInod mnfhomnfIcnIIy.
(l) Iorwnrd Wnvo Vocfor IqunfIons
!oforrIng fo IIg. 5, fho forwnrd wnvo vocfor , for fho i-/l frnnsIfIon oIomonf cnn bo
oxrossod ns dIrocfod IIno sogmonf $%:
|l2]
nnd fho vocfor %Is oxrossod: |l3]
!onrrnngIng oqunfIon |l3]: |l4]
SubsfIfufIng sfnndnrd form: |l5]
WhIch roducos fo: |l6]
(2) !ovorso Wnvo Vocfor IqunfIons
Tho rovorso wnvo vocfor , for fho i-/l frnnsIfIon oIomonf cnn bo oxrossod ns dIrocfod
IIno sogmonf &':
|l?]
nnd fho vocfor 'Is oxrossod: |l8]
!onrrnngIng oqunfIon |l8]: |l9]
SubsfIfufIng sfnndnrd form: |20]
WhIch roducos fo: |2l]
WIfh fho dorIvnfIon of oqunfIon |l6] nnd |2l], wo hnvo now dofInod fho gonornIIzod
forwnrd nnd rovorso wnvo vocfors mnfhomnfIcnIIy. Thoso vocfor dofInIfIons wIII bo usod
15
fo formuInfo fho vocfor nrnmofrIc oqunfIons of fho gonornIIzod IIno sogmonf, fho bnsIs
for fho Sinple Bi-Jirec/ionol Wote nnd fho wnvo oxnnsIons fhnf foIIow.
(3) Tho !Inonr II-dIrocfIonnI Wnvo
Tho Sinple Bi-Jirec/ionol Wote Is nII fhnf wo hnvo fhus fnr dofInod or doscrIbod; buf fhIs
wnvo forms fho bnsIs for fho Lineor, Trigrono/ic, nnd Hexogrono/ic wnvos whIch nro nII
roducfs of fho oxnnsIon of fhIs bnsIc buIIdIng bIock. Tho fIrsf sfo In fho rocoss of
wnvo oxnnsIon nnd combInnfIon fhnf ovonfunIIy Ionds fo fho 384 numbor dnfn sof Is fho
gonornfIon of fho Lineor Bi-Jirec/ionol Wote. ThIs wnvo Is roducod from fho Sinple Bi-
Jirec/ionol Wote (SBW) by sImIo concnfonnfIon I.o. InsorfIng fIvo coIos of fho SBW
ond-fo-ond wIfh fho orIgInnI, nnd roducIng sIx SBW cycIos. AccordIng fo McKonnn, fho
Lineor Wote Is nn oxrossIon of fho sIx IInos fhnf dofIno onch I-ChIng hoxngrnm - Tho
SBW fhon rorosonfs ono IIno of fho hoxngrnm, nnd fhoro nro sIx SBW connocfod ond-fo-
ond fo form fho Lineor Bi-Jirec/ionol Wote (LBW). IIg. 6 Is n grnh of fhIs oxnndod
SBW, or Lineor Bi-Jirec/ionol Wote (LBW), nnd shows fho concnfonnfIon rocoss fhnf
oxnnds fho SBW (64 vnIuos, oxcIudIng zoro) Info fho LBW (384 vnIuos, oxcIudIng zoro).
AIfhough fhIs grnh doos nof show fho vocfor sfrucfuro of IIg. 4 (fo nvoId crowdIng fho
grnh), If Is ImIIod horo. Tho LBW fhoroforo oxrossos fho snmo rocoss of dIrocfod fIow
ns doos fho SBW, n counfor-cIockwIso fIow of somo oInf onfIfy nIong fho nfh frncod by
fho forwnrd nnd rovorso wnvos In IIg. 6.
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Tho concnfonnfIon rocoss fhnf roducos fho LBW cnn bo oxrossod mnfhomnfIcnIIy ns
foIIows:
Ior: |22]
16
nnd for: |23]
Whoro: (ronouncod IIn of i) Is fho vnIuo of fho forwnrd or rovorso IInonr wnvo nf
frnnsIfIon oInf i or nf ; nnd Is fho vnIuo of fho forwnrd or rovorso
IInonr wnvo nf , whoro Is fho romnIndor whon i Is dIvIdod by 64.
Tho Lineor Bi-Jirec/ionol Wote (LBW) wIII now bo oxrossod mnfhomnfIcnIIy, nnd
oxnndod Info fho Trigrono/ic Bi-Jirec/ionol Wote (TBW), nnd Hexogrono/ic Bi-
Jirec/ionol Wote (HBW), usIng mnfhomnfIcs dorIvod from fho vocfor nrnmofrIc oqunfIon
of fho sfrnIghf IIno.
9HFWRU3DUDPHWULF(TXDWLRQVRIWKH)RUZDUGDQG5HYHUVH/LQH6HJPHQWV
(l) Tho !Inonr Iorwnrd Wnvo Vocfor
Tho porone/er Is Infroducod by dofInIng fho sfrnIghf IIno In forms of n oInf , n
vocfor , nnd n nrnmofor /. !ofor fo IIg. 5 nnd Iocnfo fho vocfors $$%nnd3. Wo
hnvo nIrondy dofInod vocfor $%In oqunfIon |l2] ns fho forwnrd wnvo vocfor (Iqn
|l6]), whIch osfnbIIshos n dIrocfIon for our IIno sogmonf. Vocfor $ Is In sfnndnrd
osIfIon (fnII osIfIonod nf fho coordInnfo sysfom orIgIn), so fhnf If Is dofInod by fho
osIfIon of Ifs hond nf oInf . Vocfor 3 Is fho vnrInbIo or movIng vocfor nnd,
IIko fho movIng oncII oInf, Ifs hond frncos fho nfh of fho sfrnIghf IIno fhnf wo nro
Inforosfod In. !of us ronnmo vocfor 3 ns fho IInonr forwnrd wnvo vocfor , nnd sInco If
Is In sfnndnrd osIfIon (fnII nf fho orIgIn so fhnf If Is dofInod by fho coordInnfos of Ifs
hond), If Is doscrIbod mnfhomnfIcnIIy by fho foIIowIng oxrossIon:
|24]
In whIch nro fho vnrInbIo coordInnfos of fho vocfor hond. ThIs vocfor cnn nIso bo
oxrossod ns fho sum of vocfors $nnd fho forwnrd wnvo vocfor ns foIIows:
|25]
Vocfor $In sfnndnrd osIfIon Is oxrossod ns
nnd from oqunfIon |l6]
17
so fhnf oqunfIon |25] cnn now bo oxrossod ns:
|26]
wIfh fho nrnmofor W hnvIng n rnngo: ovor fho x domnIn
IqunfIon |26] cnn now bo soIvod for x nnd yI, fho gonornI coordInnfos of fho vocfor , fo
doformIno fho porone/ric equo/ion of /le line doscrIbIng fho mofIon of fho forwnrd wnvo.
SoIvIng for x nnd y
I
yIoIds fho foIIowIng porone/ric equo/ione of fho IIno:
|2?]
|28]
SoIvIng |2?] nnd |28] for fho nrnmofor f wo gof:
|29]
Thoso oxrossIons for fho x nnd y vnrInbIos In oqunfIon |29], nro fho e/onJorJ forn of /le
line equo/ion, nnd show fhnf fho porone/er f bohnvos ns nn in/erpolo/ion opero/or for fho
x nnd y coordInnfos of fho forwnrd wnvo IIno sogmonf. !onrrnngIng forms for fho
vnrInbIos In oqunfIon |29] Ionds fo fho elope ,-in/ercep/ forn of fho sfrnIghf-IIno oqunfIon,
whIch Is n convonIonf form of oxrossIon for fho IIno sogmonf of Inforosf In fhIs
dovoIomonf. Tho elope ,-in/ercep/ form of fho IIno Is doformInod by soIvIng |29] for fho
vnrInbIo , whIch rosuIfs In fho oxrossIon:
|30]
ofIno: nnd , so fhnf |29] bocomos:
>@
whIch Is fho elope ,-in/ercep/ form of fho forwnrd IInonr wnvo IIno sogmonf, whoro fho
sIoo Is , nnd fho Inforcof Is . IqunfIon |3l] Is fho
vocfor-dorIvod oxrossIon fhnf Is usod fo gonornfo fho IInonr forwnrd wnvo ovor fho
domnIn . ThIs forwnrd wnvo vocfor gonornfIon rocoss Is now roonfod for
fho rovorso wnvo vocfor.
18
(2) Tho !Inonr !ovorso Wnvo Vocfor
Tho rocoss for gonornfIng fho tec/or porone/ric equo/ione for fho rovorso wnvo sogmonf
Is fho snmo ns for fho forwnrd sogmonf, buf wIfh n vocfor fhnf hns fho oosIfo sonso
(oosIfo fIow) of fho forwnrd wnvo vocfor. AgnIn, rofor fo IIg. 5 nnd fInd vocfors &, &',
nnd 4Vocfor &' hns nIrondy boon dofInod In oqunfIon |l?] ns fho rovorso wnvo vocfor,
. Vocfor 4, IIko vocfor 3, Is fho movIng vnrInbIo vocfor (fnII Is fIxod, buf hond
movos nnd frncos fho IIno of Inforosf) whIch wIII frnco fho nfh of our rovorso wnvo IIno
sogmonf. Wo now ronnmo 4 ns fho rovorso wnvo-gonornfIng vocfor nnd sInco If Is In
sfnndnrd osIfIon If cnn bo oxrossod ns:
|32]
In whIch nro fho vnrInbIo coordInnfos for fho hond of . IxrossIng ns fho
sum of & nnd fho nrnmofor scnIod , wo hnvo:
|33]
SubsfIfufIng for & nnd wo hnvo:
|34]
SoIvIng for x nnd y
!
yIoIds fho foIIowIng porone/ric equo/ione of fho IIno:
|35]
|36]
SoIvIng for fho nrnmofor f wo gof:
|3?]
fhon soIvIng for y! gIvos us fho elope ,-in/ercep/ form of fho IInonr rovorso IIno sogmonf:
|38]
19
ofIno: nnd , so fhnf |38] Is oxrossod In
fho sIoo ,-in/ercep/ forn of fho IInonr rovorso wnvo IIno sogmonf. ofIco nIso fhnf
x x
r(i) f(i)
= , nn IdonfIfy fhnf wIII bo oxIoIfod Infor. Ior fho elope ,-in/ercep/ forn
of |38] wo subsfIfufo fho doIfn ( ) oxrossIons nnd coIIocf forms:
>@
IqunfIons |3l] nnd |39] consfIfufo fho Jefining expreeeione for fho IInonr forwnrd nnd
rovorso wnvos rosocfIvoIy, nnd oqunfIon |ll] rovIdos fho corrocf vnIuo for fho subscrIf
i In oqunfIon |39]. Thoso oqunfIons cnn bo oIfhor oxnndod Info fho frIgrnmnfIc nnd
hoxngrnmnfIc bI-dIrocfIonnI wnvos (TBW nnd HBW) dIrocfIy, whIch nro fhon combInod fo
form comIox wnvos; or fhoy cnn bo fIrsf combInod Info n IInonr comIox wnvo, fhon
oxnndod Info fho frIgrnmnfIc nnd hoxngrnmnfIc comIox wnvos. IIfhor of fhoso fwo
rocoduros wIII Iond fo fho snmo fInnI 384 numbor dnfn sof, buf fho Inffor Is n moro
sfronmIInod rocoss fhnf oIImInnfos sovornI oornfIonnI sfos. Tho comIox wnvo Is
dofInod horo, ns nny wnvo fhnf Is n IInonr combInnfIon of ono or moro bI-dIrocfIonnI
wnvos, nnd Is nof oxrossod In bI-dIrocfIonnI form. So now Iof us confInuo wIfh fho
rocoss of wnvo combInnfIon, bogInnIng wIfh fho IInonr bI-dIrocfIonnI (forwnrd nnd
rovorso) wnvo.
(3) Tho !Inonr ComIox Wnvo
Ioforo bogInnIng fho mnfhomnfIcnI dovoIomonf of fho IInonr comIox wnvo, Iof us fIrsf
osfnbIIsh fho rocoduro for forwnrd nnd rovorso wnvo combInnfIon.
ofInIfIon 4:
In ordor fo roduco forwnrd nnd rovorso wnvo ondoInf (nodo) cIosuro nf zoro (0) vnIuo,
fho forwnrd nnd rovorso wnvos musf bo subfrncfod from ono nnofhor fo yIoId zoro vnIuod
ondoInfs for fho combInod sImIo wnvo. In ordor fo mnxImIzo fho numbor of osIfIvo
vnIuos for fho rosuIfnnf combInod wnvo, fho forwnrd wnvo Is subfrncfod from fho rovorso
wnvo.
Tho IInonr comIox wnvo Is fhoroforo roducod by subfrncfIng oqunfIon |3l] (fho forwnrd
wnvo IIno sogmonf), from oqunfIon |39] (fho rovorso wnvo IIno sogmonf). Tho combInod
or comIox IInonr wnvo Is fhoroby oxrossod ns:
|40]
!oIncIng fho oxrossIon wIfh fho rIghf hnnd sIdos of oqunfIon |39] nnd
|3l], wo gof:
20
y
C1
r(i)
r(i)
r(i 1)
r(i)
r(i)
(i 1)
f(i)
f(i)
f(i)
f(i)
f(i)
i
(x)
y
x
x y
y
x
x
y
x
x y
y
x
x = +

"
$
#
#

"
$
#
#
%
&
K
'
K
(
)
K
*
K

"
$
#
#

"
$
#
#
%
&
K
'
K
(
)
K
*
K
+ +

|4l]
nnd combInIng IIko forms nnd ronrrnngIng oqunfIon |4l] gIvos us:
y x
C1
r(i)
r(i)
f(i)
f(i)
r(i 1) f(i)
r(i)
r(i)
(i 1)
f(i)
f(i)
i
(x)
y
x
y
x
x y y
y
x
y
x
x = + +

"
$
#
#
%
&
K
'
K
(
)
K
*
K

"
$
#
#

"
$
#
#
%
&
K
'
K
(
)
K
*
K
+ +

|42]
!sIng fho IdonfIfy shown rovIousIy, x x
r(i) f(i)
= , oqunfIon |42] Is roducod fo fho
Jefining equo/ion for fho IInonr comIox wnvo:
y
C1
(x)
y y
x
(x) y y
y x y x
x
r(i) f(i)
f(i)
r(i 1) f(i)
r(i) (i 1) f(i) i
f(i)
=
+
+ +
+

"
$
#
#
%
&
K
'
K
(
)
K
*
K

"
$
#

"
$
#
#
%
&
K
'
K
(
)
K
*
K
+
+

>@
Whoro y
Cl
(x) Is fho lineor conplex uote func/ion nnd fho doIfn () funcfIons dofInod ns:
y (y y )
r(i) r(i) r(i 1)
=
+
,
WhIch Is fho chnngo In fho IInonr rovorso wnvo doondonf vnrInbIo ,r ovor fho x domnIn
of x x x
i (i 1)

+
y (y y )
f(i) f(i 1) f(i)
=
+
,
WhIch Is fho chnngo In fho IInonr forwnrd wnvo doondonf vnrInbIo ,f ovor fho x domnIn
of x x x
i (i 1)

+
x (x x )
f(i) (i 1) i
=
+
,
WhIch Is fho chnngo In fho Indoondonf vnrInbIo x ovor fho domnIn of fho IInonr comIox
wnvo IIno sogmonf, x x x
i (i 1)

+
i x = int( )
As dofInod In oqunfIon |ll]
SubsfIfufIng fho domnIn ondoInfs, x
I
nnd x
(I+l)
for fho x-vnrInbIo, oqunfIon |43] roducos
fo:
@x x
i
= y
C1
(x) y y
r(i) f(i)
= |44]
@x x
(i 1)
=
+
y
C1
(x) y y
r(i 1) f(i 1)
=
+ +
|45]
21
WhIch confIrms whnf wo obsorvo nf fho IInonr wnvo IIno sogmonf ondoInfs, nnd
vnIIdnfos oqunfIon |43]
([SDQVLRQRIWKH/LQHDU&RPSOH[:DYH/&:
AccordIng fo McKonnn
l0
fho TrIgrnmnfIc wnvo Is nn oxrossIon of fho frIgrnm nIr fhnf
form onch I-ChIng hoxngrnm. SInco onch hoxngrnm hns n nIr of frIgrnms, n Trigrono/ic
Wote nIr Is consfrucfod such fhnf fho fwo frIgrnmnfIc wnvos nro Incod ond-fo-ond
(concnfonnfod), nnd hnvo fho domnIn (x-nxIs rnngo) of sIx sImIo wnvo cycIos. Tho
Trigrono/ic wnvo Is nIso vIowod ns hnvIng n vnIuo of fhroo fImos fho IInonr wnvo, sInco n
frIgrnm consIsfs of fhroo IInos (frIgrnm = 3 x l IInos). SImIInrIy, fho Hexogrono/ic wnvo
Is vIowod ns nn oxrossIon of fho unIfy of onch hoxngrnm, nnd Is consfrucfod so fhnf n
sIngIo hoxngrnmnfIc wnvo occuIos fho domnIn of sIx sImIo wnvos cycIos (sIx IInos fo n
hoxngrnm), or fwo frIgrnmnfIc wnvo cycIos (fwo frIgrnms fo n hoxngrnm). AddIfIonnIIy,
sInco fho hoxngrnm confnIns sIx IInos, fho hoxngrnmnfIc wnvo Is soon ns sIx fImos ns
Inrgo ns fho sImIo wnvo (hoxngrnm = 6 x l IInos, nnd hoxngrnmnfIc wnvo = 6 x l sImIo
wnvos).
ThIs Tri-Letel Conplex Wote Is doscrIbod ns hnvIng fho snmo fhroo nosfod IovoIs of
oxrossIon ns oxhIbIfod by nn I-ChIng hoxngrnm. Tho fo IovoI Is fho Hexogrono/ic
Wote, or hoxngrnm ns n whoIo, whIch confnIns fho fwo Iowor IovoIs, fwo Trigrono/ic
Wotee nnd sIx Sinple Wote cycIos. Tho mId IovoI of oxrossIon Is fho Trigrono/ic Wote,
whIch confnIns fho sIx Sinple Wote cycIos boIow nnd Is confnInod by fho ono
hoxngrnmnfIc cycIo nbovo. Tho boffom IovoI of oxrossIon Is fho lineor uote, hnvIng sIx
sImIo wnvo cycIos fhnf nro confnInod wIfhIn fwo frIgrnmnfIc wnvo cycIos nnd ono
hoxngrnmnfIc wnvo cycIo.
If wo woro fo Iook nf fho comIox wnvo ns nnnIogous fo somo hysIcnI wnvo, bo If
oIocfromngnofIc or ncousfIc, fhon fhIs frI-IovoI wnvo sfrucfuro couId bo vIowod ns
hnrmonIc In nnfuro. Tho hoxngrnmnfIc wnvo wouId fhon corrosond fo fho wnvo
fundnmonfnI or l
sf
hnrmonIc, fho frIgrnmnfIc wnvo wouId corrosond fo fho socond
hnrmonIc (2x fho fundnmonfnI froquoncy), nnd fho IInonr wnvo wouId corrosond fo fho
sIxfh hnrmonIc (6x fho fundnmonfnI froquoncy). In fho cnso of fho Tri-Letel Conplex
Wote, howovor, fho hnrmonIc wnvos nro nof onIy froquoncy muIfIIos, fhoy nro nIso
nmIIfudo muIfIIos of fho fundnmonfnI, or hoxngrnmnfIc wnvo. AIfhough fhIs nofIon of
wnvo hnrmonIcs mny onIy bo nn InforosfIng orsocfIvo nf fhIs oInf, If mny bo usofuI
whon oxnmInIng fho wnvo fonfuros of fhoso numbor sofs usIng IourIor nnnIysIs.
7KH([SDQVLRQ3URFHVV([SUHVVHG*UDSKLFDOO\
CrnhIcnIIy sonkIng fho Trigrono/ic Conplex Wote Is sImIy n 3x mngnIfIcnfIon of fho
IInonr comIox wnvo, I.o. fho mngnIfIcnfIon of fho fIrsf fwo of Ifs sImIo wnvo cycIos. ThIs
3x mngnIfIcnfIon monns fhnf fho IInonr comIox wnvo sogmonfs nro oxnndod by n fncfor
of fhroo, In bofh fho x nnd , dIrocfIons. SImIInrIy, fho Hexogrono/ic Conplex Wote Is n
6x mngnIfIcnfIon of fho IInonr comIox wnvo, I.o. fho mngnIfIcnfIon of fho fIrsf of fho
22
sImIo wnvo cycIos. IIguro ? Is n grnhIcnI rorosonfnfIon of fhIs rocoss, nnd shows fho
3x nnd 6x mngnIfIcnfIon ovor fho fIrsf 64 frnnsIfIon Indox vnIuos (64 of 384). Tho grnh
shows ono comIofo lineor cycIo, ono-fhIrd of n /rigrono/ic wnvo cycIo, nnd ono-sIxfh of n
lexogrono/ic wnvo cycIo.
Ono sIgnIfIcnnf fonfuro fo nofIco In IIg. ?, Is fhnf /le lineor, /rigrono/ic, nnd
lexogrono/ic wnvos nro offsof from ono nnofhor fho fIrsf onk of onch wnvo IovoI Is nof
nIIgnod wIfh Ifs noIghbor. ofIco nIso, fhnf fhIs fIrsf onk nf onch wnvo IovoI (IInonr, frI,
nnd hox) occurs nf fho rImnry cIosuro oInf (I.o. fho fIrsf cIosuro oInf ns obsorvod from
wIfhIn fho onvoIoo of fho IInonr, frI, nnd hox bI-dIrocfIonnI wnvos). Ior fho lineor wnvo
fhIs cIosuro occurs nf Indox l, for fho /rigrono/ic wnvo If Is nf Indox 3, nnd for fho
lexogrono/ic wnvo If occurs nf Indox 6. ThIs Is oxncfIy fho dofInIng l-3-6 rnfIo for
lineor, /rigrono/ic, nnd lexogrono/ic wnvos, whIch fhIs grnh IIIusfrnfos woII.
Anofhor fonfuro fo nofIco nbouf IIg. ? Is fhnf fhIs wnvo offsof Is duo fo fho fncf fhnf
lineor wnvo sogmonf l Is IncIudod In fho lineor wnvo numbor sof. !omombor fhnf fhIs
fIrsf sogmonf (from Indox 0 fo Indox l) Is n rosuIf of fho wrnIng fonfuro of fho sImIo
)LJXUH
II-dIrocfIonnI wnvo frnnsIfIon 64 Is wrnod (coIod) fo frnnsIfIon zoro for fho sImIo
wnvo, or frnnsIfIon 384 Is wrnod fo zoro for fho onfIro IInonr bI-dIrocfIonnI wnvo.
Thoroforo, frnnsIfIon numbor 0 Is nof fho sfnrfIng oInf of fho wnvo, buf frnnsIfIon l Is.
onofhoIoss, Iof us mnfhomnfIcnIIy oxross fho IInonr forwnrd wnvo oxnnsIon, shown In
IIg. ?, ns foIIows:
|46]
23
Or by ronrrnngIng forms In |46]:
|4?]
!IkowIso for fho IInonr rovorso wnvo:
|48]
nnd ronrrnngIng: |49]
ThIs snmo sof of oqunfIons, |46] fhrough |49] cnn bo usod fo oxnnd fho IInonr wnvo Info
fho hoxngrnmnfIc wnvo shown In IIg. ?, by roIncIng nII numbor 3`s by 6`s. Howovor,
sInco fho ncfunI sfnrfIng oInf for fhIs wnvo sof Is nf frnnsIfIon l nnd nof frnnsIfIon 0,
fho roor oxnnsIon wIII Iook ns shown In IIg. 8. In fhIs fIguro, nIIgnmonf bofwoon
IInonr, frIgrnmnfIc, nnd hoxngrnmnfIc wnvos occurs nf frnnsIfIon Indox l; nIso n oInf of
rImnry cIosuro.
)LJXUH
IIg. 8 nIso shows fhIs oxnnsIon In forms of fho lineor, /ri, nnd lex bI-dIrocfIonnI wnvos,
In whIch fho IInonr bI-dIrocfIonnI wnvo Is oxnndod Info fho /rigrono/ic, nnd
lexogrono/ic bI-dIrocfIonnI wnvos. Howovor, fhoso bI-dIrocfIonnI wnvos nro ovonfunIIy
combInod fo form fho comIox wnvo sysfom, ns doscrIbod by oqunfIon |43] for fho IInonr
wnvo cnso. SInco nII fhroo bI-dIrocfIonnI wnvos nro fo bo oxrossod ns comIox wnvos,
24
sovornI oornfIonnI sfos cnn bo omIffod nnd fho rocoss sfronmIInod, by oxnndIng fho
IInonr comIox wnvo dIrocfIy. ConsoquonfIy, wo wIII foIIow n mnfhomnfIcnI rocoss fhnf
oxnnds fho IInonr comIox wnvo, doscrIbod by oqunfIon |43], Info fho frI nnd hox
comIox wnvos. In fho Inforosf of mnInfnInIng vIsunI cInrIfy of fhIs rocoss, howovor,
nnd of romnInIng fruo fo fho nofIon of n dIrocfod fIowIng wnvo cycIo nf nII fhroo IovoIs of
oxrossIon, wo show fho oxnndod wnvo sysfom ns bI-dIrocfIonnI In nnfuro.
IIg. 9 shows fho roor oxnnsIon of fho IInonr bI-dIrocfIonnI wnvo Info fho frIgrnmnfIc
nnd hoxngrnmnfIc bI-dIrocfIonnI wnvos, wIfh wnvo IndoxIng nf frnnsIfIon l. ThIs grnh
shows fho onfIro 384 numbor wnvo domnIn, In whIch n sIngIo lexogrono/ic wnvo cycIo
confnIns fwo /rigrono/ic wnvo cycIos nnd sIx IInonr wnvo cycIos. ThIs nofIon of nII fhroo
IovoIs of wnvo oxrossIon boIng confnInod, or nosfod In ono IovoI Is fho ncfunI fhoorofIcnI
bnsIs for fho /ri-letel uote combInnfIon fhnf roducos n sIngIo Tri-Letel Conplex Wote
/le Jo/o ee/.
)LJXUH
IIg. l0 shows fho snmo frI-IovoI bI-dIrocfIonnI wnvo sof ns IIg. 9, buf wIfh fho nvorngo
vnIuo of fho IInonr nnd /rigrono/ic bI-dIrocfIonnI wnvos nIIgnod wIfh fho nvorngo vnIuo
of fho lexogrono/ic wnvo. Thoso fwo fIguros nro mnfhomnfIcnIIy oquIvnIonf for fhIs
dovoIomonf, ns wo shnII soo.
IIg. l0 Is IncIudod horo bocnuso If Is fho form of fho Tri-Letel Bi-Jirec/ionol Wote fhnf
nonrs In fho TWZ documonfnfIon, nnd If Is obvIous fhnf If Iooks dIfforonf fhnn IIg. 9.
Tho IInonr nnd frIgrnmnfIc bI-dIrocfIonnI wnvos In IIg. l0 hnvo fhoIr nvorngo vnIuos
nIIgnod fo fho hoxngrnmnfIc wnvo nvorngo vnIuo, so fhnf fhoy movo nbouf n common IIno
fho hoxngrnmnfIc nvorngo. ThIs grnh mny Iook dIfforonf fhnn IIg. 9, buf fho fncf Is
fhnf fhoy nro IdonfIcnI mnfhomnfIcnIIy. Tho ronson Is fhnf, In ordor fo roduco fho
25
combInod comIox wnvo, fho forwnrd wnvo Is subfrncfod from fho rovorso wnvo, ns
shown In oqunfIons |40] fhrough |43]. SInco fho forwnrd nnd rovorso wnvos romnIn
)LJXUH
cIosod, or connocfod nf fhoIr ondoInfs, If doosn`f mnffor whoro nIong fho y-nxIs fhoy nro
shIffod fho rosuIfIng dIfforonco Is fho snmo. ConsoquonfIy, IIg. 9 Is oquIvnIonf fo IIg.
l0, nnd fho grnhs nonrIng In fho TWZ documonfnfIon.
So Iof us now bogIn wIfh fho mnfhomnfIcnI oxnnsIon of fho Lineor Conplex Wote, of
oqunfIon |43], Info fho Trigrono/ic nnd Hexogrono/ic Conplex Wotee, nnd fhon fInnIIy
Info fho Tri-Letel Conplex Wote fho 384 numbor Jo/o ee/.
7KH0DWKHPDWLFVRIWKH7ULJUDPDWLFDQG+H[DJUDPDWLF&RPSOH[:DYHV
(l) Tho TrIgrnmnfIc ComIox Wnvo
CIoso InsocfIon of IIgs 8 nnd 9 rovonIs fho roInfIonshI bofwoon fho funcfIons
doscrIbIng fho IInonr, frIgrnmnfIc, nnd hoxngrnmnfIc wnvos. Tho roInfIonshI bofwoon
IInonr nnd frIgrnmnfIc forwnrd wnvos Is shown In IIgs 8 nnd 9, nnd Is oxrossod
mnfhomnfIcnIIy ns foIIows:
y y
F3 F1
(3x 2) 3 (x) = |50]
Whoro fho qunnfIfy wIfhIn fho nronfhosos Is fho nrgumonf for y( ), nnd nof n
muIfIIIor. !onrrnngIng forms In |50] wo gof:
26
y y
F3 F1
(x) 3 ( )
x 2
3
=
+
|5l]
Whoro y (x)
F3
Is fho vnIuo of fho /rigrono/ic foruorJ uote nf x, nnd 3
2
3
y
F1
( )
x+
Is fhroo
fImos fho vnIuo of fho lineor foruorJ uote nf ( )
x+ 2
3
. !IkowIso, fho /rigrono/ic reteree
uote Is oxrossod In forms of fho IInonr rovorso wnvo by fho oqunfIon:
y (3x 2) 3y (x)
R3 R1
= |52]
nnd by ronrrnngIng forms wo gof:
y y
R3 R1
(x) 3
x 2
3
( ) =
+
|53]
Tho frIgrnmnfIc comIox wnvo Is dofInod In fho snmo mnnnor ns fho IInonr comIox
wnvo, wIfh fho frI forwnrd wnvo subfrncfod from fho frI rovorso wnvo ns In oqunfIon |40]
nnd oxrossod In frIgrnmnfIc forms by:
y (x) y (x) y (x)
C3 R3 F3
= |54]
!sIng |50], |52], nnd |54] wo cnn oxross fho frIgrnmnfIc comIox wnvo In forms of fho
IInonr comIox wnvo ns foIIows:
y y y y
R3 F3 R1 F1
(3x 2) (3x 2) 3 (x) 3 (x) = |55]
or oquIvnIonfIy: y y y
C3 R1 F1
( ) 3 3 (x) (x) 3 2 x = |56]
IncforIng fho rIghf sIdo of |56} gIvos: y {y y
C3 R1 F1
( ) 3 (x) (x)} 3 2 x = |5?]
SubsfIfufIng fho oxrossIon for fho IInonr comIox wnvo on fho rIghf hnnd sIdo of
oqunfIon |40], Info |5?]:
y
C3
(3x 2) lin(x) 3[ ] = |58]
fhon ronrrnngIng |58] wo gof: y x
C3
( ) lin
x
3[ ] ( )
2
3
=
+
|59]
IqunfIon |59] shows fhnf fho vnIuo of fho /rigrono/ic conplex uote nf x, Is oqunI fo
fhroo fImos fho vnIuo of fho lineor conplex uote nf x 2 3 + 1 6 . !oIncIng fho x-form
form In fho lin(x) oxrossIon of oqunfIon |43] wIfh fho oxrossIon x+ 2 3 1 6 , fhon
subsfIfufIng Info |59] gIvos us fho dofInIng oqunfIon of fho /rigrono/ic conplex uote ns
foIIows:
27
y (x) y y
C3
3
y y
x
y x y x
x
r(i) f(i)
f(i)
r(i) (i 1) f(i) i
f(i)
x 2
3
( )
r(i 1) f(i)
= + +
+ +

"
$
#
#

"
$
#

"
$
#
#
%
&
K
'
K
(
)
K
*
K

+
+
+

>@
In fhIs oxrossIon fho subscrIf i Is oxrossod ns n funcfIon of x, usIng fho rocoss
sImIInr fo fhnf whIch roducod oqunfIon |ll]. In fhIs cnso, sInco fho x form hns bocomo
x+ 2 3 1 6 , fho oxrossIon dofInIng fho boundIng subscrIf i fhon bocomos:
i x j = + = int 2 3 1 6 < A
|6l]
Whoro subscrIf i Is ronnmod ns j fo dIsfInguIsh If from fho IInonr wnvo oxrossIon
subscrIf shown In fho rovIous IInonr wnvo oqunfIons. IqunfIon |6l] dofInos subscrIf j
ns fho roundod down Infogor vnIuo of fho funcfIon x + 2 3 1 6 , fhus osfnbIIshIng fho
boundnry condIfIons for fho frIgrnmnfIc IIno sogmonf mnod by fhIs funcfIon.
IqunfIon |60] oxrossos fho Trigrono/ic Conplex Wote (TCW) ns nn oxnnsIon of fho
Lineor Conplex Wote (LCW) dIrocfIy. Howovor, fho snmo rosuIf wouId bo obfnInod If fho
IInonr forwnrd nnd rovorso wnvos hnd boon oxnndod Info fho frIgrnmnfIc forwnrd nnd
rovorso wnvos, nnd fhoso rosuIfs combInod fo for fho /rigrono/ic conplex uote. ThIs
dIrocf nronch cIonrIy oIImInnfos fwo vory dofnIIod mnfhomnfIcnI sfos. Tho snmo
sorIos of sfos wIII now bo usod fo fInd fho oxrossIon for fho lexogrono/ic conplex
uote.
(2) Tho HoxngrnmnfIc ComIox Wnvo
As for fho frIgrnmnfIc wnvo, InsocfIon of IIgs. 8 nnd 9 rovonIs fhnf fho roInfIonshI
bofwoon fho IInonr forwnrd wnvo nnd fho hoxngrnmnfIc forwnrd wnvo cnn bo oxrossod:
y y
F F1
( x ) (x)
6
6 5 6 = |62]
!onrrnngIng forms for fhIs funcfIon wo gof:
y (x) y
F F1
( )
x
6
6
5
6
=
+
|63]
Whoro y (x)
F6
Is fho vnIuo of fho hoxngrnmnfIc forwnrd wnvo nf x, nnd 6 5 6 y
F1
x+ 1 6 < A
Is
sIx fImos fho vnIuo of fho IInonr forwnrd wnvo nf x+ 5 6 1 6 < A
. !IkowIso, fho hoxngrnmnfIc
rovorso wnvo Is oxrossod In forms of fho IInonr rovorso wnvo by fho oqunfIon:
y y
R R1
( x ) (x)
6
6 5 6 = |64]
nnd by ronrrnngIng forms wo gof:
y y
R R1
(x)
x
( )
6
6
5
6
=
+
|65]
28
Tho lexogrono/ic conplex uote Is dofInod In fho snmo mnnnor ns fho lineor nnd
/rigrono/ic conplex uotee, wIfh fho hox forwnrd wnvo subfrncfod from fho hox rovorso
wnvo ns In oqunfIon |40] nnd |54], nnd oxrossod In hoxngrnmnfIc forms by:
y y y
C R F
(x) (x) (x)
6 6 6
= |66]
!sIng |62], |64], nnd |66] wo cnn oxross fho lexogrono/ic conplex uote In forms of fho
lineor conplex uote ns foIIows:
y y y y
R F R1 F1
( x ) ( x ) (x) (x)
6 6
6 5 6 5 6 6 = |6?]
or oquIvnIonfIy: y y y
C R1 F1
( ) (x) (x)
6
6 5 6 6 x = |68]
IncforIng fho rIghf sIdo of |68] gIvos: y {y y
C R1 F1
( ) (x) (x)}
6
6 5 6 x = |69]
SubsfIfufIng fho oxrossIon for fho IInonr comIox wnvo on fho rIghf hnnd sIdo of
oqunfIon |40], Info |69]:
y { }
C
( x ) lin(x)
6
6 5 6 = |?0]
fhon ronrrnngIng |?0] wo gof: y x
C
( ) lin
x
{ ( )}
6
6
5
6
=
+
|?l]
IqunfIon |?l] shows fhnf fho vnIuo of fho lexogrono/ic conplex uote nf x, Is oqunI fo sIx
fImos fho vnIuo of fho lineor conplex uote nf x+ 5 6 1 6 < A
. !oIncIng fho x-form form In
fho y (x)
C1
oxrossIon of oqunfIon |43] wIfh fho oxrossIon x+ 5 6 1 6 < A
, fhon subsfIfufIng
Info |?l] gIvos us fho dofInIng oqunfIon of fho lexogrono/ic conplex uote ns foIIows:
y (x)
C
y y
x
y x y x
x
r(i) f(i)
f(i)
r(i) (i 1) f(i) i
f(i)
x
( ) y y
r(i 1) f(i) 6
6
5
6
= + +
+ +

"
$
#
#

"
$
#

"
$
#
#
%
&
K
'
K
(
)
K
*
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+
+
+

>@
In fhIs oxrossIon fho subscrIf i Is oxrossod ns n funcfIon of x, usIng fho rocoss
sImIInr fo fhnf whIch roducod oqunfIons |ll] nnd |6l]. In fhIs cnso, sInco fho x form hns
bocomo x + 5 6 1 6 , fho oxrossIon dofInIng fho boundIng subscrIf i fhon bocomos:
i x k = + = int 5 6 1 6 < A
|?3]
Whoro subscrIf i Is ronnmod ns / fo dIsfInguIsh If from fho IInonr nnd frIgrnmnfIc wnvo
oxrossIon subscrIfs shown In fho dofInIng wnvo oqunfIons. IqunfIon |?3] dofInos
subscrIf / ns fho roundod down Infogor vnIuo of fho funcfIon x + 5 6 1 6 , fhus osfnbIIshIng
fho boundnry condIfIons for fho lexogrono/ic IIno sogmonf mnod by fhIs funcfIon.
29
As wIfh fho Trigrono/ic Conplex Wote (TCW) oxrossod In |60], oqunfIon |?2] oxrossos
fho Hexogrono/ic Conplex Wote (HCW) ns nn oxnnsIon of fho Lineor Conplex Wote
(LCW) dIrocfIy. SImIInrIy, fho snmo rosuIf wouId bo obfnInod If fho IInonr forwnrd nnd
rovorso wnvos hnd boon oxnndod Info fho hoxngrnmnfIc forwnrd nnd rovorso wnvos, nnd
fhoso rosuIfs combInod fo form fho lexogrono/ic conplex uote. WIfh fho lineor |43],
/rigrono/ic |60], nnd lexogrono/ic |?2] conplex uotee now dofInod mnfhomnfIcnIIy nnd
oxrossod grnhIcnIIy, wo nro now In n osIfIon fo combIno fhom fo form fho Tri-Letel
Conplex Wote, or 384 numbor Jo/o ee/.
(3) Tho CombInod TrI-!ovoI ComIox Wnvo
ow fhnf fho fhroo IovoIs of TineWote oxrossIon hnvo boon doscrIbod nnd dofInod
mnfhomnfIcnIIy, wo nro now In n osIfIon fo Infogrnfo fhoso fhroo IovoIs Info n sIngIo
unIfnry sysfom of oxrossIon. Tho Tri-Letel Conplex Wote Is soon ns nn Infogrnfod
whoIo, nnd nnnIogous fo fho I-ChIng hoxngrnm fhnf funcfIons ns n hoIIsfIc onfIfy, buf
confnIns fho IndIvIdunI oxrossIon of hoxngrnm, frIgrnm, nnd IIno (yIn or ynng). In ordor
fo osfnbIIsh fhIs frI-IovoI oxrossIon mnfhomnfIcnIIy, wo combIno fho comIox wnvos of
fho IInonr, frIgrnmnfIc, nnd hoxngrnmnfIc IovoIs of oxrossIon. Tho gonornI oqunfIon
oxrossIng fho summnfIon of fho fhroo wnvo IovoIs Is wrIffon ns foIIows:
y (x) lin(x) tri(x) hex(x)
T
= + + |?4]
SubsfIfufIons In |?4] for IIn(x), frI(x), nnd hox(x) from oqunfIon |43], |60], nnd |?2] gIvo
us:
y (x) y (x) y (x) y (x)
T C1 C3 C6
= + + |?5]
nnd furfhor subsfIfufIons from |43], |60], nnd |?2] gIvo us fho dofInIng oxrossIon for fho
Tri-Letel Conplex Wote:
y (x)
T
y y
x
x
y x y x
x
r(i) f(i)
f(i)
r(i) (i 1) f(i) i
f(i)
(y y )
r(i 1) f(i)
=
+ +

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#

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#
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%
&
K
'
K
(
)
K
*
K
+ +
+
+
+

+ + +
+
+
+

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$
#
#
+
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'
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(
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+
+ 3 y y
y y
x
x 2
3
y x y x
x
r(j) f(j)
f(j)
r(j) (j 1) f(j) j
f(j)
( )
r(j 1) f(j)

+ + +
+
+
+

"
$
#
#

"
$
#

"
$
#
#
%
&
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'
K
(
)
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+
+
6 y y
y y
x
x 5
6
y x y x
x
r(k) f(k)
f(k)
r(k) k
r(k 1) f(k)
(k 1) f(k)
f(k)
( )

>@
30
IqunfIon |?6] Is fho dofInIng oqunfIon for fho Tri-Letel Conplex Wote. ThIs oxrossIon
fnkos ono from fho IndIvIdunI oIomonfs of fho lineor conplex uote, u fo fho /rigrono/ic
nnd lexogrono/ic conplex uotee, nnd fInnIIy fo fho /ri-letel conplex uote. ofIco fhnf
fho subscrIfs j for fho frIgrnmnfIc socfIon, nnd / for fho hoxngrnmnfIc socfIon of
oqunfIon |?6] hnvo roIncod fho subscrIf i In oqunfIons |60] nnd |?2], ns fhoy hnvo boon
dofInod In oqunfIons |6l] nnd |?3]. Wo now hnvo n comIofo nnd woII-dofInod funcfIon
for our Tri-Letel Conplex Wote, or Jo/o ee/.
IqunfIon |?6] roducos n frI-IovoI wnvo numbor sof fhnf confnIns somo nognfIvo vnIuos.
Tho 384 numbor Jo/o ee/, on fho ofhor hnnd, Is fho sof of osIfIvo ronI numbors In fho
domnIn 0 384 x . ThIs monns fhnf nrf of fho rnw dnfn sof roducod by oqunfIon
|?6] IIos oufsIdo fho ,-vnIuo domnIn fhnf Is fhoughf fo bo fho roor oxrossIon of fhIs
wnvoform. Ono rocoduro fhnf Is wIdoIy usod for convorfIng nognfIvo vnIuos of somo
nrbIfrnry wnvoform, Info osIfIvo vnIuos, Is fho uso of fho nbsoIufo vnIuo oornfor. If ono
vIows fhIs /ri-letel conplex uote ns somo kInd of InformnfIon cnrryIng sIgnnI, IIko nn
nmIIfudo moduInfod rndIo wnvo, for oxnmIo, fhon n vnIId rocoduro for rocossIng such
n sIgnnI Is fho nIIcnfIon of fho nbsoIufo vnIuo oornfor. In fho rf sIgnnI rocossIng
cnso, fho rocoIvod moduInfod-cnrrIor wnvoform Is nssod fhrough nbsoIufo vnIuo cIrcuIfry
(rocfIfIor) so fhnf fho nognfIvo vnIuos of fho wnvo nro convorfod fo osIfIvo vnIuos. ThIs
ncfunIIy Imrovos fho sIgnnI fo noIso rnfIo of fho cnrrIor onvoIoo, whIch Is fho
InformnfIon cnrryIng moduInfIon sIgnnI. ThIs rocfIfIod sIgnnI Is fhon rocossod by n
dofocfor cIrcuIf fhnf oxfrncfs fho InformnfIon cnrryIng moduInfIon wnvo from fho cnrrIor
wnvo. AIfhough fho frI-IovoI wnvo nnd fho rndIo wnvo nro nof sfrIcfIy nnnIogous, fhoy
nonr sImIInr onough fo mnko n InusIbIo nrgumonf for fho nIIcnfIon of fho nbsoIufo
vnIuo oornfor horo. ThIs oornfIon Is oxrossod ns:
y y
DW T
ABS = |??]
Whoro: y
DW
Is fho Do/o Wote fhnf Is grnhod In IIg. ll, nnd dofInod ns fho nbsoIufo
vnIuo of fho Tri-Letel Conplex Wote ns oxrossod In oqunfIon |?3]. ThIs numbor sof Is
usod ns Inuf dnfn for fho TineWote Zero soffwnro, whIch orforms nn InfInIfo sorIos
oxnnsIon fhnf Moyor cnIIs n froc/ol /roneforn
ll
, fo gonornfo fho TImoWnvo vIowod on
fho comufor scroon.
6WDQGDUGDQG5HYLVHG'DWD6HW&RPSDULVRQV
WIfh oqunfIon |?3] nnd |?4], nnd fho grnh In IIg. ll, wo hnvo comIofod fhIs formnIIzod
dovoIomonf of fho TWZ Jo/o ee/. Wo nro now In n osIfIon fo comnro fhoso rosuIfs
wIfh fhoso of fho sfnndnrd dovoIomonf roorfod by McKonnn nnd Moyor In /le 1ntieiIle
LonJecope nnd fho Tine1xplorer mnnunI, ns woII ns nddross fho Issuos rnIsod by fho
Wo//ine OIjec/ion.
IIg. l2 Is n grnh of bofh fho e/onJorJ nnd retieeJ Jo/o ee/e, nnd If shows somo
romnrknbIo sImIInrIfIos ns woII ns sIgnIfIcnnf dIfforoncos. Ono InforosfIng fonfuro of fhIs
grnh, Is fho nnfuro of onch wnvo nf Ifs rosocfIvo ondoInfs. !ocnII fhnf fho vnIuo of fho
31
wnvo nf x = 0 wIII bo dIscnrdod bocnuso If Is n duIIcnfo or wrn of fho vnIuo nf x = 3S4.
ThIs wIII nof offocf fho roInfIvo vnIuos of fho fwo wnvos nf x = 3S4, bocnuso fhoy nro bofh
zoro-vnIuod nf fhIs ondoInf. Howovor, fho vnIuo of onch wnvo nf x = 1 Is nof fho snmo,
wIfh fho e/onJorJ uote hnvIng n vnIuo of l0 whIIo fho retieeJ uote vnIuo Is zoro.
)LJXUH
Why doos fhIs mnffor, you mny nsk, sInco fhoro nro mnny obvIous dIfforoncos bofwoon
fho fwo wnvos whnf Is fho sIgnIfIcnnco of fhIs dIfforonco Ior fho e/onJorJ uote, If hns
boon nrguod fhnf fho zoro vnIuo nf fho ond of fho wnvoform ImIIos somo kInd of
sInguInrIfy nf fho ond of fho rocoss or nf fho ond of fImo. ThIs retieeJ uote Is
ImIyIng, howovor, fhnf fhoro mny bo sInguInrIfIos nf Io/l onds of fho confInuum. ThIs Is
AIso nn nrgumonf for n cloeeJ sysfom fhnf mny bo undorgoIng somo kInd of cycIIc
ronownI rocoss orhns onch cycIo oxrossIng ovor hIghor ordorod sfnfos of comIox
form, or Aotel/,.
Thoro nro concofs omorgIng from fho fIoId of qunnfum cosmoIogy fhnf mny doscrIbo nn
nnnIogous cycIIc rocoss. ThIs Is fhoory In whIch unIvorsos nro fronfod IIko qunnfum
nrfIcIos fhnf InhnbIf n Inrgor, or hIghor dImonsIonnI domnIn cnIIod n nul/iteree. MIchIo
Knku
l2
, n fhoorofIcnI hysIcIsf nnd co-foundor of sfrIng fIoId fhoory, hns doscrIbod n
rocoss whoro unIvorsos omorgo from fho zoro-oInf, or vncuum fIoId, go fhrough nn
ovoIufIonnry rocoss, fhon orhns rofurn fo fho zoro-oInf fIoId nf fho ond of fho cycIo.
ThIs cycIo mny fhon roonf IfsoIf, ossIbIy wIfh Incronsod comIoxIfy nnd Aotel/,. So
couId fhIs bo sImIInr fo fho rocoss fhnf fho TineWote nnd Aotel/, Tleor, nffomf fo
rovonI Iorhns furfhor InvosfIgnfIon Info fho nnfuro of fho TImoWnvo wIII shod somo
IIghf on fhoso quosfIons.
32
)LJXUH
Anofhor sIgnIfIcnnf fonfuro of IIg. l2 Is fho nnronf ngroomonf of fho fwo wnvos In fho
Iowor froquoncy domnIn. Iroquoncy confonf of nny wnvoform oxrossos IfsoIf ns
vnrInfIons In fho rnfo of chnngo of Ifs vnIuo ns fho wnvo rongnfos In somo ronIm, fhnf
couId bo oIfhor n snco or fImo domnIn, or bofh. So fho sIoo of n wnvoform nf nny gIvon
oInf, or Ifs gonornI shno, cnn rovonI froquoncy confonf (fho mngnIfudo nnd rnfo of
socIfIc undorIyIng rocossos). IxnmInnfIon of fho wnvo nIr In IIg. l2 shows fhnf fhoro
Is n common Iowor froquoncy rocoss occurrIng for onch wnvoform. Tho hIghor froquoncy
rocossos nonr ns roInfIvoIy shorfor durnfIon onks rIdIng uon fho sIowor rocoss.
Tho Iowosf froquoncy rocoss occurrIng In fhoso wnvoforms cnn bo soon by drnwIng nn
ImngInnry IIno bofwoon fho hIghosf of nII fho onks ns ono movos ovor fho domnIn of fho
wnvoforms. SIIghfIy hIghor froquoncy comononfs cnn bo soon by drnwIng fhnf
ImngInnry IIno ovor fho onks nnd vnIIoys uon whIch fho shnrosf nnd shorfosf
durnfIon onks rIdo. Tho grnhs do dIffor In fho hIghor froquoncy domnIn ns cnn bo soon
by fho sfooor sIoos of fho Inrgosf e/onJorJ uote frnnsIfIons. ThIs couId vory woII bo
duo fo hIgh froquoncy noIso rosonf In fho e/onJorJ dnfn sof bocnuso of fho Imboddod
mnfhomnfIcnI orrors.
Tho Iow froquoncy, or Iong durnfIon rocossos, nro fhoso fhnf mny occur on fho scnIo of
mIIIonnIn or ovon bIIIIons of yonrs, whorons fho hIghor froquoncy rocossos mny occur on
fho scnIo of n humnn IIfofImo. CouId If bo fhnf fho Iowosf froquoncy rocoss Is fho
sIgnnfuro of somo cronfIvo rIncIIo nf work, bo If sfrnngo nffrncfor, zoro-oInf fIoId, or
eeclo/on. CouId fhIs cronfIvo onorgy, bo orfurbIng fho fnbrIc of snco-fImo In such n
wny ns fo frIggor fho cronfIon nnd consorvnfIon of hIghor ordorod sfnfos somofhIng IIko
fho grnvIfnfIonnI onorgy of n nssIng nonrby sfnr frIggorIng fho formnfIon of n comofs
from fho Orf cIoud Is fhIs Iowosf froquoncy rocoss fhon n kInd of ground sfnfo, uon
whIch nII hIghor froquoncy rocossos oxross fhomsoIvos Iorhns In fImo fhoso
quosfIons wIII bo nnswornbIo, nIfhough corfnInIy nof fodny.
33
An obvIous fonfuro of IIg. l2 fhnf cIonrIy shows In fhIs grnh, Is fho dIfforonco In fho
nvorngo wnvo vnIuo bofwoon sfnndnrd nnd rovIsod wnvos. Tho nvorngo wnvo vnIuo for
fho sfnndnrd wnvo Is somowhnf gronfor fhnn fho nvorngo vnIuo of fho rovIsod wnvo. ThIs
dIfforonco In nvorngo wnvo vnIuo nonrs fo bo fho rosuIf of dIfforoncos In fho hIghor
froquoncy comononfs of fho wnvo nIr, orhns duo fo noIso In fho sfnndnrd wnvo fhnf
Is roducod by fho mnfhomnfIcnI orrors fhnf nro rosonf. Thoso hIgh froquoncy
comononfs of fho sfnndnrd wnvo show u ns fho sfoo onks fhnf rIso woII nbovo fho
onks In fho rovIsod wnvo. In fho IourIor nnnIysIs fhnf foIIows, fhoso Inrgo onks nonr
ns hIgh froquoncy noIso fhnf ndds rnndomnoss fo fho wnvo. Tho Imncf of fhIs dIfforonco
on fho fInnI TImoWnvo, Is fo shIff fho nvorngo IovoI of novoIfy uwnrd (Iowor vnIuos)
from fhnf oxrossod by fho sfnndnrd wnvo. In ofhor words, fho rovIsod wnvo oxrossos n
rocoss wIfh somowhnf hIghor IovoIs of novoIfy, fhnn doos fho sfnndnrd wnvo. SInco
Aotel/, Isn`f n cnIIbrnfod rocoss, If`s nof ossIbIo fo doformIno whnf fho moro
ronsonnbIo IovoI of Aotel/, wouId bo. AII fhnf cnn bo oxrossod fhon, Is roInfIvo
ovoIfy.
Ono fInnI fonfuro of IIg. l2 fhnf roquIros somo dIscussIon, Is fho corroInfIon numbor nf
fho fo of fho grnh. In ordor fo doformIno nnd qunnfIfy fho dogroo of Infordoondonco,
or Infor-roInfodnoss of fho sfnndnrd nnd rovIsod wnvoforms, n mnfhomnfIcnI oornfIon
cnIIod correlo/ion wns orformod wIfh fhoso fwo numbor sofs. Tho numbor nf fho fo of
fho grnh Is fho rosuIf of fhnf nnnIysIs n vnIuo of 0.564. A corroInfIon of l.0 wouId
monn fhnf fho wnvoforms nro IdonfIcnI, whorons n corroInfIon of zoro wouId IndIcnfo no
funcfIonnI roInfIonshI bofwoon fho fwo. AddIfIonnIIy, n corroInfIon of l wouId IndIcnfo
fhnf fho wnvoforms woro mIrror Imngos of ono nnofhor n onk rofIocfod by n frough ofc.
In fhIs cnso n corroInfIon of 0.564 IndIcnfos fhnf fhoso fwo wnvoforms show n sIgnIfIcnnf
IovoI of Infordoondonco, nIfhough fnr from IdonfIcnI. ThIs IovoI of corroInfIon couId bo
consIdorod IIkoIy for fwo numbor sofs fhnf shnro n common orIgIn, ns woII ns shnrIng
mnny of fho snmo dovoIomonfnI rocoduros.
'DWD:DYHDQG5DQGRP1XPEHU6HW&RPSDULVRQV
Ono mofhod for nssossIng fho InformnfIon cnrryIng ofonfInI of fho Do/o Wote, nnd
convIncIng onosoIf fhnf If Is nof n rnndom rocoss, Is fo comnro If wIfh n dnfn sof fhnf
hns boon rnndomIy gonornfod. SovornI such ronJon uote sofs woro consoquonfIy
roducod fo bo comnrod wIfh fho rovIsod nnd sfnndnrd Do/o Wote numbor sofs dIrocfIy,
nnd fo nIso uso ns Inuf fo fho TWZ soffwnro fo gonornfo rnndom soodod TImoWnvos.
IIg. l3 Is n grnh of fho rovIsod Do/o Wote wIfh n rnndom wnvo sof ovorIny, nnd If
cIonrIy shows fhnf fhoso numbor sofs bonr IIffIo rosombInnco fo ono nnofhor. CorroInfIon
nnnIysIs of fho fwo sofs shows n corroInfIon of 0.03, or ossonfInIIy un-corroInfod ns ono
wouId oxocf for nny rnndom numbor sof. IIg. l3 nIso nonrs fo show fhnf fho rovIsod
Do/o Wote Is n vory dIfforonf fyo of numbor sof from fho rnndom wnvo sof, nnd If
nonrs fo showIng somo kInd of InformnfIon cnrryIng rocoss. Is fhIs In fncf fho cnso, or
doos If jusf nonr fhnf wny
IxnmInnfIon of fho pouer epec/ro for fho Jo/o nnd ronJon uotee shown In IIgs. l2 nnd
l3 cnn rovonI somofhIng nbouf fho nnfuro of fhoso fhroo wnvoforms nnd fhoIr
34
roInfIonshI. Tho convorsIon of fImo, or snco domnIn wnvoforms Info froquoncy domnIn
wnvoforms (froquoncy socfrum or owor socfrum) Is orformod usIng n mnfhomnfIcnI
oornfIon cnIIod n IourIor frnnsform. WIfh fhIs mofhod, n froquoncy socfrum cnn bo
roducod, whIch cnn foII us how much owor Is confnInod In onch of fho froquoncy
comononfs (hnrmonIcs) of n gIvon wnvoform, nnd fhoroby rovIdIng fho froquoncy
dIsfrIbufIon of fho wnvo owor. ThIs dIsfrIbufIon wouId fyIcnIIy bo dIfforonf for
InformnfIon cnrryIng wnvoforms fhnn for rnndom, or noIso sIgnnIs. Tho rnndom, or noIso
sIgnnI socfrum Is fyIcnIIy fInf ovor fho sIgnnI bnndwIdfh, nnd offon dIsfInguIshnbIo
from nn InformnfIon cnrryIng sIgnnI socfrum fhnf oxhIbIfs 1 f (f = froquoncy) bohnvIor.
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IourIor frnnsform oornfIons woro orformod on fho dnfn sofs shown In IIgs. l2 nnd l3,
wIfh fho rosuIfs shown In IIg. l4. Tho fo grnh of IIg. l4 IncIudos Iofs for fho
e/onJorJ nnd retieeJ Do/o Wote owor socfrn, whIIo fho boffom grnh dIsInys fho
IonJon Wote owor socfrum. Tho coIorod IInos drnwn fhrough onch of fho socfrn nro
pouer func/ion curvo-fIfs, fhnf show fho froquoncy roII-off chnrncforIsfIcs of onch wnvo.
ofIco fhnf fho fwo owor socfrn In fho fo grnh oxhIbIf froquoncy roII-off (owor IovoI
docronsos wIfh IncronsIng froquoncy), whorons fho Iowor grnh owor socfrum oxhIbIfs
n fInf froquoncy rosonso (owor IovoI Is froquoncy Indoondonf). ThIs froquoncy roII-off
Is chnrncforIsfIc of InformnfIon cnrryIng sIgnnIs, whorons fho fInf rosonso Is
chnrncforIsfIc of noIso or rnndom sIgnnIs.
Tho retieeJ Jo/o uote socfrum, shown In fho fo grnh In groon, Is oxhIbIfIng fho
nonrIy orfocf 1 f froquoncy rosonso fhnf Is fyIcnI for nn InformnfIon cnrryIng
wnvoform. On fho ofhor hnnd, fho e/onJorJ Jo/o uote owor socfrum shown In bIuo,
oxhIbIfs froquoncy roII-off, buf wIfh n fInffor rosonso fhnf Is nof 1 f . In fncf, fho fInffor
35
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froquoncy rosonso of fho e/onJorJ Jo/o uote Is fho IIkoIy rosuIf of hIgh froquoncy noIso
fhnf Incronsos fho owor nf fho fnII ond of fho socfrum nnd rovonfs n sfooor roII-off.
ThIs Is somofhIng fhnf shouId bo oxocfod from fho dIsforfod e/onJorJ Jo/o uote wIfh
Imboddod mnfhomnfIcnI orrors, whIch wouId fond fo ndd rnndomnoss fo fho wnvo. Tho
sIgnnfuro of such rnndomnoss cnn bo soon In fho IonJon Wote owor socfrum, shown
In fho Iowor grnh In rod. ThIs Iof shows fho fyIcnIIy fInf froquoncy rosonso of n
36
rnndom, or noIso sIgnnI wIfh no InformnfIon confonf. AnronfIy, fho grnhs In IIg. l4
nro showIng fhnf fho sfnndnrd nnd rovIsod dnfn uotee nro dofInIfo InformnfIon cnrryIng
wnvoforms, buf fhnf fho dIsforfod e/onJorJ Jo/o uote hns Imboddod hIgh froquoncy
noIso fhnf fInffons Ifs rosonso. ThIs Is ossonfInIIy whnf IIgs. l2 nnd l3 nro showIng ns
woII.
6WDQGDUG5HYLVHGDQG5DQGRP*HQHUDWHG7LPH:DYH5HVXOWV
(l) Tho TImoWnvo Zoro Scroon Sof ComnrIsons
Onco fho Do/o Wote, or 384 numbor dnfn sof hns boon gonornfod, If bocomos fho Inuf
dnfn for fho TineWote Zero soffwnro nckngo. As monfIonod rovIousIy, fho soffwnro
orforms whnf hns boon cnIIod n froc/ol /roneforn, or oxnnsIon of fho 384 dnfn numbor
sof fo roduco fho TineWote vIowod on fho comufor scroon ns n grnh of Aotel/,. In
ordor for fhIs frncfnI oxnnsIon fo bo orformod roorIy, fho soffwnro roquIros fhnf fho
384 numbor dnfn sof shown In IIg. l0 bo rovorsod, such fhnf dnfn oInf 384 bocomos
dnfn oInf l nnd dnfn oInf 0 Is dIscnrdod (sInco If`s n duIIcnfo or wrn of dnfn oInf
384).
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37
Throo sonrnfo dnfn sofs woro usod In ordor fo gonornfo fho TineWotee noodod for
comnrIson fho e/onJorJ dnfn sof, fho retieeJ dnfn sof, nnd n ronJon dnfn sof. Tho
rosuIfs of somo of fhoso TineWote comnrIsons wIII bo shown In fho grnhs fhnf foIIow,
bogInnIng wIfh fho dofnuIf TineWote grnhs fhnf nro IncIudod wIfh fho Tine1xplorer
soffwnro ns ro-comufod wnvoforms.
IIgs. l5n nnd l5b show fho TineWote fhnf Is sforod by fho soffwnro ns Scroon l, nnd If
covors fho orIod bofwoon l942 nnd 20l2. IIg. l5n shows bofh fho TineWote rosuIfIng
from fho e/onJorJ Jo/o sof on fho Ioff, nnd fhnf for fho retieeJ Jo/o sof on fho rIghf. On
fho ofhor hnnd, IIg. l5b Is fho TineWote gonornfod by fho ronJon dnfn sof, nnd If
cIonrIy bonrs IIffIo rosombInnco fo fho grnhs of IIg. l5n.
ThIs Is fho TineWote grnh fhnf McKonnn hns cnIIod hIsfory`s frncfnI mounfnIn,
bocnuso of Ifs mounfnIn-IIko shno. Thoro nro sovornI fonfuros fo nofIco horo, wIfh fho
fIrsf boIng fhnf fhoso fwo Iofs hnvo romnrknbIy sImIInr shnos obvIousIy nof IdonfIcnI,
buf fhoro Is cIonrIy n common domInnnf rocoss nf work. Anofhor common fonfuro of
sIgnIfIcnnco shown In fhoso fwo grnhs, Is fhnf fho mnjor doconf Info Aotel/, (onk of fho
mounfnIn) bogIns somofImo In l96?. IInnIIy, ns monfIonod onrIIor, fho TineWote
roducod by fho retieeJ Do/o Wote numbor sof, shows n hIghor nvorngo IovoI of Aotel/,
for fhIs fImo orIod (Iowor vnIuos), fhnn doos fho TineWote roducod by fho e/onJorJ
38
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Jo/o sof. ThIs Aotel/, dIfforonco Is fho IIkoIy rosuIf of fho e/onJorJ uote dIsforfIon,
cnusod by fho Imboddod mnfhomnfIcnI orrors fhnf roduco sIgnIfIcnnf hIgh froquoncy
noIso In fho wnvo. As shown In IIg. l4, fho hIgh froquoncy comononfs of fho rovIsod
dnfn wnvo nro Iowor fhnn fho sfnndnrd wnvo by nn ordor of mngnIfudo.
IIg. l6n shows fho sfnndnrd nnd rovIsod TImoWnvo grnhs for Scroon 4 of fho TWZ
dIsIny. AgnIn, fhoso fwo Iofs nro quIfo sImIInr In forms of fhoIr nonrnnco, nnd soom
fo bo showIng ovIdonco of somo common undorIyIng rocoss. Tho dIfforoncos mny bo duo
fo fho fncf fhnf fho sfnndnrd numbor sof roducos moro hIgh froquoncy noIso bocnuso of
fho Imboddod orrors In fho numbor sof. Tho corroInfIon bofwoon fhoso fwo grnhs wns
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found fo bo 0.?3l, nof ns hIgh ns Scroon l, buf sfIII n sIgnIfIcnnf corroInfIon nonofhoIoss.
On fho ofhor hnnd, fho rnndom dnfn sof TImoWnvo shown In IIg. l6b, shows vory IIffIo
corroInfIon wIfh oIfhor of fho grnhs In IIg. l6n. ThIs Is oxocfod, sInco rnndom numbor
sofs nro by dofInIfIon, un-corroInfod wIfh nny ofhor numbor sof.
A comIofo sof of comnrIsons IIko fhoso shown In IIgs. l5 nnd l6 woro orformod on nII
fho TImoWnvo Zoro scroon sofs (Scroons l-l0) wIfh vory sImIInr rosuIfs. Tho corroInfIon
rosuIfs for fho TWZ Scroon sof comnrIsons rnngod from n Iow of 0.?3 fo n hIgh of 0.98
39
wIfh nn nvorngo corroInfIon of 0.86, showIng fhnf fho sfnndnrd nnd rovIsod TImoWnvos
In fhIs scroon sof woro romnrknbIy sImIInr. ThIs wns nof fho cnso for ofhor TImoWnvos
fhnf woro oxnmInod, whIch wIII bo shown Infor. In ofhor cnsos of TineWote comnrIson,
fho dIfforoncos bofwoon fho sfnndnrd nnd rovIsod wnvos, nonrs fo show fhnf fho
retieeJ TineWote oxrossos n Aotel/, rocoss hnvIng boffor nIIgnmonf wIfh known
hIsforIcnI rocoss somofhIng ono wouId oxocf from n moro rocIso formnIIznfIon
rocoss. Moro nnnIysIs Is corfnInIy In ordor, buf fho dnfn fhus fnr sooms fo mnko fhnf
cnso.
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(2) ComnrIsons for Ofhor SIgnIfIcnnf HIsforIcnI IorIods
SovornI ofhor TImoWnvo orIods hnvIng hIsforIcnI sIgnIfIcnnco woro oxnmInod for
comnrIson, buf fho fwo roorfod horo nro fho orIods from l895-l925, nnd from l935-
l955. Tho fIrsf orIod IncIudos mnjor ndvnncos In hysIcs nnd fochnoIogy, ns woII ns n
worId wnr; nnd fho socond orIod IncIudos fho dovoIomonf nnd uso of nucIonr wonons,
ns woII ns fwo mnjor wnrs. IIg. l? Is n grnh of fho TImoWnvo comnrIson for fho l895-
40
l925 orIod, nnd ngnIn fhoso Iofs nro romnrknbIy sImIInr In form. SovornI sIgnIfIcnnf
dnfos nro mnrkod wIfh groon nnd rod nrrows fo sIgnIfy Aotel nnd HoIi/uol honomonn.
Tho fIrsf oworod fIIghf hnons nf KIffyhnwk on ocombor l?, l903; foIIowod by
IInsfoIn`s SocInI Thoory of !oInfIvIfy (ST!) on Juno 30, l905; ConornI !oInfIvIfy In
l9l5, nnd fho WorId Wnr I orIod of l9l4-l9l8. Tho ovonfs fhnf wouId bo consIdorod
novoI (mnnnod fIIghf nnd bronkfhroughs In hysIcs) nII occur nf ovoIfy froughs or
ovoIfy dosconfs. Tho HnbIfunI honomonon (wnr), on fho ofhor hnnd, nonrs fo drIvo
whnf sooms fo bo n vory novoI orIod, bnck Info hnbIf. Whon bofh novoI nnd hnbIfunI
honomonon nro occurrIng sImuIfnnoousIy, fhoy bofh InfIuonco fho shno of fho
TImoWnvo. WWI mny hnvo drIvon fho wnvo furfhor Info hnbIf fhnn If dId, If If woron`f
for fho sImuIfnnoous occurronco of vory novoI honomonn. Ior oxnmIo, fho work on fho
ConornI Thoory of !oInfIvIfy occurs In fho mIdsf of WorId Wnr I wIfh Ifs snmo 'O!I
hnbIfunI nnfuro. Tho moro novoI rocoss of n sIgnIfIcnnf ndvnncomonf In scIonfIfIc
knowIodgo, ncfunIIy nonrs fo suross whnf wouId bo n mnjor nsconf Info hnbIf, nnd
ncfunIIy drIvIng fho wnvo Info novoIfy froughs.
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ofIco fhnf fho sfnndnrd TImoWnvo on fho Ioff doosn`f show fho rogrossIon Info hnbIf
durIng fho IIrsf WorId Wnr fho rovIsod TImoWnvo cIonrIy doos. ThIs Is ono cnso In
whIch fho rovIsod TImoWnvo nonrs fo rovIdo n boffor doscrIfIon of fho ovoIfy
rocoss fhnn doos fho sfnndnrd TImoWnvo. Howovor, fhIs Is somofhIng fhnf shouId bo
oxocfod for n rocoss wIfh n moro rocIso nnd consIsfonf mnfhomnfIcnI modoI.
41
IIg. l8 shows fho l9l5 fImo orIod, for whIch fho fwo wnvos oxhIbIf n subsfnnfInI
dIsngroomonf. WIfh fho oxcofIon of n brIof fwo-monfh orIod, fho sfnndnrd TImoWnvo
shows n sfondy dosconf Info ovoIfy. Tho rovIsod TImoWnvo, howovor, shows moro of
whnf ono mIghf oxocf for n Innof ombroIIod In gIobnI confIIcf. AddIfIonnIIy, fho rovIsod
TImoWnvo shows sovornI Insfnncos whoro fho doformInod mnrch Info hnbIf Is oIfhor
sIowod or fomornrIIy rovorsod; nnd wIfh fho ubIIcnfIon of fho gonornI fhoory In onrIy
l9l6, fho IovoI of ovoIfy bocomos foo gronf for fho forcos of hnbIf, nnd fho wnvo Iungos.
ThIs fIguro rovIdos n good oxnmIo of how fho sfnndnrd nnd rovIsod TImoWnvos cnn
oxhIbIf bohnvIornI dIvorgonco, nnd how fhIs dIvorgonco fonds fo nffIrm fho Imrovod
nccurncy of fho rovIsod wnvoform. !of us now fnko n Iook nf nnofhor orIod fhnf mosf of
us nro fnmIIInr wIfh fho orIod fhnf IncIudos WorId Wnr II, nucIonr onorgy
dovoIomonf, nnd fho Koronn Wnr.
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IIguro l9 shows fho sfnndnrd nnd rovIsod TImoWnvo comnrIson grnhs for fho orIod
l935-l955, nnd fhoro nro obvIous sImIInrIfIos nnd cIonr dIfforoncos bofwoon fho fwo
wnvos. Iofh grnhs show fhnf WWII bogIns nnd onds durIng sfoo nsconfs Info hnbIf,
buf fhoy doscrIbo somowhnf dIvorgIng rocossos, for much of fho mIddIo orIod of fho
wnr. Tho rovIsod TImoWnvo shows fhnf n vory novoI rocoss Is nnronfIy nf work for
much of fho orIod of fho wnr. Tho sfnndnrd TImoWnvo doos show novoI InfIuoncos, buf
If Is noIfhor ns consIsfonf nor drnmnfIc ns for fho rovIsod TImoWnvo. Somo vory ofonf
novoI rocoss sooms fo bo occurrIng durIng much of fho wnr orIod, nnd fhnf rocoss mny
bo surossIng n mnjor nsconf Info hnbIf fhnf mIghf ofhorwIso bo hnonIng. CouId fhIs
novoI rocoss bo fho dovoIomonf of nucIonr scIonco nnd fochnoIogy, ovonfunIIy IondIng
42
fo fho roducfIon nnd uso of nucIonr wonons Thnf mny bo nn offonsIvo nofIon, buf Iof`s
fnko n cIosor Iook nf If.
Tho dovoIomonf of nucIonr scIonco Is ronIIy nbouf bocomIng moro nwnro nnd
knowIodgonbIo of n rocoss fhnf owors fho sun nnd fho sfnrs moro nwnro of jusf how n
vory oworfuI nsocf of nnfuro works. Whnf ono fhon doos wIfh such knowIodgo Is n
dIfforonf rocoss onfIroIy nnd InrgoIy n mnffor of conscIousnoss nnd mnfurIfy. As wo
cnn soo from fho rovIsod TImoWnvo grnh, fho momonf fhnf fhIs knowIodgo Is convorfod
fo wonons fochnoIogy fho nucIonr oxIosIon nf TrInIfy SIfo In ow MoxIco fho wnvo
bogIns n sfoo nsconf Info hnbIf.
Tho uso of fhIs nwosomo owor ngnInsf ofhor humnn boIngs In HIroshImn nnd ngnsnkI
occurs shorfIy nffor fho fosf nf TrInIfy SIfo, nnd occurs on n vory sfoo nscondIng sIoo of
hnbIf. Iorhns fho rocoss of bocomIng moro nwnro of nnfuro, nnd oursoIvos Is vory
novoI Indood. If Is fho sncrod knowIodgo of fho shnmnn, who rofurns from nn ImmorsIon
Info nn nsocf of nnfuro, wIfh guIdnnco or honIIng for hor or hIs ooIo. Wo soom fo hnvo
Iosf fho sonso of sncrod knowIodgo wIfh Ifs nccomnnyIng rosonsIbIIIfy, somowhoro
nIong fho wny. Iorhns If Is fImo fo rognIn fhnf sonso, nnd rocInIm rosonsIbIIIfy for our
knowIng.
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Tho rovIsod TImoWnvo of IIg. l9 nIso shows fho orIod of fho Koronn wnr ns n vory sfoo
nsconf Info hnbIf, nIfhough somofhIng occurrIng onrIy In l952 dId momonfnrIIy rovorso
fho hnbIfunI frond.
43
&RUUHODWLRQ'DWDDQG7LPH:DYH&RPSDULVRQV
CorroInfIon nnnIysIs wns orformod for nII fho dnfn sofs comnrod In fhIs roorf, ns woII
ns fho romnInIng oIghf TWZ scroon sofs nof shown horo, nnd soIocfod fImo orIods. ThIs
fyo of nnnIysIs nIIows us fo oxnmIno fho roInfIonshI bofwoon dnfn sofs, nnd osfImnfo
fhoIr dogroo of Infordoondonco I.o. how sImIInr fhoIr InformnfIon confonf Is. Tho
rosuIfs of fhoso nnnIysos nro shown grnhIcnIIy In IIg. 20, nnd fhoy IncIudo fho fon
TineWote scroons IncIudod wIfh fho TWZ soffwnro, nIno soIocfod hIsforIcnI wIndows,
nnd fho 384 numbor dnfn sofs. In nII cnsos shown, fho rovIsod nnd rnndom dnfn sofs nro
boIng corroInfod (comnrod) wIfh fho sfnndnrd dnfn sof. SInco nny numbor sof corroInfod
wIfh IfsoIf, hns n corroInfIon cooffIcIonf of ono, fho bIuo IIno nf fho fo of fho grnh
rorosonfs fho sfnndnrd dnfn soIf-corroInfIon.
!ocnII fhnf n corroInfIon of l sIgnIfIos numbor sofs fhnf hnvo IdonfIcnI InformnfIon
confonf, n corroInfIon of zoro sIgnIfIos no common InformnfIon confonf, nnd n corroInfIon
of l monns fhnf fho numbor sofs InformnfIon confonf oxhIbIf mIrror Imngo bohnvIor
wnvo onks fo wnvo vnIIoys ofc. Tho groon IIno In fho grnh shows fho dogroo of
corroInfIon bofwoon fho rovIsod wnvoform nnd fho sfnndnrd wnvoform, for onch of fho
sonrnfo TImoWnvos fhnf woro oxnmInod. Tho rod IIno shows fho corroInfIon IovoI
bofwoon wnvos gonornfod by fho rnndom soodod dnfn sofs, nnd fhoso gonornfod by fho
sfnndnrd dnfn sof. Tho fIrsf oInf of onch IIno, Is fho corroInfIon cooffIcIonf for onch of fho
384 numbor dnfn sofs oxnmInod ronJon, retieeJ, nnd e/onJorJ dnfn sofs.
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44
Tho fIrsf fonfuro fo nofIco nbouf fho retieeJ nnd e/onJorJ dnfn sof corroInfIons shown In
IIg. 20, Is fho fncf fhnf fho rovIsod 384 numbor dnfn sof shows n corroInfIon wIfh fho
sfnndnrd numbor sof of nbouf 60 - n comnrIson fhnf Is shown In IIg. l2. ThIs Is n
sIgnIfIcnnf cross-IInkIng of InformnfIon confonf, buf somofhIng fhnf ono mIghf oxocf for
numbor sofs wIfh n common bnso nnd vory sImIInr dovoIomonfnI rocoduros. Tho noxf
fonfuro of sIgnIfIcnnco Is fho fncf fhnf fho corroInfIon bofwoon fho retieeJ nnd e/onJorJ
TineWotee, for nII fon TWZ scroon sofs, Is boffor fhnn ?0 nnd ns hIgh ns 98, showIng
n vory hIgh IovoI of Infordoondonco. Tho fImo orIods rorosonfod by fhoso fon
TImoWnvo scroons, rnngos from 4 yonrs fo 36,000 yonrs, whIch Is InboIod on fho grnh.
Tho durnfIon of fhoso TImoWnvo orIods mny hnvo n bonrIng on fho IovoI of corroInfIon,
ns wo shnII soo In n momonf.
IogInnIng wIfh fho orIod l895-l925, fho grnh shows moro scnffor In fho corroInfIon
bofwoon sfnndnrd nnd rovIsod dnfn sofs, whIch rnngos from nbouf 98 down fo 8, wIfh
ono nnfI-corroInfIon of 95. ofIco fhnf fho corroInfIon nonrs worso for vory shorf
fImo orIods, ono fo fwo monfhs or so. Ono ossIbIo oxInnnfIon Is fhnf fho vory shorf
fImo orIod TImoWnvos nro gonornfod by n vory fow dnfn oInfs In ofhor words n Iow
wnvo snmIIng froquoncy or rnfo. A smnII, nnd undor-snmIod Inuf dnfn sof wouId ndd
n hIghor IovoI of noIso fo fho wnvo sIgnnI, nnd consoquonfIy roduco fho hIghor dnfn
scnffor obsorvod. Tho snmIIng fhoorom, from InformnfIon fhoory, sfnfos fhnf nIInsIng
(noIso gonornfIon) bogIns fo occur whon fho sIgnnI snmIIng rnfo bocomos Ioss fhnn fwIco
fho hIghosf froquoncy comononf of fho snmIod sIgnnI. ThIs Is corfnInIy somofhIng fhnf
mny bo occurrIng In fho mnfhomnfIcs of TImoWnvo gonornfIon.
AddIfIonnIIy, ns monfIonod rovIousIy, fhIs dIfforonco couId bo fho consoquonco of hnvIng
nn Imrovod modoI of fho rocoss. If Is Imorfnnf fo romombor fhrough nII of fhIs
comnrIson nnnIysIs, fhnf fho sfnndnrd dnfn sof Is gonornfod by n rocoss wIfh Imboddod
fInws - nof onough fo dosfroy fho InformnfIon confonf of fho wnvo sIgnnIs, buf onough fo
cnuso somo dIsforfIon of fhnf InformnfIon confonf. ThIs corroInfIon nnnIysIs Is
InforosfIng, rImnrIIy bocnuso If Ionvos fho sfnndnrd TineWote Infncf, moro or Ioss buf
fho Imorfnnf oInf fo romombor Is fhnf ovon wIfh Iow corroInfIon fho retieeJ Jo/o ee/
nonrs fo roduco n boffor TineWote.
If Is robnbIo fhnf fho vnrInfIons wo obsorvo In IIg. 20 nro fho rosuIf of Io/l fho
dIsforfIon of fho InformnfIon confonf of fho 384 numbor Jo/o ee/, ns n rosuIf of
mnfhomnfIcnI orrors, onJ fho Iow dnfn wnvo snmIIng rnfo fhnf occurs for shorf durnfIon
TineWotee (nn unoxnmInod buf InusIbIo fhosIs). If Is nIso Imorfnnf fo oInf ouf horo,
fhnf whon wo do soo sIgnIfIcnnf dIfforoncos In fho TImoWnvos gonornfod by fho e/onJorJ
nnd retieeJ dnfn sofs, fhoso dIfforoncos hnvo rovonIod n retieeJ TImoWnvo of gronfor
nccurncy. Howovor, If Is Imorfnnf fhnf wo oxnmIno n sIgnIfIcnnf vnrIofy of nddIfIonnI
TImoWnvo orIods, fo gnfhor moro sfnfIsfIcs on fho funcfIonIng of fho rovIsod wnvo; buf
fho dnfn In hnnd so fnr, soom fo bo suggosfIng fhnf fho mnfhomnfIcnI formnIIznfIon of fho
dnfn sof gonornfIng rocoss, doos Imrovo fho TineWote nccurncy.
Anofhor sIgnIfIcnnf fonfuro of fho rovIsod dnfn corroInfIon Iof In IIg. 20 fhnf shouId bo
monfIonod horo, Is fho fncf fhnf fho corroInfIon cooffIcIonf for fho l9l5 orIod Is nonrIy -
l, sIgnIfyIng nn nnfI-corroInfIon or mIrror Imngo roInfIonshI bofwoon fho wnvos. ThIs Is
45
fho TImoWnvo comnrIson fhnf Is shown If IIg. l8. If ono woro fo Inco nn ImngInnry
fwo-sIdod mIrror bofwoon fho sfnndnrd nnd rovIsod TImoWnvo grnhs, fhon fho
rofIocfIon on oIfhor sIdo of fho mIrror wouId cIosoIy rosombIo fho wnvo fhnf Is on fho
ofhor sIdo honco fho doscrIfIon of nnfI-corroInfIon ns n mIrror Imngo roInfIonshI.
AIso nofIco, fhnf n groon doffod IIno mnrks fho nvorngo of nII fho sfnndnrd/rovIsod wnvo
corroInfIons nf nbouf ?0.
Tho rod IIno of IIg. 20 shows fho corroInfIon of fho rnndom numbor gonornfod wnvos,
wIfh fho sfnndnrd dnfn sof. Iy dofInIfIon, fho rnndom dnfn sofs shouId show IIffIo or no
corroInfIon wIfh oIfhor fho sfnndnrd or rovIsod dnfn sofs, nor wIfh nny ofhor rnndom
numbor sof. In sovornI cnsos In IIg. 20, fhIs furns ouf fo bo fruo, buf fhoro nro nIso
sovornI cnsos In whIch fho rnndom sof corroInfIon Is nof nonr zoro, ns ono wouId oxocf.
In gonornI, fho rod IIno Iof of IIg. 20 shows n much Iowor IovoI of corroInfIon wIfh fho
e/onJorJ numbor sof fhnn doos fho retieeJ sof ns oxocfod. Inch dnfn oInf on fho rod
IIno, howovor, Is ncfunIIy nn nvorngo of oIfhor fwo, or sovon rnndom numbor sof
corroInfIons. In ofhor words, oIfhor fwo or sovon rnndom numbor corroInfIons woro
nvorngod fo roduco onch oInf on fho rod IIno grnh. If furns ouf fhnf mosf of fho
sIxfoon corroInfIon oInfs roducod by nvorngIng onIy fwo rnndom sofs, hnvo much moro
scnffor fhnn do fho four oInfs roducod by nvorngIng sovon rnndom sof corroInfIons.
Tho 384 numbor ronJon Jo/o ee/, nnd fho orIods l895-l925, l905, nnd l9l5, woro nII
roducod by nvorngIng sovon rnndom sof corroInfIons. Tho vIoIof doffod IIno runnIng
fhrough fho rnndom numbor sof corroInfIons, Is fho nvorngo corroInfIon IovoI for nII fho
rnndom sofs shown, nnd If shows n vory Iow nvorngo corroInfIon of nbouf 5.
If Is nIso ossIbIo fhnf fho snmo rocoss roosod for roducIng fho Inrgor corroInfIon
scnffor of fho rovIsod dnfn sof, couId bo nf work for fho rnndom dnfn sofs I.o. shorf
durnfIon fImo orIods wIfh Iow snmIIng froquoncIos, couId bo cnusIng dnfn scnffor duo
fo noIso. If n smnII numbor of fho 384 Jo/o file oInfs nro usod fo gonornfo n elor/ perioJ
TineWote, fhon fhoro Is n much hIghor robnbIIIfy of corroInfIon bofwoon fho rnndom
sofs nnd fho TImoWnvo numbor sofs. WIfhouf furfhor InvosfIgnfIon, howovor, fhIs Is n
socuInfIvo, If InusIbIo fhosIs.
Tho grnhs of IIg. 20 do show fhnf fho e/onJorJ nnd retieeJ dnfn sofs nnd fhoIr
dorIvnfIvo TineWotee nro romnrknbIy woII corroInfod. In fho rogIons whoro fho
corroInfIon wonkons, or bronks down onfIroIy, fho rovIsod TineWote nonrs fo show n
ovoIfy rocoss fhnf Is In cIosor ngroomonf wIfh known hIsforIcnI rocoss. In nddIfIon,
fho Iofs In IIg. 20 mny bo rovonIIng n rocoss whoroby shorf orIod TineWotee roduco
snmIIng noIso fhnf wonkons fho corroInfIon. ThIs dnfn suorfs fho vIow, fhnf fho
InformnfIon confonf of fho e/onJorJ TineWote Is somowhnf dIsforfod, buf nof dosfroyod;
nnd suggosfs fhnf fho retieeJ TineWote nnd Ifs pieceuiee lineor func/ion Is nbIo fo
corrocf fhIs dIsforfIon, nnd rovIdos nn Imrovod oxrossIon of fho ovoIfy rocoss.
&RQFOXGLQJ5HPDUNV
Tho dovoIomonf of fho 384 numbor Jo/o ee/ from fho sof of Iire/ OrJer of Difference
(IO) Infogors hns boon oxrossod ns n rocoss fhnf Is pieceuiee lineor In nnfuro. ThIs
rocoss InvoIvos fho combInnfIon nnd oxnnsIon of sfrnIghf-IIno sogmonfs, whIch cnn bo
46
oxrossod mnfhomnfIcnIIy ns n pieceuiee lineor func/ion. Tho e/onJorJ Jetelopnen/ hns
boon doscrIbod by McKonnn nnd Moyor In fho TineWote Zero documonfnfIon nnd In
ofhor roorfs. Iuf fhIs rocoss IncIudos n rocodurnI sfo cnIIod fho hnIf fwIsf, fhnf Is
nof consIsfonf wIfh fho sfrucfuro of IocowIso IInonr mnfhomnfIcs; nnd consoquonfIy
roducos n dIsforfIon of fho IO InformnfIon confonf. WnfkIns oInbornfod on fhIs In
somo dofnII, In hIs woII-documonfod oxoso on fho nnfuro of fho lolf /uie/, In whIch ho
doscrIbod fho dIsforfIons nnd InconsIsfoncIos InvoIvod. Ho fhon concIudod fhnf fhIs
dIsforfIon wouId rondor fho TineWote monnIngIoss, ns n ronIIsfIc grnhIcnI doIcfIon of
fho Aotel/, rocoss ns hnd boon doscrIbod by McKonnn. I mnInfnIn fhnf fhIs concIusIon
wns romnfuro, nnd nnronfIy Incorrocf.
Tho retieeJ Jetelopnen/ of fho 384 numbor Jo/o ee/ IncIudos fho uso of mnfhomnfIcs fhnf
corrocfIy oxrossos fho pieceuiee lineor dovoIomonf rocoss, nnd fhoroforo roducos nn
undIsforfod oxnnsIon of fho IO numbor sof. Tho TineWote fhnf rosuIfs from fhIs
oxnnsIon rocoss, Is fhon nn nccurnfo rofIocfIon of fho IO numbor sof, rovIdod fho sof
cnn bo doscrIbod or modoIod by n IocowIso IInonr funcfIon. Tho pieceuiee lineor func/ion
doscrIbod horo, mny onIy bo nn nroxImnfIon fo somo moro conplex func/ion fhnf hns
yof fo bo found. In fncf, I wouId nrguo fhnf fhIs Is quIfo IIkoIy for n honomonon or
rocoss of fhIs nnfuro, whIch furfhor sfudy mny shod somo IIghf on. If wo nssumo fhnf
fho retieeJ TineWote Is n ronsonnbIy nccurnfo rofIocfIon of fho InformnfIon confonf of
fho IO numbor sof, fhon fho e/onJorJ TineWote shouId hnvo n dogroo of nccurncy
roorfIonnI fo Ifs dogroo of corroInfIon wIfh fho retieeJ TineWote. As wo hnvo soon
fhus fnr, fhoso fwo TineWotee show nn oteroge correlo/ion of nbouf ?0, so fhnf fho
e/onJorJ uote hns nn nvorngo nccurncy of nbouf ?0 whon comnrod wIfh fho retieeJ
uote. Howovor, wo hnvo nIso soon fhIs corroInfIon ns hIgh ns 98, or ns Iow ns 6, wIfh
ono cnso of n mIrror Imngo or nnfI-corroInfIon of 0.94.
ThIs work hns sorvod fo cInrIfy nnd formnIIzo fho rocoss by whIch fho 384 numbor
TImoWnvo Jo/o ee/ Is gonornfod. ThIs hns boon dono by showIng fhnf fho rocoss Is
doscrIbnbIo wIfhIn fho frnmowork of IocowIso IInonr mnfhomnfIcs In gonornI, nnd vocfor
mnfhomnfIcs In nrfIcuInr. Inch sfo hns boon doIInonfod nnd formnIIzod
mnfhomnfIcnIIy, fo gIvo fho rocoss cInrIfy nnd confInuIfy. Tho formnIIzod nnd rovIsod
dnfn sof sorvos ns fho foundnfIon of fho TineWote gonornfod by fho TineWote Zero
soffwnro, whIch Is vIowod ns n grnhIcnI doIcfIon of n rocoss doscrIbod by fho obb nnd
fIow of n honomonon cnIIod Aotel/,. Aotel/, Is fhoughf fo bo fho bnsIs for fho cronfIon
nnd consorvnfIon of hIghor ordorod sfnfos of comIox form In nnfuro nnd fho unIvorso.
Tho rosuIfs roorfod horo mnko no fInnI cInIms ns fo fho vnIIdIfy of fho TineWote ns If Is
oxrossod by Aotel/, Tleor,, nor doos If cInIm fhnf fho curronf TineWote Is fho bosf
doscrIfIon of fhIs Aotel/, rocoss. If doos show fhnf fho roor mnfhomnfIcnI fronfmonf
of fho IO numbor sof, roducos n TineWote fhnf nonrs fo bo moro consIsfonf wIfh
known hIsforIcnI rocoss. ThIs consIsfoncy Is gonornI, howovor, nnd moro work noods fo
bo dono fo oxnmIno fho socIfIc rofIocfIons or rojocfIons fhnf fho TImoWnvo mny bo
rovonIIng. If Aotel/, Tleor, Is n vnIId hyofhosIs, rofIocfIng n ronI honomonon In
nnfuro, fhon ono wouId oxocf fhnf If Is vorIfInbIo In socIfIc wnys.
If hns nIso soomod nrorInfo fo oxnmIno somo of fho sfos In fhIs wnvo dovoIomonf
rocoss In forms of fhoIr corrosondonco wIfh oIomonfs of hIIosohy nnd scIonco. Tho
47
fIow of YIn nnd Ynng onorgy rofIocfod In fho oxrossIon of fho forwnrd nnd rovorso bI-
dIrocfIonnI wnvos, for oxnmIo, fInds hIIosohIcnI corrosondonco In n nnfurnI cycIo of
IIfo-donfh-robIrfh, or In fho rocoss of fho shnmnnIc journoy ImmorsIon, ongngomonf,
nnd rofurn. Corrosondonco cnn nIso bo found In scIonco, In fho fIoIds of cosmoIogy,
nsfronomy, nsfrohysIcs, nnd qunnfum hysIcs fho IIfo cycIos nnd mofIon of honvonIy
bodIos, qunrks, nnd unIvorsos; fho hnrmonIc nnd hoIogrnhIc nnfuro of IIghf nnd wnvo
mochnnIcs; nnd fho cycIIc frnnsformnfIon of mnffor fo onorgy, nnd onorgy fo mnffor. Tho
rofIocfIon of nII nnfurnI honomonn nnd rocossos ovor fho confInuum of oxIsfonco, from
fho smnIIosf scnIos u fo fho Inrgosf scnIos, musf suroIy IncIudo whnfovor rocoss Is
occurrIng In fho I-ChIng ns woII. Tho quosfIon Is, nro wo nro cIovor nnd conscIous onough
fo docIhor nnd oxross If corrocfIy nnd nrorInfoIy
$FNQRZOHGJHPHQWV
I wouId IIko fo fhnnk Terence McKenna, for brIngIng fhIs InfrIguIng nnd rovocnfIvo nofIon
Info fho coIIocfIvo, nnd for fho courngo nnd forosIghf shown, by hIs wIIIIngnoss fo oon
hImsoIf nnd hIs Idons fo scrufIny nnd boundnry dIssoIufIon. If fhoro Is nny roIovnnco or
monnIng fo bo found In fho TineWote or Aotel/, Tleor,, fhon If Is suroIy somofhIng fhnf
Is Inrgor fhnn ho, or nny of us; nnd If Is nIso somofhIng fhnf Is roorIy In fho domnIn of
nII humnn oxorIonco, wIfh onch of us n wIfnoss, nrfIcInnf, nnd confrIbufor.
I wouId nIso IIko fo oxross my fhnnks nnd nrocInfIon fo Mathew Watkins for hIs work In
oxosIng fho mnfhomnfIcnI InconsIsfoncIos, vngnrIos, nnd rocodurnI orrors of fho
sfnndnrd TineWote Jo/o ee/ dovoIomonf, nnd chnIIongIng n fhoory fhnf mny hnvo
bocomo fnr foo sodonfnry nnd Inbrod for Ifs own good. Whnfovor fho fInnI oufcomo of fhIs
ondonvor of Aotel/, Tleor,, ho hns sof fho onforrIso on Ifs roor courso of oon nnd
crIfIcnI InquIry.
I nm nIso gronfIy Indobfod fo Peter Meyer for hIs skIII nnd forosIghf In cronfIng n TWZ
soffwnro nckngo fhnf Is fIoxIbIo, nccossIbIo, nnd frIondIy fo fho sorIous InvosfIgnfor.
WIfhouf hIs OS vorsIon of TImoWnvo Zoro soffwnro, fhIs work wouId hnvo boon much
moro dIffIcuIf If nof ImossIbIo. Ho hns cronfod n soffwnro nckngo fhnf mnkos fhoso
nofIons ronIIsfIcnIIy fosfnbIo, In n roInfIvoIy sfrnIghfforwnrd mnnnor. ThIs mndo If
ossIbIo for mo fo oxnmIno fho offocfs of fho rovIsod dnfn sof on fho TImoWnvo IfsoIf, ns
woII ns fncIIIfnfIng fho oxnmInnfIon of fho dofnIIod sfrucfuro of fho wnvo In work fo
foIIow.
My fhnnks nIso fo Dan Levy for hIs offor fo ubIIsh fhIs work on hIs !ovIfy sIfo, ns woII ns
hosfIng nn ucomIng TineWote mnfhomnfIcnI nnnox fo Aotel/, Tleor,. I wnnf nIso fo
ncknowIodgo Brian Crissey nf IIuo Wnfor IubIIshIng for hIs hoI In InfogrnfIng fho now
rocoss Info fho TImoWnvo Zoro soffwnro nckngos nnd documonfnfIon.

1
Computer Software program written by Meyer and others, based on a mathematical relationship exhibited by the I-
Ching, formulated and reported by T. McKenna and D. McKenna, the Invisible Landscape, Harper San Francisco, 1993,
p. 121
2
T. McKenna, the Invisible Landscape, p. 140
3
M. Watkins, Autopsy for a Mathematical Hallucination, Terence McKennas Hyperborea at www.levity.com
48

4
T. McKenna, Time Explorer Manual, p60, the Invisible Landscape, pp. 140-142
5
P. Meyer, http://www.magnet.ch/serendipity/twz/kws.html
6
DeltaPoint, Inc., 22 Lower Ragsdale Dr., Monterey, CA 93940, (408) 648-4000
7
Microsoft Corp., One Microsoft Way, Redmond, WA 98052
8
McKenna, TimeExplorer Manual, PP. 60-63, http://www.levity.com/eschaton/waveexplain.html
9
H.B. Anderson, Analytic Geometry with Vectors, p71, McCutchan Publishing Corp., Berkeley, Ca. 1966
10
T. McKenna, TimeExplorer software manual, pp. 62-63
11
P. Meyer, TimeExplorer software manual, pp. 85-91
12
M. Kaku, What Happened BEFORE the Big Bang?, Astronomy, May 1996, pp. 34-41
[John Sheliak] sheliak@dsrt.com
[Terence McKenna] syzygy@ultraconnect.com
[return to Levity] http://www.levity.com/eschaton/
Filename: MathWave PaperII
Directory: D:\DG4\TWZ Files
Template: C:\Program Files\Microsoft
Office\Templates\NORMAL.DOT
Title: Delineation, Specification, and Formalization of the TWZ
Data Set Generation Process - Philosophical, Procedural, and Mathematical
Subject:
Author: John Sheliak
Keywords:
Comments:
Creation Date: 11/11/97 9:50 PM
Change Number: 2
Last Saved On: 11/11/97 9:50 PM
Last Saved By: John Sheliak
Total Editing Time: 1 Minute
Last Printed On: 11/11/97 9:51 PM
As of Last Complete Printing
Number of Pages: 48
Number of Words: 13,802 (approx.)
Number of Characters: 78,672 (approx.)